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Stochastic partial differential andintegro-differential equations

Konstantinos Anastasios Dareiotis

Doctor of PhilosophyUniversity of Edinburgh

2015

Declaration

I declare that this thesis was composed by myself and that the work contained thereinis my own, except where explicitly stated otherwise in the text.

(Konstantinos Anastasios Dareiotis)

3

4

Acknowledgements

I would like to thank my supervisor, Professor István Gyöngy, for his guidance, pa-

tience and support during my PhD. I feel very grateful that I had the chance to be

one of his students and to gain an ε> 0 of his knowledge.

I would also like to thank my fellow students in the Probability and Stochastic

Analysis group. Especially, I would like to thank Máté Gérencser and James-Michael

Leahy, for our collaborations and friendship.

Lastly, I would like to thank my family, for their love and their sacrifices that

made it possible.

5

To the memory of my beloved grandmother,

Françoise

6

Abstract

In this work we present some new results concerning stochastic partial differential

and integro-differential equations (SPDEs and SPIDEs) that appear in non-linear

filtering. We prove existence and uniqueness of solutions of SPIDEs, we give a com-

parison principle and we suggest an approximation scheme for the non-local inte-

gral operators. Regarding SPDEs, we use techniques motivated by the work of De

Giorgi, Nash, and Moser, in order to derive global and local supremum estimates,

and a weak Harnack inequality.

7

8

Contents

Acknowledgements 5

Abstract 7

1 Introduction 11

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Outline of the results and structure of the thesis . . . . . . . . . . . . . . 11

1.3 Notation and useful lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Stochastic Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Stochastic partial integro-differential equations 17

2.1 Solvability and Comparison Principle . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Existence, uniqueness and Comparison Theorems . . . . . . . . 18

2.1.2 Itô’s formula for the square of the norm of the positive part . . . 24

2.1.3 The main estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.4 Proof of Theorems 2.1.1, 2.1.2, 2.1.3 and 2.1.4 . . . . . . . . . . . 36

2.2 Numerical approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.1 A spatial discretization scheme . . . . . . . . . . . . . . . . . . . . 39

2.2.2 Estimates for the approximation of the operators . . . . . . . . . 45

2.2.3 Proof of Theorem 2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Stochastic Partial differential equations 55

3.1 Global Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 The global L∞-estimates . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.2 Itô’s formula for the Lp -norm and energy-type estimates . . . . 59

3.1.3 Proof of Theorem 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Application to semilinear equations . . . . . . . . . . . . . . . . . . . . . 68

3.3 Weak Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3.1 Formulation and main results . . . . . . . . . . . . . . . . . . . . 73

3.3.2 Martingale growth estimates and Itô’s formula . . . . . . . . . . . 74

3.3.3 Local supremum estimates . . . . . . . . . . . . . . . . . . . . . . 78

9

3.3.4 Proof of Theorem 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.3.5 Proof of Theorem 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . 85

10

Chapter 1

Introduction

1.1 Motivation

The present thesis is focused on qualitative properties of solutions of stochastic par-

tial differential equations (SPDEs), and stochastic partial integro-differential equa-

tions (SPIDEs), as well as on developing numerical methods for solving them.

The theory of SPDEs finds applications in many scientific fields, such as physics,

biology, chemistry and finance. The main motivation for studying this type of equa-

tions is the filtering of partially observable processes. There, under consideration

is a system whose evolution is modeled by a stochastic differential equation (sig-

nal process). One has access only to a noisy partial observation of the signal (ob-

servation process), and the classic problem is to estimate the probability density

function of the signal at time t by using the information obtained from the observa-

tion process up to time t . It turns out that the distribution of the best estimate (in

mean square sense) of the signal at t given the observation until t satisfies a nonlin-

ear SPDE, called Kushner-Shiryaev equation, which can be transformed to a linear

SPDE, called the Zakai equation. The Zakai and the Kushner-Shiryaev equations

have been extensively studied in the past decades in the case when the signal and

observation are diffusion processes. In applications, however, one should often deal

with more general signal and observation models where jumps in the signal or/and

in the observation jumps may occur. This motivates recent interest in models with

Itô-Lévy processes. In this case, the corresponding Zakai equation is a linear SPIDE.

1.2 Outline of the results and structure of the thesis

The results of this thesis are organized in two chapters, Chapter 2 and 3. Each chap-

ter is divided into sections. Each section is then divided into subsections. The main

11

results of each section are stated in the first subsection and their proofs are given in

the last one. In the intermediate subsections, technical lemmas (which are of their

own interest) are proved, in order to be used for the proofs of the main results.

In more detail:

In Chapter 2, we deal with two types of stochastic partial integro-differential

equations driven by Lévy noise. In Section 2.1 we prove existence and uniqueness

of solutions in L2-spaces for these two classes of equations. We also prove an Itô

formula for the square of the L2-norm of the positive part, which is then applied

in order to obtain a comparison principle. In Section 2.2 we suggest a discretiza-

tion scheme (in space) for the non-local integral operators, which is combined with

a finite difference scheme for the differential operators, to obtain that the rate of

convergence of the scheme to the solution is one.

In Chapter 3 we turn our attention to stochastic partial differential equations

driven by Wiener processes. Although the notion of SPDEs is less general than SPI-

DEs, we have decided to present our results concerning SPDEs in Chapter 3, since

they seem to be more interesting from analytical point of view, and their proofs are

more technical. The results in Chapter 3 rely on the De-Giorgi-Nash-Moser theory

of elliptic and parabolic PDEs. In Section 3.1 we present global L∞-estimates for the

solution of the Cauchy problem with zero boundary condition. We apply then these

estimates in Section 3.2 in order to construct solutions for a certain class of semi-

linear SPDEs in Section 3.2. In Section 3.3 we obtain local L∞-estimates for convex

functions of solutions of equations with no boundary conditions and then we use

these estimates to obtain a weak Harnack inequality for solutions of SPDEs.

1.3 Notation and useful lemmas

Let us introduce some basic notation that will be used through the rest of this the-

sis. Let (Ω,F ,P ) be a probability space equipped with a right-continuous filtration

(Ft )t≥0, such that F0 contains all P-zero sets. We consider aσ-finite measure space

(Z ,Z ,ν). Let N (d t ,d z) be an Ft− Poisson random measure on [0,∞)× Z . We as-

sume that its compensator is d tν(d z) and we use the notation

N (d t ,d z) = N (d t ,d z)−d tν(d z).

We also consider a sequence of independent real valued Ft -Wiener processes w kt ∞k=1.

The notation P is used for the predictable σ-algebra on Ω× [0,T ].

A process ξ= (ξt )t∈[0,T ] with values in a topological space X will be called càdlàg

if with probability one the trajectories of ξ are continuous from the right in t ∈ [0,T )

12

and have limits from the left at every t ∈ (0,T ] in the topology of X . If a process

ξ takes values in a Banach space Y , it will be called strongly càdlàg if it is càdlàg

with respect to the norm topology in Y . For a càdlàg process ξ = (ξt )t∈[0,T ] we will

write ξt− = lims↑t ξs , for t ∈ (0,T ] and ξt− = ξ0 for t = 0. Also when we deal with

(stochastic) integrals that are càdlàg in the time variable t , with the notation∫ t

0 ·, we

mean∫

(0,t ] ·.If X is a topological space then B(X ) is the Borel σ-algebra on X . If X is a

normed linear space then |x|X denotes the norm of x ∈ X , X ∗ is the dual of X , and

⟨x∗, x⟩ denotes the action of x∗ ∈ X ∗ on x ∈ X . If A is a set, then I A will denote the

indicator function of A. The notation Q stands for the whole spaceRd , for an integer

d ≥ 1, or for a bounded Lipschitz domain in Rd . We write

∂i u := ∂u

∂xi, ∂i j u := ∂2u

∂xi∂x j, for i , j = 1, ...,d ,

for the first and second order partial derivatives of a function u defined on Q. Let

us also denote by ∂0 the identity operator and ∂−i =−∂i for i = 1, ...,d . For a multi-

index α= (α1, ...,αd ) ∈Nd of order |α| :=α1 + ...+αd , we write

∂αu := ∂|α|u∂α1 x1...∂αd xd

.

We write C∞c (Q) for the set of all smooth functions that are compactly supported in

Q. As usual, for an integer m ≥ 0 and a real p ≥ 1, we denote by W mp (Q) the space of

functions u ∈ Lp (Q), whose generalized derivatives up to order m lie in Lp (Q). We

set H m(Q) := W m2 (Q) and we write H 1

0 (Q) for the closure of C∞c (Q) in H 1(Q) under

the norm

|u|H 1 =( d∑

i=1|∂i u|2L2

+|u|2L2

)1/2.

The inner product in L2(Q) will be denoted by (·, ·). We will use the notation H−1(Q)

for the dual of H 10 (Q). The space of all square-summable sequences will be denoted

by `2.

Lest us now introduce the concept of a Gel’fand triple. Let (H , (·, ·)H ) be a sepa-

rable Hilbert space and let (V , | · |V ) be a separable reflexive Banach space such that

V is continuously and densely embedded in H . By identifying H with its dual H∗ by

the help of the inner product (·, ·)H in H , we get

V ,→ H ≡ H∗ ,→V ∗,

where H∗ ,→V ∗ is the adjoint embedding of V ,→ H . It also follows that the embed-

13

ding H∗ ,→V ∗ is continuous and dense. Notice that ⟨v∗, v⟩ = (v∗, v)H when v∗ ∈ H .

The triad (V , H ,V ∗) will be called a Gel’fand triple.

Finally, we note that unless otherwise indicated, the summation convention is

used with respect to repeated integer-valued indices throughout this thesis.

1.4 Stochastic Evolution Equations

The SPDEs and SPIDEs under consideration, will be understood as stochastic evolu-

tion equations on the Gel’fand triple (H 10 (Q),L2(Q), H−1(Q)). A variational approach

for stochastic evolution equations driven by continuous martingales where intro-

duced in [43], and by semimartingales, in [22]. The ones that we will present here

are a particular case of [22]. Let (V , H ,V ∗) be a Gel’fand triple. Let us also consider

the mappings A :Ω× [0,T ]×V → V ∗, Bk :Ω× [0,T ]×V → H , for k ∈N+, and C :

Ω× [0,T ]×V → L2(Z ,ν; H) that are P×B(V )-measurable, and an F0−measurable

random variable ψ with values in H . We pose the following conditions:

Assumption 1.4.1. There exist constants L ≥ 0, γ > 0, and a predictable process

g = (g t )t∈[0,T ], such that

I) “Semicontinuity of A ”: For any v , v1, v2 ∈V , and (ω, t ) ∈Ω× [0,T ], the func-

tion ⟨v,At (v1 +λv2)⟩ is continuous in λ on R ,

II) “Monotonicity ”: For any v1, v2 ∈V , and (ω, t ) ∈Ω× [0,T ],

2⟨At (v1)−At (v2), v1 − v2⟩+∞∑

k=1|Bk

t (v1)−Bkt (v2)|2H

+∫

Z|Ct (z, v1)−Ct (z, v2)|2Hν(d z) ≤ L|v1 − v2|2H ,

III) “Coercivity ”: For any v ∈V , and (ω, t ) ∈Ω× [0,T ],

2⟨At (v), v⟩+∞∑

k=1|Bk

t (v)|2H +∫

Z|Ct (z, v)|2Hν(d z) ≤ g t +L|v |2H −γ|v |2V ,

IV) “ Restriction of growth on A ” : For any v ∈V , and (ω, t ) ∈Ω× [0,T ],

|At (v)|2V∗ ≤ g t +L|v |2V .

V) E |ψ|2H +E∫ T

0 g t d t <∞.

14

Let us consider now the problem

dut =At (ut )d t +Bkt (ut )d w k

t +∫

ZCt (z,ut−)N (d t ,d z) (1.4.1)

u0 =ψ (1.4.2)

We will say that u is a solution of (1.4.1)-(1.4.2) if

• u = (ut )t∈[0,T ] is an adapted, strongly càdlàg H-valued process,

• ut ∈V for dP ⊗d t-almost every (ω, t ) ∈Ω× [0,T ], and E∫ T

0 |ut |2V d t <∞,

• for each φ ∈V , almost surely

(ut ,φ) =ψ+∫ t

0⟨At (us),φ⟩d s +

∫ t

0(Bk

t (us),φ)d w ks

+∫ t

0

∫Z

(Ct (z,us−),φ)N (d s,d z),

for any t ∈ [0,T ].

Two solutions u and v will be considered identical if with probability one, ut = vt

for all t ∈ [0,T ] (as elements of H).

The following theorem is a simple consequence of Theorems 2.9 and 2.10 from

[22].

Theorem 1.4.1. Under Assumption 1.4.1, there exists a unique solution of (1.4.1)-

(1.4.2). Moreover, the following estimate holds:

E supt∈[0,T ]

|ut |2H +E∫ T

0|ut |2V d t ≤C

(E |ψ|2H +E

∫ T

0g t d t

),

where C is a constant depending only on L and γ.

15

16

Chapter 2

Stochastic partial integro-differential

equations

As already mentioned, stochastic partial integro-differential equations play an im-

portant role in non-linear filtering of jump-diffusion processes. For more informa-

tion on the subject, and in particular, derivation of the Zakai equation, we refer the

reader to [20] and [21]. In this chapter, we introduce the notion of SPIDE, we prove

existence and uniqueness of the solutions, we derive a comparison principle, and

we give a numerical approximation scheme.

2.1 Solvability and Comparison Principle

Our goal in this section is to prove existence and uniqueness of solutions, as well as

comparison principles, for stochastic partial integro-differential equations driven

by Lévy processes. The existence and uniqueness of solutions, is a simple conse-

quences of the results concerning general stochastic evolutions equations driven

by semi-martingales that exist in [22]. For the comparison principles, we need to

obtain an Itô formula for the square of the L2-norm of the positive part of (possi-

bly) discontinuous semimartingales with values in H−1 that have dP ⊗d t-versions

in H 10 . Our formula extends an Itô formula from [44] proved for continuous semi-

martingales.

Comparison principles are powerful tools and play important role in PDE the-

ory. Comparison theorems for SPDEs are known in various generalities in the liter-

ature. To the best of our knowledge, the first results on comparison of solutions of

SPDEs appear in [38] and [15]. Recent results appear in [44], [13], [11] and [12]. In

[11] and [13] quasi linear SPDEs, and in [12] quasi-linear SPDEs with obstacle are

studied. In the above publications, the equations under consideration are driven by

17

Wiener processes, or cylindrical Wiener processes, and only differential operators

are present. Here, in Theorems 2.1.2 and 2.1.4, we present comparison theorems

for two classes of quasilinear SPIDEs, linear versions of which, arise in non-linear

filtering.

The results in this section are from [7], a joint work with István Gyöngy.

2.1.1 Existence, uniqueness and Comparison Theorems

In this section we will be dealing with two types of equations. In order to introduce

these equations, together with the space (Z ,Z ) and the random measure N (d t ,d z)

of Chapter 1, let us consider another measurable space (F,F), an Ft−Poisson ran-

dom measure M(d t ,dζ) on [0,∞)×F and two σ−finite measures π(1), π(2) on F . We

assume that the compensator of M(d t ,dζ) is d tπ(2)(dζ) and we write

M(d t ,dζ) = M(d t ,dζ)−d tπ(2)(dζ).

First we consider the equation

dut (x) =(Lt ut (x)+ ft (x,ut (x),∇ut (x))+∂i f i

t (x))

d t

+Gkt (u)(x)d w k

t +∫

Zg t (x, z,ut−(x))N (d t ,d z),

(2.1.1)

for (t , x) ∈ [0,T ]×Q, with initial condition

u0(x) =ψ(x), x ∈Q, (2.1.2)

where

Lt u(x) = ∂ j (ai jt (x)∂i u(x))+I (1)

t ut (x),

I (1)t u(x) =

∫F

[u(x + ct (x,ζ))−u(x)− ct (x,ζ) ·∇u(x)]mt (x,ζ)π(1)(dζ),

Gkt (u)(x) =φi k

t (x)∂i u(x)+σkt (x,u(x)).

In the formulas above as well as in (2.1.1), the summation takes place over i , j ∈1, ...,d and k ∈N. We make the following assumptions. Let K > 0 denote a constant.

Assumption 2.1.1.

i) The coefficients ai j , are real-valued P ⊗B(Q) measurable functions on Ω×[0,T ]×Q and are bounded by K for every i , j = 1, ...,d . The coefficient φi = (φi k )∞k=1

18

is an l2-valued P⊗B(Q)-measurable function onΩ×[0,T ]×Q for every i = 1,2, ...,d ,

such that ∑i

∑k|φi k

t (x)|2 ≤ K for all ω, t and x.

ii) f is a real valued P ⊗B(Q)⊗B(R)⊗B(Rd ) -measurable function on Ω×[0,T ]×Q ×R×Rd , and σ = (σk )∞k=1 is a P ⊗B(Q)⊗B(R)-measurable function on

Ω×[0,T ]×Q×R, with values in l2. The function g is defined onΩ×[0,T ]×Q×Z ×Rwith values in R and it is P ⊗B(Q)⊗Z ⊗B(R)-measurable. We assume that there

exists a predictable process ht with values in L2(Q), such that for all ω, t , x, z,r,r ′

| ft (x,r,r ′)|2 +∑k|σk

t (x,r )|2 +∫

Z|g t (x, z,r )|2ν(d z)

≤ K |r |2 +K |r ′|2 +|ht (x)|2,

and

E∫ T

0|ht |2L2(Q)d t <∞.

iii) ψ is an F0-measurable random variable in L2(Q) with E |ψ|2L2<∞.

iv) There exists a constant Å> 0 such that for all ω, t , x and for all ξ= (ξ1, ...ξd ) ∈Rd we have

ai jt (x)ξiξ j − 1

2φi k

t (x)φ j kt (x)ξiξ j ≥ Å|ξ|2.

v) For all ω, t , x, z,r1,r2

∑k|σk

t (x,r1)−σkt (x,r2)|2 ≤ K |r1 − r2|2.

vi) For all i ∈ 1, ...,d, f i are L2(Q)-valued P-measurable functions onΩ×[0,T ]

such that

E∫ T

0

d∑i=1

| f it |2L2(Q) d t <∞ .

We will refer to the constant Å as the parabolicity constant. If Å, as in our case, is

strictly positive, the corresponding equation will be called non-degenerate, while if

it is zero, then the equation will be called degenerate. For solvability of degenerate

SPIDEs we refer the reader to [49] and [4].

Assumption 2.1.2. The function ft (x,r,r ′) is continuous in r , for eachω, t , x and r ′.

Assumption 2.1.3. For all ω, t , x,r1,r2,r ′1,r ′

2

2(r1 − r2)( ft (x,r1,r ′1)− ft (x,r2,r ′

1))

19

+∫

Z|g t (x, z,r1)− g t (x, z,r2)|2ν(d z) ≤ K |r1 − r2|2,

and

| ft (x,r1,r ′1)− ft (x,r1,r ′

2)| ≤ K |r ′1 − r ′

2|.

Assumption 2.1.4. The function r +g t (x, z,r ) is non-decreasing in r for allω, t , x, z.

Assumption 2.1.5. The function c mapsΩ×[0,T ]×Rd ×F into Rd , it is P⊗B(Rd )⊗F-measurable, and there exists an F-measurable real function c on F such that

(i) |ct (x,ζ)| ≤ c(ζ), for all ω, t , x,ζ,

(ii)∫

F c2(ζ)∧ c(ζ) π(1)(dζ) ≤ K ,

(iii) |ct (x,ζ)− ct (y,ζ)| ≤ c(ζ)|x − y |, for all ω, t , x, y,ζ.

Assumption 2.1.6. The function m mapsΩ×[0,T ]×Rd×F intoR, it is P⊗B(Rd )⊗F-

measurable, and we have

(i) 0 ≤ mt (x,ζ) ≤ K , for all ω, t , x,ζ,

(ii)|mt (x,ζ)−mt (y,ζ)| ≤ K |x − y |, for all ω, t , x, y,ζ.

Assumption 2.1.7. The functions c lt (x,ζ), l = 1, ...,d , are twice continuously differ-

entiable in x, for each ω, t ,ζ, and

(i) |∂i c lt (x,ζ)| ≤ K , |∂i j c l

t (x,ζ)| ≤ K , for all i , j , l = 1, ...,d ,

(ii) K −1 ≤ |det (I+θ∇ct (x,ζ))|for all ω, t , x,ζ and θ ∈ [0,1], where I denotes the identity matrix.

Remark 2.1.1. Denote by Tθ,t ,ζ the mapping x 7→ x +θct (x,ζ), for fixed ω, t ,θ and ζ.

By virtue of the inverse function theorem, it follows from (ii) of Assumption 2.1.7

that Tθ,t ,ζ is a local diffeomorphism. In addition, by the first inequality in (i) and

by (ii) of Assumption 2.1.7, there exists a constant γ > 0, such that the norm of the

matrix (I+θ∇ct (x,ζ))−1 is uniformly bounded by γ. Hence, by Hadamard’s theorem

(see, eg, Theorem 5.1.5 in [1]), Tθ,t ,ζ is a g l obal diffeomorphism, for fixed ω, t ,θ

and ζ. We denote by Jθ,t ,ζ the inverse of Tθ,t ,ζ. Notice that for fixed ω, t ,θ,ζ and for

all j = 1, ...,d , the functions J jθ,t ,ζ(x) are twice continuously in x, and their first and

second order derivatives are uniformly bounded.

We want to give a meaning to equation (2.1.1) as a stochastic evolution equation

on the Gel’fand triple (H 10 (Q),L2(Q), H−1(Q)), where the embedding H 1

0 (Q) ,→ L2(Q)

is the identity.

To this end, let us see how the operator I (1)t acts on functions u ∈ H 1

0 (Q): Under

Assumptions 2.1.5 through 2.1.7, I (1)t is a bounded linear operator from H 1

0 (Q) into

H−1(Q) for fixed (ω, t ), and for all u, v ∈ H 10 (Q) the process ⟨I (1)

t u, v⟩ is predictable.

To see this, consider first the case Q =Rd . For u ∈C∞c (Rd )(even for u ∈W 2

2 (Rd )) one

20

can easily see that I (1)t u(x) is a function in L2(Rd ). For δ ∈ (0,1), let us also denote

by I (1δ) and I (1δ) the operators defined as I (1) but with F replaced by Fδ = ξ ∈ F :

c(ξ) < δ and by F cδ= F \ Fδ respectively. Then for v ∈ C∞

c (Rd ) we have by Taylor’s

formula

(I (1)t u, v) = (I (1δ)

t u, v)+ (I (1δ)t u, v)

=∫ 1

0(1−θ)

∫Fδ

∫Rd∂ki u(Tθ,t ,ζ)c i

t (x,ζ)ckt (x,ζ)mt (x,ζ)v(x)d xπ(1)(dζ)dθ

+∫

F cδ

[u(x + ct (x,ζ))−u(x)− ct (x,ζ) ·∇u(x)]mt (x,ζ)v(x) d x π(1)(dζ)

=∫ 1

0(θ−1)

∫Fδ

∫Rd∂i u(x +θct (x,ζ))∂ j (q i j

t (x,ζ,θ)v(x))d xπ(1)(dζ)dθ

+∫

F cδ

[u(x + ct (x,ζ))−u(x)− ct (x,ζ) ·∇u(x)]mt (x,ζ)v(x) d x π(1)(dζ),

(2.1.3)

where the last equality is obtained by integration by parts, and q i j is given by

q i jt (x,ζ,θ) :=

d∑l=1

c lt (x,ζ)c i

t (x,ζ)mt (x,ζ)∂l J jθ,t ,ζ(Tθ,t ,ζ(x)).

Due to Assumptions 2.1.5 through 2.1.7 for a constant N = N (d ,K ),

(I (1)t u, v) ≤ N |u|H 1(Rd )|v |H 1(Rd ),

which shows that I (1)t extends uniquely to a bounded linear operator from H 1 to

H−1, and the duality product ⟨I (1)t u, v⟩ is given by the right-hand side of (2.1.3).

In case Q is a bounded Lipschitz domain, one can define the action of I (1)t u on

v ∈ H 10 (Q) again by (2.1.3), where u and v this time are extended to zero outside of

Q. For further study of these operators we refer to [17].

Definition 2.1.1. A strongly càdlàg (continuous if ν ≡ 0) adapted process u with

values in L2(Q) is called a solution of the problem (2.1.1)-(2.1.2) if

i) ut ∈ H 10 (Q) for dP ⊗d t almost every (ω, t ) ∈Ω× [0,T ],

ii) E∫ T

0 |ut |2H 10 (Q)

d t <∞,

iii) for all ϕ ∈ H 10 (Q) we have almost surely

21

(ut ,ϕ) =(ψ,ϕ)+∫ t

0

((ai j

s ∂i us + f jt ,∂− jϕ)+ ( fs(us ,∇us),ϕ)+⟨I (1)

s us ,ϕ⟩)

d s

+∫ t

0(φi k

s ∂i us +σks (us),ϕ)d w k

s

+∫ t

0

∫Z

(gs(z,us−),ϕ)N (d z,d s)

for all t ∈ [0,T ], where recall that (·, ·) is the inner product in L2(Q).

Theorem 2.1.1. Let Assumptions 2.1.1 through 2.1.3 and 2.1.5 through 2.1.7 hold.

Then there exists a unique solution of the problem (2.1.1)-(2.1.2). Moreover the fol-

lowing estimate holds

E supt∈[0,T ]

|ut |2L2(Q) +E∫ T

0|u|2

H 10 (Q)

d t ≤ N

(E |φ|2L2(Q) +E

∫ T

0(|ht |2L2(Q) +

d∑i=1

| f it |2L2(Q))d t

),

where N is a constant depending only on d ,κ,K and T .

After some preliminaries we will see that Theorem 2.1.1 follows easily from The-

orem 1.4.1 (i.e. Theorems 2.9 and 2.10 from [22]).

Together with (2.1.1)-(2.1.2) let us also consider the problem

d vt (x) =(Lt vt (x)+Ft (x, vt (x),∇vt (x))+∂i f i

t (x))

d t

+Gkt (v)(x)d w k

t +∫

Zg t (x, z, vt−(x))N (d t ,d z), (2.1.4)

v0(x) =Ψ(x), (2.1.5)

where F satisfies ii) from Assumption 2.1.1 andΨ is an F0-measurable random vari-

able in L2(Q).

Theorem 2.1.2. Suppose that Assumptions 2.1.1 and 2.1.4 through 2.1.7 hold. Let

u and v be solutions of the problems (2.1.1)-(2.1.2) and (2.1.4)-(2.1.5) respectively.

Suppose that either f or F satisfy Assumption 2.1.3. Let f ≤ F and ψ ≤ Ψ. Then

almost surely, for all t ∈ [0,T ] we have ut (x) ≤ vt (x) for almost every x ∈Q.

Remark 2.1.2. For equations driven by continuous noise, under the assumptions

posed for existence-uniqueness of the solutions, one gets the comparison principle

with no extra conditions. Notice that this is not the case when jump-type noise is

present. Namely, Assumption 2.1.4 needs to hold. Notice also that this assumption

22

is not satisfied even by linear functions of the form f (r ) = Cr , when C < −1. How-

ever, Assumption 2.1.4 cannot be omitted in Theorem 2.1.2. Consider for example

the SDE

ut = 1−∫ t

02us−d Ns ,

where Nt is a Poisson process with intensity one. Let τ be the time that the first

jump of N occurs. Then P (τ≤ T ) > 0. Since ut = e−2t on [0,τ), one can see that on

the set τ≤ T we have u(τ) =−e−2τ < 0.

The second equation that we will deal with is

dut (x) =(Lt ut (x)+ ft (x,ut (x),∇ut (x))+∂i f i

t (x))

d t

+Gkt (ut )(x)d w k

t +∫

FSt ,ζut−(x) M(d s,dζ) (2.1.6)

for (t , x) ∈ [0,T ]×Rd , with initial condition

u0(x) =ψ(x), x ∈Rd , (2.1.7)

where

Lt u(x) = Lt u(x)+I (2)t u(x),

I (2)t u(x) =

∫F

[λt (x +bt (ζ),ζ)u(x +bt (ζ))−λt (x,ζ)u(x)

−bt (ζ) ·∇(λt (x,ζ)u(x))]π(2)(dζ), (2.1.8)

St ,ζu(x) =λt (x +bt (ζ),ζ)u(x +bt (ζ))−λt (x,ζ)u(x) (2.1.9)

+ (λt (x,ζ)−1)u(x).

Obviously, if we ask later for some of the previous assumptions to hold for equation

(2.1.6), we mean with g ≡ 0.

Assumption 2.1.8. The function b mapsΩ×[0,T ]×F intoRd , it is P⊗F-measurable,

and there exists an F-measurable real function b on F , such that for allω, t and ζwe

have

|bt (ζ)| ≤ b(ζ),∫

Fb2(ζ)∧ b(ζ) π(2)(dζ) ≤ K .

The function λ maps Ω× [0,T ]×Rd ×F to [0,∞), is P⊗B(Rd )⊗F-measurable, it is

twice continuously differentiable in x for all ω, t ,ζ, and we have

|λt (x,ζ)|+ |∇λt (x,ζ)|+ |∇2λt (x,ζ)| ≤ K ,

23

|1−λt (x,ζ)| ≤ b(ζ), for all ω, t , x,ζ.

It is easy to see that due to Assumption 2.1.8 for every t ∈ [0,T ] and ω ∈ Ω the

mapping I (2)t , defined in the same way as I (1)

t , is a bounded linear operator from

H 10 to H−1, and ⟨I (2)u, v⟩ is a predictable process for any u, v ∈ H 1.

The solution of the problem (2.1.6)-(2.1.7) is understood in the same sense as

that of (2.1.1)-(2.1.2), and we have the following existence and uniqueness result.

Theorem 2.1.3. Let Assumptions 2.1.1 through 2.1.3 and 2.1.5 through 2.1.8 hold.

Then there exists a unique solution of the problem (2.1.6)-(2.1.7). Moreover the fol-

lowing estimate holds

E supt∈[0,T ]

|ut |2L2(Q) +E∫ T

0|u|2

H 10 (Q)

d t ≤ N

(E |φ|2L2(Q) +E

∫ T

0(|ht |2L2(Q) +

d∑i=1

| f it |2L2(Q))d t

),

where N is a constant depending only on d ,κ,K and T .

We also consider the problem

d vt (x) =(Lt vt (x)+Ft (x, vt (x),∇vt (x))+∂i f i

t (x))

d t

+Gkt (v)(x)d w k

t +∫

FSt ,ζv(x) M(d s,dζ), (2.1.10)

v0(x) =Ψ(x), (2.1.11)

where F and Ψ are as in (2.1.4)-(2.1.5).

Theorem 2.1.4. Suppose that Assumptions 2.1.1, and 2.1.5 through 2.1.8 hold. Let u

and v solve (2.1.6)-(2.1.7) and (2.1.10)- (2.1.11) respectively. Suppose that either f or

F satisfy Assumption 2.1.3. Let f ≤ F and ψ≤Ψ. Then almost surely, for all t ∈ [0,T ]

we have ut (x) ≤ vt (x) for almost every x ∈Rd .

2.1.2 Itô’s formula for the square of the norm of the positive part

In order to prove our comparison principles, we want to obtain an Itô’s formula for

|u+t |2L2(Q), where ut is an H−1(Q)-valued semimartingale taking values in H 1

0 (Q) for

dP ⊗d t almost every (ω, t ) ∈Ω×[0,T ]. Our approach to obtain it is similar to that in

[44]. To state the formula we set

V := H 10 (Q), H := L2(Q), V ∗ := H−1(Q),

and we consider the following processes

v :Ω× [0,T ] →V , v∗ :Ω× [0,T ] →V ∗, hk :Ω× [0,T ] → H ,

24

K :Ω× [0,T ]×Z → H ,

for integers k ≥ 1, where v is progressively measurable, v∗ and hk are Ft -adapted,

measurable in (ω, t ), and K is P ⊗Z measurable. We consider also ψ, an F0-

measurable random variable in H .

We make the following assumption.

Assumption 2.1.9.

(i) Almost surely

∫ T

0

(|vt |2V +|v∗

t |2V ∗ +∑k|hk

t |2H +∫

Z|Kt (z)|2Hν(d z)

)d t <∞,

(ii) for each φ ∈V and for dP ⊗d t-almost every (ω, t ), we have

(vt ,φ) = (ψ,φ)+∫ t

0⟨v∗

s ,φ⟩d s +∫ t

0(hk

s ,φ)d w ks

+∫ t

0

∫Z

(Ks(z),φ)N (d s,d z).

Theorem 2.1.5. Suppose that Assumption 2.1.9 is satisfied. Then there exists a set

Ω ⊂Ω of probability one, and an H-valued strongly càdlàg adapted process ut such

that ut = vt for dP ⊗d t-almost every (ω, t ). Moreover for ω ∈ Ω, t ∈ [0,T ] we have

i ) ut =ψ+∫ t

0v∗

s d s +∫ t

0hk

s d w ks +

∫ t

0

∫Z

Ks(z)N (d s,d z), (2.1.12)

i i ) |u+t |2H = |ψ+|2H +2

∫ t

0⟨v∗

s ,u+s ⟩d s +2

∫ t

0(hk

s ,u+s )d w k

s

+2∫ t

0

∫Z

(Ks(z),u+s−)N (d z,d s)+

∫(0,t ]

∑k|Ius>0hk

s |2H d s

+∫ t

0

∫Z|(us−+Ks(z))+|2H −|u+

s−|2H −2(Ks(z),u+s−)H N (d z,d s).

To prove Theorem 2.1.5 we need two lemmas.

Lemma 2.1.6. Let (X ,Σ,µ) be a measure space, and let un ,u ∈ L1(X )such that un → u

in L1(X ) as n → ∞. Then there exists a subsequence un(k)∞k=1 and a function v ∈

L1(X ) such that for all k ≥ 1 we have |un(k)(x)| ≤ v(x) for all x ∈ X , and un(k)(x) →u(x) for µ-almost every x as k →∞.

Proof. There exists a subsequence un(k) such that

|un(k) −u|L1(X ) ≤ 1/2k for k ≥ 1.

25

Set v(x) = |u(x)| +∑k |un(k)(x)−u(x)|. Then v has the desired properties. More-

over,∑

k |un(k+1) −un(k)|L1(X ) <∞, which implies that un(k) converges µ-almost ev-

erywhere.

The next lemma is from [13].

Lemma 2.1.7. Let Q be a bounded Lipschitz domain in Rd . Take φn ∈C∞c (Q), n ∈N,

with

i) 0 ≤φn ≤ 1

ii) φn = 1 on x ∈Q,r (x) ≥ 1/n

iii) φn = 0 on x ∈Q,r (x) ≤ 1/2n

iv) |(φn)xi | ≤C n,

where C is a constant and r (x) = di st (x,∂Q). Then φn v → v in H 10 (Q) for all v ∈

H 10 (Q), and for some constant C we have

supn

|φn v |H 10 (Q) ≤C |v |H 1

0 (Q), ∀v ∈ H 10 (Q).

Remark 2.1.3. One can easily see the existence of a sequence (φn)n∈N satisfying the

conditions of the previous lemma. Then note that φ2n also satisfies i)-iv). Hence,

φ2n v → v in H 1

0 (Q), for all v ∈ H 10 (Q), and for some constant C we have

supn

|φ2n v |H 1

0 (Q) ≤C |v |H 10 (Q), ∀v ∈ H 1

0 (Q).

We introduce now the functions αδ(r ), βδ(r ) and γδ(r ) on R, for δ> 0, given by

aδ(r ) =

1 if r > δrδ

if 0 ≤ r ≤ δ0 if r < 0,

βδ(r ) =∫ r

0aδ(s)d s, γδ(r ) =

∫ r

0βδ(s)d s.

For all r ∈Rwe haveαδ(r ) → Ir>0, βδ(r ) → r+ and γδ(r ) → (r+)2/2 as δ→ 0. Also, for

all r,r1,r2 and δ, the following inequalities hold

|αδ(r )| ≤ 1, |βδ(r )| ≤ |r |, |γδ(r )| ≤ r 2

2,

|γδ(r1 + r2)−γδ(r1)−βδ(r1)r2| ≤ |r2|2. (2.1.13)

We are now ready to prove Theorem 2.1.5.

26

Proof of Theorem 2.1.5. We only prove ii) since the rest of the assertions are proved

in [26], in greater generality. First we prove the statement when Q = Rd . We have

that equality (2.1.12) is satisfied if and only if, almost surely, for all ϕ ∈ V and t we

have

(ut ,ϕ) = (ψ,ϕ)+∫

(0,t ]⟨v∗

s ,ϕ⟩d s +∫ t

0(hk

s ,ϕ)d w ks

+∫ t

0

∫Z

(Ks(z),ϕ)N (d s,d z). (2.1.14)

Let φ be a mollifier with compact support and set φε(x) := ε−dφ(x/ε). For fixed x,

the function φε(x − ·) is in V , so we can plug it in (2.1.14) instead of ϕ, to get that

almost surely, for all t ∈ [0,T ]

uεt (x) = uε

0(x)+∫ t

0v∗ε

s (x)d s +∫ t

0hkε

s (x)d w ks

+∫ t

0

∫Z

K εs (z, x)N (d s,d z),

where for g ∈ V ∗ we use the notation g ε(x) := ⟨g ,φε(x − ·)⟩. Note that uε0 is F0 ⊗

B(Rd ) measurable. Also uε, v∗ε and hkε are jointly measurable in (t ,ω, x), Ft ⊗B(Rd ) measurable for each t , and K ε is P ⊗Z ⊗B(Rd ) measurable. It is also easy

to see that there exists a constant Cε, depending on ε, such that for all t ,ω, x, z

|uεt (x)| ≤Cε|ut |H , |uε

0(x)| ≤Cε|u0|H , |v∗εt |H ≤Cε|v∗

t |V ∗

|v∗εt (x)| ≤Cε|v∗

t |V ∗ , |hkεt (x)| ≤Cε|hk

s |H ,

|K εt (x, z)| ≤Cε|Kt (z)|H .

One can also check that for a constant C , for all ε

|uεt |H ≤C |ut |H , |uε

0|H ≤C |u0|H , |K εt (z)|H ≤C |Kt (z)|H

|hkεt |H ≤C |hk

s |H , |v∗εt |V ∗ ≤C |v∗

t |V ∗ , |uεt |V ≤C |ut |V .

Now let αδ,βδ,γδ be as before, and fix x. By Itô’s formula (see for example [35] or

[60]), for each x there exists a set Ωx of full probability, such that for all ω ∈Ωx and

27

t ∈ [0,T ] we have

γδ(uεt (x)) =γδ(uε

0(x))+∫ t

0βδ(uε

s (x))v∗εs (x)d s

+∑k

∫ t

0βδ(uε

s (x))hkεs (x)d w k

s

+ 1

2

∑k

∫ t

0αδ(uε

s (x))|hkεs (x)|2d s

+∫ t

0

∫Zβδ(uε

s−(x))K εs (z, x)N (d s,d z)

+∫ t

0

∫Zγδ(uε

s (x)+K εs (z, x))

−γδ(uεs−(x))−βδ(uε

s−(x))K εs (z, x)N (d z,d s). (2.1.15)

One can redefine the stochastic integrals such that (2.1.15) holds for all (ω, t , x). In-

tegrating (2.1.15) overRd , taking appropriate versions of the stochastic integrals and

using the Fubini and the stochastic Fubini theorems we get for each t ∈ [0,T ],∫Rdγδ(uε

t (x))d x =∫Rdγδ(uε

0(x))d x +∫ t

0

∫Rdβδ(uε

s (x))v∗εs (x)d x d s

+∫ t

0

∫Rdβδ(uε

s (x))hkεs (x)d x d w k

s

+ 1

2

∑k

∫(0,t ]

∫Rdαδ(uε

s (x))|hkεt (x)|2 d x d s

+∫ t

0

∫Z

∫Rdβδ(uε

s−(x))K εs (z, x)d xN (d s,d z)

+∫ t

0

∫Z

∫Rdγδ(uε

s−(x)+K εs (z, x))

−γδ(uεs−(x))−βδ(uε

s−(x))K εs (z, x)d xN (d z,d s) (a.s.). (2.1.16)

For a stochastic Fubini theorem we refer to [45], noting that the Fubini theorem

there can be extended easily, by obvious changes in its proof, to our situation. Since

each term in the above equation is a càdlàg process in t , we see that (2.1.16) holds

almost surely, for all t ∈ [0,T ]. We claim that for each t ∈ [0,T ] both sides of (2.1.16)

28

converges in probability as ε→ 0 to give that∫Rdγδ(ut (x))d x =

∫Rdγδ(u0(x))d x +

∫ t

0⟨v∗

s ,βδ(us)⟩d s

+∫ t

0

∫Rdβδ(us(x))hk

s (x)d xd w ks

+ 1

2

∑k

∫ t

0

∫Rdαδ(us(x))|hk

s (x)|2d xd s

+∫ t

0

∫Z

∫Rdβδ(us−(x))Ks(z, x)d xN (d s,d z)

+∫ t

0

∫Z

∫Rdγδ(us−(x)+Ks(z, x))

−γδ(us−(x))−βδ(us−(x))Ks(z, x)d xN (d z,d s). (2.1.17)

holds almost surely for each t ∈ [0,T ]. We are going to show that each term in

(2.1.16) converges in probability to the corresponding one in (2.1.17). Since for

any sequence εk ↓ 0 we have uεkt → ut in L2(Rd ) for every ω ∈ Ω, by the equality

a2 − b2 = (a − b)(a + b) we have (uεkt )2 → u2

t in L1(Rd ). Thus for every ω ∈ Ω by

Lemma 2.1.6 there exist g ∈ L1(Rd ) and a subsequence, denoted again by εk , such

that for all k ≥ 1

|γδ(uεkt (x))| ≤ (uεk

t (x))2

2≤ g (x)

2for all x.

Since γδ(uεkt (x)) → γδ(ut (x)) for almost every x, as k →∞, by Lebesgue’s theorem

on dominated convergence we obtain∫Rdγδ(uεk

t (x))d x →∫Rdγδ(ut (x))d x as k →∞.

Thus, for ε ↓ 0 the left-hand side of (2.1.16) converges to the left-hand side of (2.1.17)

for every ω ∈Ω, and hence also in probability, for each t ∈ [0,T ]. To see the conver-

gence of the second term in the right-hand side of (2.1.16) we fix (s,ω) such that

us ∈V . Then it is straightforward to check that

|βδ(uεs )−βδ(us)|V → 0, as ε→ 0.

Taking into account the well-known fact that there exist f 0s and f i

s ∈ L2(Rd ) for i =1, ...,d such that

v∗s = f 0

s +∂i f is ,

we have

v∗εs = f 0ε

s +∂i f iεs ,

29

which gives

|v∗s − v∗ε

s |V ∗ ≤d∑

i=0| f iε

s − f is |H → 0, as ε→ 0.

Hence we conclude∫Rdβδ(uε

s (x))v∗εs (x)d x = ⟨v∗ε

s ,βδ(uεs )⟩→ ⟨v∗

s ,βδ(us)⟩.

Notice that there is a constant C such that∣∣∣∫Rnβδ(uε

s (x))v∗εs (x)d x

∣∣∣≤C (|us |2V +|v∗s |2V ∗)

for all ε> 0, ω ∈Ω and s ∈ [0,T ]. Therefore, almost surely∫ t

0

∫Rdβδ(uε

s (x))v∗εs (x)d xd s →

∫ t

0⟨v∗

s ,βδ(us)⟩d s for all t .

For the sum of the stochastic integrals against the Wiener processes we just note

that almost surely for all s ∈ [0,T ]

∑k

∣∣∣∫Rdβδ(uε

s (x))hkεs (x)d x −

∫Rdβδ(us(x))hk

s (x)d x∣∣∣2 → 0 as ε ↓ 0,

and ∑k

∣∣∣∫Rdβδ(uε

s (x))hkεs (x)d x −

∫Rdβδ(us(x))hk

s (x)d x∣∣∣2

≤ 4supt≤T

|ut |2L2

∑k|hk

s |2L2 for all ε> 0.

Hence almost surely

∫ T

0

∑k

∣∣∣∫Rdβδ(uε

s (x))hkεs (x)d x −

∫Rdβδ(us(x))hk

s (x)d x∣∣∣2

d s → 0,

which implies that for ε ↓ 0∫ t

0

∫Rdβδ(uε

s (x))hkεs (x)d x d w k

s →∫ t

0

∫Rdβδ(us(x))hk

s (x)d x d w ks

in probability, uniformly in t ∈ [0,T ]. Note that for each k we have

∣∣∣∫Rdαδ(uε

s (x))|hkεs (x)|2 −αδ(us(x))|hk

s (x)|2d x∣∣∣

≤∫Rd

|(hkεs (x))2 − (hk

s (x))2|d x

30

+∫Rd

|hks (x)|2|αδ(uε

s (x))−αδ(us(x))|d x → 0.

for each ω ∈Ω and s ∈ [0,T ]. Moreover,∣∣∣∫Rdαδ(uε

s (x))|hkεs (x)|2d x

∣∣∣≤ |hks |2H ,

where the right-hand side is almost surely integrable on [0,T ]. Hence the almost

sure convergence of the fourth term in the right-hand side of (2.1.16) follows. By

the inequalities in (2.1.13), similar arguments show the convergence of the last two

terms in probability. We conclude that for each t ∈ [0,T ] equation (2.1.17) holds

almost surely. Since the stochastic processes in both sides of (2.1.17) are càdlàg pro-

cesses, equation (2.1.17) holds almost surely for all t ∈ [0,T ].

Now by letting δ→ 0 in (2.1.17), using arguments similar to the previous ones,

and keeping in mind the inequalities (2.1.13) and the fact that for all v ∈V

|βδ(v)− v+|V → 0, |βδ(v)|V ≤ |v |V ,

we can finish the proof of the theorem for Q =Rd .

We reduce the case of a bounded Lipschitz domain Q to that of the whole space

by using the sequence φn from Lemma 2.1.7. Remember that φn has compact sup-

port in Q. Thus for a function η on Q we denote by φnη, not only the function de-

fined on Q by the multiplication of φn and η, but also its extension to zero outside

of Q. Notice that when u satisfies (2.1.12) on Q, then φnu satisfies

φnut =φnu0 +∫ t

0φn v∗

s d s +∫ t

0φnhk

s d w ks

+∫ t

0

∫YφnKs(z)N (d s,d z)

on the whole Rd , where the functional φn v∗ is defined by

⟨φn v∗s , g ⟩ := ⟨v∗

s ,φn g ⟩Q

for g ∈ H 1(Rd ). The notation ⟨·, ·⟩Q means the duality product between H 10 (Q) and

H−1(Q). Notice that ⟨v∗s ,φn g ⟩Q is well defined, since the restriction of φn g to Q

31

belongs to H 10 (Q). Then by the result in the case of the whole space, we have

∫Qφ2

n |u+t |2d x =

∫Q|φnu+

0 |2d x +2∫ t

0⟨v∗

s ,φ2nu+

s ⟩Q d s

+2∫ t

0

∫Qφ2

nhks u+

s d xd w ks

+∫ t

0

∫Q

∑k|Iφn us>0φnhk

s |2d xd s

+∫ t

0

∫Z

∫Q

2Ks(z)φ2nu+

s−d xN (d s,d z)

+∫ t

0

∫Z

∫Q|φn(us−+Ks(z))+|2 −|φnu+

s−|2

−2Ks(z)φ2nu+

s−d xN (d z,d s), (2.1.18)

since φn is supported in Q. It is now easy to take n →∞ here to finish the proof of

the theorem. We only note that for the second term on the right-hand side we have

by Lemma 2.1.7 and Remark 2.1.3

⟨v∗s ,φ2

nu+s ⟩Q →⟨v∗

s , u+s ⟩Q for all ω, s,

and for a constant C ,

⟨v∗s ,φ2

nu+s ⟩Q ≤C |v∗

s |V ∗ |us |V for all n.

2.1.3 The main estimates

In this section we present some lemmas that we will need for the proofs of Theorems

2.1.1 through 2.1.4. The following is well known (see, e.g., [50], or exercise 1.3.19 in

[41], or some more general results in [58]).

Lemma 2.1.8. Let u ∈ W 1p (Q). Let un ∈ W 1

p (Q) such that |un −u|W 1p→ 0 as n →∞.

Then we have |u+n −u+|W 1

p→ 0.

For the next three lemmas, we assume that Assumptions 2.1.5 through 2.1.8

hold. For u ∈C∞c (Rd ), let us define the quantities,

%t (u) :=∫Rd

∫F

(λt (x +bt (ζ))u (x +bt (ζ)))2 − (λt (x,ζ)u(x))2

−2bt (z) ·∇ (λt (x,ζ)u (x))λt (x,ζ)u (x)π(2)(dζ)d x,

32

%t (u) :=∫Rd

∫F

[(λt (x +bt (ζ))u (x +bt (ζ)))+]2 − [(λt (x,ζ)u(x))+]2

−2bt (z) ·∇ (λt (x,ζ)u (x))λt (x,ζ)u+ (x)π(2)(dζ)d x.

Lemma 2.1.9. For any u ∈C∞c (Rd ), ω ∈Ω, t ∈ [0,T ] and ε> 0 we have∫

RdI (1)

t u2(x) d x ≤ε|u|2H 1(Rd )

+N (ε)|u|2L2(Rd )

, (2.1.19)∫Rd

I (1)t (u+)2(x) d x ≤ε|u+|2

H 1(Rd )+N (ε)|u+|2

L2(Rd ), (2.1.20)

%t (u) ≤ε|u|2H 1(Rd )

+N (ε)|u|2L2(Rd )

, (2.1.21)

%t (u) ≤ε|u+|2H 1(Rd )

+N (ε)|u+|2L2(Rd )

, (2.1.22)

where the constant N (ε) depends only on ε, K and d.

Proof. We prove (2.1.20). Clearly,∫Rd

I (1)t (u+)2(x)d x =

∫Rd

I (1δ)t (u+)2(x)d x +

∫Rd

I (1δ)t (u+)2(x)d x. (2.1.23)

The first term on the right-hand side is equal to

∫ 1

0(1−θ)

∫Fδ

∫Rd∂i j (u+)2(x +θct (x,ζ))

×c it (x,ζ)c j

t (x,ζ)mt (x,ζ)d xπ(1)(dζ)dθ

= E1(t ,δ)+E2(t ,δ),

where

E1(t ,δ) =∫ 1

0(1−θ)

∫Fδ

∫Rd

2∂i u+(x +θct (x,ζ))∂ j u+(x +θct (x,ζ))

×c it (x,ζ)c j

t (x,ζ)mt (x,ζ)d xπ(1)(dζ)dθ,

E2(t ,δ) =∫ 1

0(1−θ)

∫Fδ

∫Rd

2u+(x +θct (x,ζ))∂i j u(x +θct (x,ζ))

×c it (x,ζ)c j

t (x,ζ)mt (x,ζ)d xπ(1)(dζ)dθ.

Using Assumptions 2.1.5, 2.1.6 and 2.1.7, we see after a change of variables that

|E1(t ,δ)| ≤C (δ)C |u+|2H 1(Rd )

,

where C (δ) = ∫Fδ

c2(ζ)π(d z) and C is a constant depending only on K and d . For E2

33

we have

E2(t ,δ) =∫ 1

0(1−θ)

∫Fδ

∫Rd

2∂ j (∂i u(x +θct (x,ζ))

×q i jt (x,ζ,θ)u+(x +θct (x,ζ))d xπ(1)(dζ)dθ.

By integration by parts and using the Assumptions 2.1.5, 2.1.6 and 2.1.7 again we see

that

|E2(t ,δ)| ≤C (δ)C |u+|2H 1(Rd )

.

For the second term in (2.1.23) by Young’s inequality and Assumptions 2.1.6, 2.1.5,

we have ∫Rd

I (1δ)t (u+)2(x) d x ≤ γ|u|2

H 1(Rd )+C (γ)|u|2

L2(Rd ),

for all γ> 0, where C (γ) depends only on γ and K . Putting these estimates together

and choosing δ and γ sufficiently small, we finish the proof of (2.1.20). One can

repeat the same calculation with c replaced by b, m = 1 and λu in place of u to get

(2.1.22). Also (2.1.19) and (2.1.21) can be proved in the same way.

Lemma 2.1.10. For any u ∈ H 10 (Q), ω ∈Ω, t ∈ [0,T ] and ε> 0 we have

2⟨I (1)t u,u⟩ ≤ ε|u|2

H 10 (Q)

+N (ε)|u|2L2(Q), (2.1.24)

2⟨I (1)t u,u+⟩ ≤ ε|u+|2

H 10 (Q)

+N (ε)|u+|2L2(Q), (2.1.25)

where the constant N (ε) depends only on ε and K and d.

Proof. We prove (2.1.25). It suffices to prove it for Q = Rd . Due to Lemma 2.1.8 and

the continuity of the operator I (1)t : H 1 → H−1, we may and will also assume that

u ∈C∞c (Rd ). Notice that for any α,β ∈R

2(β−α)α+ ≤ (β+)2 − (α+)2 − (β+−α+)2 ≤ (β+)2 − (α+)2. (2.1.26)

Consequently, for any α,β,γ ∈R

2(β−α−γ)α+ ≤ (β+)2 − (α+)2 −2γα+.

Using this with α = u(x), β = u(x + ct (x,ζ)) and γ = ct (x,ζ)∇u(x), and taking into

account that 2∇uu+ =∇(u+)2, we can easily see that

2⟨I (1)t u,u+⟩ = 2(I (1)

t u,u+) ≤∫Rd

I (1)t (u+)2(x) d x.

Hence (2.1.25) follows from Lemma 2.1.9. One can prove (2.1.24) in the same way,

by using the inequality 2(β−α)α≤β2 −α2, instead of (2.1.26).

34

For u ∈ H 1(Rd ) we set

µt (u) :=∫

F

∫Rd

[(λt (x +bt (ζ),ζ)u(x +bt (ζ))]2 − [u(x)]2

−2u(x)[λt (x +bt (ζ),ζ)u(x +bt (ζ))−u(x)]d xπ(2)(dζ),

ρt (u) :=2⟨I (2)t u,u⟩+µt (u), (2.1.27)

µt (u) :=∫

F

∫Rd

[(λt (x +bt (ζ),ζ)u(x +bt (ζ)))+]2 − [u+(x)]2

−2u+(x)[λt (x +bt (ζ),ζ)u(x +bt (ζ))−u(x)]d xπ(2)(dζ),

ρt (u) :=2⟨I (2)t u,u+⟩+ µt (u). (2.1.28)

Using the simple inequality |[(x+y)+]2−[x+]2−2x+y | ≤ 2|y |2, and Assumption 2.1.8

one can see that µt (u) is continuous in u ∈ H 1(Rd ). It can be shown similarly that

µt (u) is continuous in u ∈ H 1(Rd ).

Lemma 2.1.11. For any u ∈ H 1(Rd ), (ω, t ) ∈Ω×R+ and ε> 0 we have

ρt (u) ≤ ε|u|2H 1(Rd )

+N (ε)|u|2L2(Rd )

, (2.1.29)

ρt (u) ≤ ε|u+|2H 1(Rd )

+N (ε)|u+|2L2(Rd )

. (2.1.30)

Proof. Since (2.1.29) can be shown in the same way as (2.1.30), we only prove the

latter one. Clearly it suffices to prove it for u ∈C∞c (Rd ). A simple calculation shows

that

ρt (u) = %t (u)+∫

F

∫Rd

(λt (x,ζ)−1)2[u+(x)]2d xπ(2)(dζ)

+∫

F

∫Rd

2bt (ζ) ·∇(u(x)λt (x,ζ))u+(x)(λt (x,ζ)−1)d xπ(2)(dζ)

By Young’s inequality, Assumption 2.1.8 and (2.1.22) we get that

ρt (u) ≤ ε|u+|2H 1(Rd )

+N (ε)|u+|2L2(Rd )

.

Lemma 2.1.12. Let Assumption 2.1.3 hold. Then for all (ω, t ) ∈Ω× [0,T ], u ∈ H 10 (Q)

and ε> 0 we have

2( ft (u,∇u)− ft (v,∇v),u − v)+∫

Z|g t (z,u)− g t (z, v)|2L2(Q)ν(d z)

≤ ε|u − v |2H 1

0 (Q)+N (ε)|u − v |2L2(Q), (2.1.31)

2( ft (u,∇u)− ft (v,∇v), (u − v)+)+∫

Z|Iu>v (g t (z,u)− g t (z, v))|2L2(Q)ν(d z)

35

≤ ε|(u − v)+|2H 1

0 (Q)+N (ε)|(u − v)+|2L2(Q), (2.1.32)

where N (ε) depends only on ε and K .

Proof. We show (2.1.32). Using the second part of Assumption 2.1.3 and Young’s

inequality we have

2( ft (u,∇u)− ft (v,∇v), (u − v)+) ≤ K

ε|(u − v)+|2L2(Q) +ε|∇(u − v)+|2L2(Q)

+∫

Q( ft (x,u,∇u)− ft (x, v,∇u))(u(x)− v(x))+d x.

This combined with Assumption 2.1.3 gives (2.1.32). Inequality (2.1.31) can be shown

in the same way.

2.1.4 Proof of Theorems 2.1.1, 2.1.2, 2.1.3 and 2.1.4

We are now ready to proceed with the proofs of the main theorems.

Proof of Theorems 2.1.1 and 2.1.3. We prove Theorem 2.1.1. It suffices to show that

conditions I) through IV) from Assumption 1.4.1 are satisfied. The growth condition

of the operator

uA7−→ Lt u + ft (u,∇u)+∂i f i

can be verified easily. Notice that for every ω, t and x, the function ft (x,r,r ′) is

continuous in (r,r ′). Using this, ii) from Assumption 2.1.1 and the fact that Lt is

a bounded linear operator from H 10 (Q) into H−1(Q), we see that A is semicontinu-

ous. Now, by ii) and iv) from Assumption 2.1.1, the boundedness of φ and (2.1.24)

we see that for a θ > 0 and a constant C we have

2⟨Lt u,u⟩+2(u, ft (u,∇u))−2( f it ,Di u)+∑

k|Gk

t (u)|2L2(Q) +∫

Z|g t (u)|2L2(Q)ν(d z)

≤−θ|u|2H 1

0 (Q)+C |u|2L2(Q) +C |ht |2L2(Q) +C

d∑i=1

| f it |2L2(Q).

for all t ,ω and u ∈ H 10 (Q). This shows that the coercivity condition is satisfied. Using

i), iv), v) from Assumption 2.1.1 and (2.1.24) we see that for all (t ,ω) and γ> 0

2⟨Lt u −Lt v,u − v⟩+∑k|Gk (u)−Gk (v)|2L2(Q)

≤ (γ−Å)|u − v |2H 1

0 (Q)+C (γ)|u − v |2L2(Q),

36

for all u, v ∈ H 10 (Q), where Å is the ellipticity constant form part (iv) of Assumption

2.1.1. Combining this with (2.1.31) we have that the monotonicity condition is also

satisfied. The proof of Theorem 2.1.3 goes in the same way. We omit the details, we

only note that one also has to use (2.1.29).

Proof of Theorem 2.1.2. Without loss of generality we can assume that Assumption

2.1.3 is satisfied by f . For the difference h = u − v we have

ht =h0 +∫ t

0Lshs + fs(us ,∇us)−Fs(vs ,∇vs)d s

+∫ t

0φki

s ∂i hs +σks (us)−σk

s (vs)d w ks

+∫ t

0

∫Z

g (s, z,us−)− g (s, z, vs−))N (d s,d z).

By Theorem 2.1.5 we have

|h+t |2L2 =

∫ t

0A(1)

s + A(2)s + A(3)

s +2⟨I (1)s hs ,h+

s ⟩ d s +mt

for a martingale mt , where

A(1)s =

∫Q

−2ai j

s (x)∂i h+s (x)∂ j h+

s (x)

+∑k

∣∣∣Ihs>0

∑iφki

s (x)∂i hs(x)+ Ihs>0(σks (x,us(x))−σk

s (x, vs(x)))∣∣∣2

d x (2.1.33)

A(2)s = 2

∫Q

( fs(x,us ,∇us)−Fs(x, vs ,∇vs))h+s (x)d x (2.1.34)

A(3)s =

∫Z

∫Q

[hs(x)+ gs(x, z,us−(x))− gs(x, z, vs−(x))]+2 −|hs(x)+|2

−2h+s (x)[gs(x, z,us(x))− gs(x, z, vs(x))]d xν(d z).

One can easily see that for every ε > 0, there exist C (ε) > 0 depending only on ε, K

and d , such that

A(1)s ≤ (−Å+ε)|h+

s |2H 10 (Q)

+C (ε)|h+s |2L2(Q).

By Assumption 2.1.4 we obtain

A(3)s =

∫Z

∫Q

Ihs>0|gs(x, z,us)− gs(x, z, vs)|2d xν(d z).

Hence, by (2.1.32) we have

A(2)s + A(3)

s ≤ ε|h+s |2H 1

0 (Q)+C (ε)|h+

s |2L2(Q).

37

Combining these estimates and using (2.1.25) we have a constant C such that, al-

most surely

|h+t |2L2(Q) ≤C

∫ t

0|h+

s |2L2(Q) d s +mt for all t ∈ [0,T ].

Then we have,

E |h+t |2L2(Q) ≤C

∫ t

0E |h+

s |2L2(Q) d s <∞ for all t ∈ [0,T ],

and the result follows by Gronwall’s lemma.

Proof of Theorem 2.1.4. We assume again that Assumption 2.1.3 is satisfied by f . For

the difference h = u − v we have

ht =h0 +∫ t

0Lshs + fs(us ,∇us)−Fs(vs ,∇vs) d s

+∫ t

0φki

s ∂i hs +σks (us)−σk

s (vs)d w ks +

∫ t

0

∫F

Ss,ζhs−M(d s,dζ)

By Theorem 2.1.5 we have

|h+t |2L2(Rd )

=∫ t

0A(1)

s + A(2)s + ρs(hs)+⟨I (1)

s hs ,h+s ⟩ d s +mt

for a martingale mt . Here A(1), A(2) are as in (2.1.33), (2.1.34) (with the integration

over Rd instead of Q), and ρ is defined in (2.1.28). By using the same arguments as

in the previous proof, this time also using (2.1.30), we bring the proof to an end.

2.2 Numerical approximation

In the previous section we showed that the equations under consideration are solv-

able. However, to obtain an explicit solution is usually impossible, and therefore we

are interested in approximating them numerically.

Various methods have been developed to solve SPDEs numerically (see, for ex-

ample, [19], [24], [23] and [29]). Among the various methods considered in the lit-

erature is the method of finite differences. For second order linear SPDEs driven

by continuous martingale noise it is well-known that the error of approximation in

space is proportional to the parameter h of the finite difference (see, e.g., [34]). In

[29], I. Gyöngy and A. Millet consider abstract discretization schemes for stochas-

tic evolution equations driven by continuous martingale noise in the variational

framework and, as a particular example, show that the rate of convergence of an

Euler-Maruyuma (explicit and implicit) finite difference scheme is of order one in

38

space and one-half in time. More recently, it was shown by I. Gyöngy and N.V.

Krylov that under certain regularity conditions, the rate of convergence in space

of a semi-discretized finite difference approximation of a linear second order SPDE

can be accelerated to any order by Richardson’s extrapolation method. For the non-

degenerate case, we refer to [27] and [28], and for the degenerate case, we refer to

[25]. In [30] and [31], was proved that the same method of acceleration can be ap-

plied to time-discretized SPDEs.

Concerning equations involving non-local operators, in spatial dimension one,

a finite difference scheme for degenerate integro-differential equations (determin-

istic) has been studied in [3]. The authors in [3] first approximate the integral op-

erator near the origin with a second derivative operator. The resulting PDE is then

non-degenerate and has an integral operator of order zero. The error of this approx-

imation is obtained by means of the probabilistic representation of the solution of

both the original equation and the non-degenerate equation. In the second step

of their approximation, the authors consider a finite difference scheme and obtain

pointwise error estimates of order one in space. As a consequence of the two-step

approximation scheme, there are two separate errors for the approximation

In this section, we consider a non-degenerate linear SPIDE and we give a spa-

tial approximation scheme whose error is proportional to the parameter of the dis-

cretization. The approximations of the non-local integral operators that we suggest

are natural and they fit in the finite-difference framework. We are able to treat the

singularity of the integral operators near the origin directly, and therefore we only

have to control one error. To obtain error estimates for our approximations, we

use the approach of [34], where the discretized equations are first solved as SDEs

in Sobolev spaces over Rd and an error estimate is obtained in Sobolev norms. Af-

ter obtaining error estimates in Sobolev norms, the Sobolev embedding theorem is

used to obtain pointwise error estimates.

For a space-time discretization of these equations we refer to [8], where using

the spatial approximation presented here, in combination with an Euler scheme in

time we fully discretize the SPIDE under consideration, in a joint work with James-

Michael Leahy.

2.2.1 A spatial discretization scheme

In this section, N (d t ,d z) and ν(d z) are as in the previous section, with Z =Rd , and

in addition we consider a σ-finite measure µ on Rd . On the cylinder [0,T ]×Rd , we

39

consider the stochastic integro-differential equation

dut (x) =(Lt ut (x)+I ut (x)+

d∑i=0

∂i f it (x)

)d t + (

M%t ut (x)+ g%t (x)

)d w%

t

+∫Rd

J (z)ut−(x) N (d z,d t ) (2.2.35)

with initial condition

u0(x) =ψ(x), x ∈Rd ,

where

Ltφ(x) = ai jt (x)∂i jφ(x), M

%t φ(x) = bi%

t (x)∂iφ, (sum over i , j ∈ 0, ...,d)

Iφ(x) =∫Rd

(φ(x + z)−φ(x)−∂iφ(x)zi

)µ(d z) (sum over i ∈ 1, ...,d),

(2.2.36)

J (z)φ(x) =φ(x + z)−φ(x),

and where for i , j ∈ 0, ...,d and % ∈ N+, ai jt , ft , bi%

t , and g%t are random functions

depending on (t , x) ∈ [0,T ]×Rd .

Through this section we will assume that∫Rd

|z|2 ∧|z| µ(d z) <∞,∫Rd

|z|2 ∧1 ν(d z) <∞. (2.2.37)

Let m ≥ 0 be an integer.

Assumption 2.2.1. For i , j ∈ 0, . . . ,d, ai jt = ai j

t (x) are real-valued functions that

are P ⊗B(Rd )-measurable and bit = (bi%

t (x))∞%=1 are `2-valued functions that are

P ⊗B(Rd )-measurable. Moreover,

(i) for each (ω, t ) ∈Ω× [0,T ], the functions ai j are max(m,1)-times continuously

differentiable in x for i , j ∈ 1, ...,d and m-times continuously differentiable in

x for i = 0 or j = 0. For each (ω, t ) ∈Ω×[0,T ], the functions bi are m-times con-

tinuously differentiable in x for i ∈ 0, ...,d. Moreover, there exists a constant

Km ≥ 0 with such that for all (ω, t , x) ∈Ω× [0,T ]×Rd ,

|∂αai jt | ≤ Km , ∀|α| ≤ max(m,1), ∀i , j ∈ 1, ...,d

|∂αa0it |∨ |∂αai 0

t |∨ |∂αbit |`2 ∨|∂αbt |`2 ≤ Km , ∀|α| ≤ m, ∀i 0, ...,d.

(ii) there exists a positive constant Å > 0 such that for all (ω, t , x) ∈Ω× [0,T ]×Rd

40

and ξ ∈Rd

d∑i , j=1

(2ai j

t −bi%t b j%

t

)ξiξ j ≥ Å|ξ|2.

We define the following spaces:

Hm := L2(Ω× [0,T ],P ; H m), Hm(`2) := L2(Ω× [0,T ],P ; H m(`2))

Hm2 (ν) := L2(Ω× [0,T ]×Rd ,P ⊗B(Rd ),dP ⊗d t ⊗dν; H m).

Assumption 2.2.2. The random functionψ is a H m-valued F0-measurable random

variable, f i ∈Hm for i ∈ 0, ...,d, and g ∈Hm(`2). Furthermore,

κ2m := E |ψ|2H m +E

∫ T

0

(d∑

i=0| f i

s |2H m +|gs |2H m (`2)

)d s <∞.

For a real-valued twice continuous differentiable function u on Rd , it is easily

seen that

u(x + z)−u(x)−d∑

i=1∂i u(x)zi =

∫ 1

0

d∑i , j=1

∂i j u(x +θz)zi z j (1−θ)dθ, x ∈Rd .

Due to (2.2.37), there exists δ> 0 such that∫|z|≤δ

|z|2ν(d z) < Å

12,

∫|z|≤δ

|z|2µ(d z) < Å

12,

∫|z|>δ

|z|ν(d z)+µ(|z| ≥ δ)+ν(|z| ≥ δ) <∞. (2.2.38)

Let us fix such δ and notice that I =Iδ+Iδc , where

Iδφ(x) =∫|z|≤δ

∫ 1

0

d∑i , j=1

∂i jφ(x +θz)zi z j (1−θ)dθ

and where Iδc is defined as in (2.2.36) with integration over |z| > δ instead of Rd .

The solution of equation (2.2.35) is understood as in the previous section, that

is, an L2(Rd )-valued strongly càdlàg adapted process u such that

(i) ut ∈ H 1 for dP ⊗d t-almost every (ω, t ) ∈Ω× [0,T ];

(ii) E∫ T

0 |us |2H 1 d s <∞;

(iii) there exists a set Ω⊂Ω of probability one such that for all (ω, t ) ∈ [0,T ]×Ω and

41

φ ∈C∞c (Rd ),

(ut ,φ) = (ψ,φ)+∫ t

0

((∂ j us ,∂−i (ai j

s φ))+⟨I us ,φ⟩+ ( fi ,∂−iφ)

)d s

+∫ t

0

(bi%

s ∂i us + g%s ,φ)

d w%s +

∫ t

0

∫Rd

(us−(·+ z)−us−,φ

)N (d z,d s).

(2.2.39)

The proof of the following theorem is a simple consequence of Theorem 1.4.1,

and can be found in [8].

Theorem 2.2.1. If Assumptions 2.2.1 and 2.2.2 hold with m ≥ 0, then there exist a

unique solution u of (2.2.35). Furthermore, u is a strongly càdlàg H m-valued process

with probability one and there exists a constant N depending only on d ,m,Å,Km , and

ν such that

E supt≤T

|ut |2H m +E∫ T

0|us |2H m+1 d s ≤ Nκ2

m . (2.2.40)

We turn our attention to the discretisation of equation (2.2.35). For each h ∈R \ 0 and standard basis vector ei , i ∈ 1, . . . ,d, of Rd we define the first-order dif-

ference operator δh,i by

δh,iφ(x) = φ(x +hei )−φ(x)

h,

for all real-valued functions φ on Rd . We define δh,0 to be the identity operator.

Notice that for all ψ,φ ∈ L2(Rd ) and i ∈ 1, ...,d, we have

(φ,δ−h,iψ) =−(δh,iφ,ψ).

We introduce the grid Gh := hzk : zk ∈ Zd ,k ∈ N with meshsize |h|. Let `2(Gh) be

the set of real-valued functions φ on Gh such that

|φ|2`2(Gh ) := |h|d ∑x∈Gh

|φ(x)|2 <∞.

We approximate the operators L and M % by

L ht φ(x) = ai j

t (x)δh,iδ−h, jφ(x),

Mh,%t φ(x) = bi%

t (x)δh,iφ(x).

In order to approximate the operator I , we approximate Iδ and Iδc separately. For

42

k ∈N and h 6= 0, let Ahk be the rectangle

(z1

k h − |h|2

, z1k h + |h|

2

]× ...×

(zd

k h − |h|2

, zdk h + |h|

2

],

where zik , i = 1, ...,d are the coordinates of zk ∈Zd , and set

B hk := Ah

k ∩ |z| ≤ δ, B hk := Ah

k ∩ |z| ≥ δ.

We approximate Iδc by

I hδc u(x) :=

∞∑k=0

((u(x +hzk )−u(x)) ζhk −

d∑i=1

ξihkδh,i u(x)

), (2.2.41)

where

ζhk :=µ(B hk ) and ξi

hk :=∫

B hk

ziµ(d z).

We continue with the approximation of the operator Iδ. Notice that

Iδφ(x) =∞∑

k=0

∫B h

k

∫ 1

0

d∑i , j=1

zi z j∂i jφ(x +θz)(1−θ)dθµ(d z),

where there are only a finite number of non-zero terms in the infinite sum over k.

The closest point in Gh to any point z ∈ B hk is clearly hzk . This simple observation

leads us to the following (intermediate) approximation of Iδφ(x):

∞∑k=0

∫ 1

0

d∑i , j=1

∫B h

k

zi z j∂i jφ(x +θhzk )(1−θ)dθµ(d z).

However, in order to ensure that our approximation is well-defined for functions

φ ∈ `2(Gh), we need to approximate the integral over θ ∈ [0,1]. For fixed h 6= 0 and

k ∈ N∪ 0, there exist σ(h,k) ∈ N, r h,kl ∈ N∪ 0, for l ∈ 0, . . . ,σ(h,k)− 1, and real

numbers (θh,kl )σ(h,k)

l=0 satisfying 0 = θh,k0 ≤ ·· · ≤ θh,k

σ(h,k) = 1, such that the line segment

θhzk θ∈[0,1] is contained in the set ∪σ(h,k)−1l=0 Ah

r h,kl

, and for θ ∈ (θh,kl ,θh,k

l+1), we have

θhzk ∈ Ahr h,k

l

. In particular, for k = 0, we have σ(h,0) = 1, r h,00 = 0, θh,0

0 = 0, and

θh,10 = 1. Since the diagonal of a d-dimensional hypercube with side length |h| has

lengthp

d |h|, for each k ∈N∪ 0, z ∈ B hk , and l ∈ 0, . . . ,σ(h,k)−1 we have,

|θz −hzr h,kl

| ≤ |θz −θhzk |+ |θhzk −hzr h,kl

| ≤p

d |h|, (2.2.42)

43

for all θ ∈ (θh,kl ,θh,k

l+1). Set

ζi jh,k =

∫B h

k

zi z jµ(d z), θh,kl =

∫ θh,kl+1

θh,kl

(1−θ)dθ

and define the operator

I hδ φ(x) =:

∞∑k=0

σ(h,k)−1∑l=0

θh,kl

d∑i , j=1

ζi jh,kδh,iδ−h, jφ(x +hzr h,k

l). (2.2.43)

There are only finitely many non-zero terms in the infinite sum over k. We set I h =I hδ+I h

δc . Let us now introduce the following martingales:

mk,i ,ht =

∫ t

0

∫B h

k

zi N (d z,d t ), mk,ht = N (B h

k , ]0, t ]), (2.2.44)

and let us consider the equation

duht (x) =

(L h

t uht (x)+I hut (x)+

d∑i=0

δh,i f it (x)

)d t +

∞∑%=1

(M

h,%t uh

t (x)+ g%t (x))

d w%t

+∞∑

k=1

d∑i=1

(σ(h,k)−1∑l=0

θh,kl δh,i uh

t (x + zr h,kl

))dmk,i ,h

t +∞∑

k=1

(uh

t (x +hzk )− uht (x)

)dmk,h

t ,

(2.2.45)

with initial condition

uh0 (x) =ψ(x), (2.2.46)

for (t , x) ∈ [0,T ]×Gh .

Remark 2.2.1. It is known that for n > d/2, H n is embedded into Cb(Rd ), the space

of continuous bounded functions on Rd . Whenever a function is in H n and n > d/2,

we always take its continuous version. Moreover , for |h| < 1, there exists a constant

N depending only on d and n such that for all φ ∈ H n ,

|φ|`2(Gh ) ≤ N |φ|2H n . (2.2.47)

For a proof of this embedding we refer to [27]. By (2.2.47), under Assumptions 2.2.1

(i) and 2.2.2 with m > d/2, (2.2.45) is an Itô equation in `2(Gh)− with Lipschitz con-

tinuous coefficients, and therefore, it has unique càdlàg `2(Gh)−valued solution.

Theorem 2.2.2. Let Assumptions 2.2.1 and 2.2.2 hold with m > 2+d/2, and let u and

uh be the solutions of (2.2.35) and (2.2.45), respectively. Then the following estimate

44

holds

E supt≤T

supx∈Gh

|ut (x)− uht (x)|2 +E sup

t≤T|ut − uh |`2(Gh ) ≤ N |h|2, (2.2.48)

where N is a constant depending only d ,m,Å,Km ,κ2m , and ν.

2.2.2 Estimates for the approximation of the operators

In this section, we present some results that will be needed for the proof of Theorem

2.2.2. Let us introduce the operators

J hδ (z)φ(x) :=

∞∑k=0

IB hk

(z)σ(h,k)−1∑

l=0

d∑i=1

θh,kl ziδh,iφ(x +hzr h,k

l), (2.2.49)

J hδc (z)φ(x) :=

∞∑k=0

IB hk

(z)[φ(x +hzk )−φ(x)], (2.2.50)

J h(z)φ(x) :=J hδ (z)φ(x)+J h

δc (z)φ(x). (2.2.51)

where θk,hl := θh,k

l+1 −θh,kl , and consider the following Itô equation in L2(Rd ),

duht (x) = (L h

t uht (x)+I huh

t (x)+ ft (x))d t + (M h,%t uh

t (x)+ g%t )d w%t

+∫Rd

J h(z)uht (x)N (d z,d t ). (2.2.52)

We will show later that if uh is a solution of (2.2.52) and uh ∈ H m with m > d/2, then

it can be restricted to the grid Gh , and it agrees with uh .

The following is very well known and a proof for a more general statement can

be found in [27].

Lemma 2.2.3. Let m ≥ 0 be an integer. Then for all u ∈ H m+2, v ∈ H m+3 and i , j ∈1, ..,d we have

|δh,i u −∂i u|H m ≤ 1

2|h||u|H m+2 ,

|δh,iδ−h, j u −∂i j u|H m ≤ N |h||u|H m+3 ,

where N depends only on d and n.

Lemma 2.2.4. Let m ≥ 0 be an integer. For all u ∈ H m+3 we have

|I u −I hu|H m ≤ N |h||u|H m+3 , (2.2.53)

where the constant N depends only on d, n and ν.

Proof. It suffices to show (2.2.53) for u ∈ C∞c (Rd ). We begin with m = 0. A simple

45

calculation shows that

Iδc u(x)−I hδc u(x)

=∞∑

k=0

∫B h

k

(u(x + z)−u(x +hzk )−

d∑i=1

zi (∂i u(x)−δhi u(x))

)µ(d z)

=∞∑

k=0

∫B h

k

∫ 1

0

d∑i=1

(zi −hzik )∂i u(x +hzk +θ(z −hzk ))dθµ(d z)

−∞∑

k=0

∫B h

k

d∑i=1

zi (∂i u(x)−δhi u(x))µ(d z).

By Minkowski’s inequality, we get

|Iδc u −I hδc u|L2 ≤

∞∑k=0

∫B h

k

d∑i=1

|zi −hzik ||∂i u|L2µ(d z)

+∞∑

k=0

∫B h

k

d∑i=1

|zi ||∂i u(x)−δhi u(x)|L2µ(d z)

≤ N |h||u|H 3 +Nd∑

i=1|∂i u(x)−δh

i u(x)|L2 ,

since |z −hzk | ≤ |h|pd/2. Thus, by Lemma 2.2.3, we have

|Iδc u −I hδc u|L2 ≤ N |h||u|H 3 . (2.2.54)

We also have

Iδu(x)−I hδ u(x) =

∞∑k=0

∫B h

k

σ(h,k)−1∑l=0

∫ θh,kl+1

θh,kl

d∑i , j=1

zi z j [∂i∂ j u(x +θz)

−δh,iδ−h, j u(x +hzr h,kl

)](1−θ)dθµ(d z). (2.2.55)

Notice that for each i , j ∈ 1, ...d,

∂i∂ j u(x +θz)−δh,iδ−h, j u(x +hzr h,kl

)

=∂i∂ j u(x +θz)−∂i∂ j u(x +hzr h,kl

)

+∂i∂ j u(x +hzr h,kl

)−δh,iδ−h, j u(x +hzr h,kl

)

=∫ 1

0

d∑q=1

(θzq −hzq

r h,kl

)∂q∂i∂ j u(x +hzr h,k

l+ρ(θz −hzr h,k

l))

dρ

+∂i j u(x +hzr h,kl

)−δh,iδ−h, j u(x +hzr h,kl

).

46

and also by 2.2.42, we have |θzq −hzq

r h,kl

| ≤ N |h|. Hence, substituting the above rela-

tion in (2.2.55), using Minkowski’s inequality, (2.2.38), and Lemma 2.2.3, we obtain

|Iδu −I hδ u|L2 ≤ |h|N |u|H 3 . (2.2.56)

Combining (2.2.54) and (2.2.56), we have (2.2.53) for m = 0. The case m > 0 follows

from the case m = 0, since for a multi-index α, we have

∂α(I u −I hu) =I∂αu −I h∂αu.

Lemma 2.2.5. Let m ≥ 0 be an integer. For all u ∈ H m+2 we have∫Rd

|J h(z)u −J (z)u|2H mν(d z) ≤ N |h|2|u|2H m+2 , (2.2.57)

where the constant N depends only on n,d and ν.

Proof. It suffices to prove the lemma for u ∈C∞c (Rd ) and m = 0. We have

Jδ(z)u(x)−J hδ (z)u(x) =

∞∑k=0

IB hk

(z)σ(h,k)−1∑

l=0

∫ θh,kl+1

θh,kl

d∑i=1

zi (∂i u(x +θz)−δh,i u(x +hzr h,kl

))dθ.

Notice that

∂i u(x +θz)−δh,i u(x +hzr h,kl

)

=∫ 1

0

d∑j=1

∂i∂ j u(x +θz +ρ(θz −hzr h,kl

))(θz j −hz j

r h,kl

)dρ

+∂i u(x +hzr h,kl

)−δh,i u(x +hzr h,kl

).

Thus, by Remark 2.2.42 and Lemma 2.2.3, we get

|J hδ (z)u −Jδ(z)u|2L2

≤ I|z|≤δ|z|2N |h|2|u|2L2,

and hence by (2.2.38), we obtain∫Rd

|J hδ (z)u −Jδ(z)u|2L2

ν(d z) ≤ N |h|2|u|2L2. (2.2.58)

47

We also have

|Jδc (z)u(x)−J hδc (z)u(x)| =

∞∑k=0

IB hk

(z)|u(x + z)−u(x +hzk )|

≤∞∑

k=0IB h

k(z)

∫ 1

0

d∑i=1

|∂i u(x +hzk +ρ(z −hzk ))|zi −hzik |dρ.

Consequently,

|J hδc (z)u −Jδc (z)u|2L2

≤ I|z|>δN |h|2|u|2H 1 ,

which implies that ∫Rd

|J hδc (z)u −Jδc (z)u|2L2

ν(d z) ≤ N |h|2|u|2H 1 . (2.2.59)

Combining (2.2.59) and (2.2.58), we have (2.2.57) for m = 0. The case m > 0 follows

from the case m = 0, since for a multi-index α, we have

∂α(Ju −J hu) = J∂αu −J h∂αu.

Lemma 2.2.6. Let Assumption 2.2.1 hold. Then for any φ ∈ L2(Rd ) we have

2(φ,L ht φ)+

∞∑%=1

|M h,%t φ|2L2

≤−Å2

d∑i=1

|δh,iφ|2L2+N |φ|2L2

, (2.2.60)

Qt (φ) := 2(φ,L ht φ)+2(φ,I hφ)+

∞∑%=1

|M h,%t φ|2L2

+∫Rd

|J h(z)φ|2L2ν(d z) ≤−Å

4

d∑i=1

|δh,iφ|2L2+N |φ|2L2

, (2.2.61)

∫Rd

|J h(z)φ|2L2ν(d z) ≤ Å

12

d∑i=1

|δh,iφ|2L2+N |φ|2L2

, (2.2.62)

where N depends only on Å,K1,d , and ν.

Proof. Inequality (2.2.60) is proved in [27]. For (2.2.61) it suffices to show that

2(φ,I hφ)+∫Rd

|J h(z)φ|2L2ν(d z) ≤ Å

4

d∑i=1

|δh,iφ|2L2+N |φ|2L2

.

We have

2(φ,I hδ φ) =

∞∑k=0

∫B h

k

σ(k,h)−1∑l=1

θh,kl zi z j

∫Rdδh,iδ−h, jφ(x +hzr h,k

l)φ(x)d xµ(d z).

48

Since ∫Rdδh,iδ−h, jφ(x +hzr h,k

l)φ(x)d x =

∫Rdδh,iφ(x +hzr h,k

l)δh, jφ(x)d x,

we get by Hölder’s inequality

2(φ,I hδ φ) ≤

∫|z|≤δ

|z|2d∑

i=1|δh,iφ|2L2

µ(d z) ≤ Å

12

d∑i=1

|δh,iφ|2L2. (2.2.63)

We calculate

2(φ,I hδcφ) = 2

∞∑k=0

∫B h

k

∫Rd

[φ(x +hzk )−φ(x)− ziδh,iφ(x)]φ(x)d xµ(d z)

≤ 2∫|z|≥δ

zi |δh,iφ|L2 |φ|L2µ(d z) ≤ 2∫|z|≥δ

|z|µ(d z)( d∑

i=1|δh,iφ|2L2

)1/2|φ|L2

≤ Å

12

d∑i=1

|δh,iφ|2L2+N |φ|2L2

, (2.2.64)

where the last follows from Young’s inequality. We also have

|J hδ (z)φ|2L2

=∫Rd

( ∞∑k=0

IB hk

(z)σ(k,h)−1∑

l=0θh,k

l ziδh,iφ(x + zr h,kl

))2

d x

≤∞∑

k=0IB h

k(z)|z|2

d∑i=1

|δh,iφ|2L2,

and

|J hδc (z)φ|2L2

≤ 4∞∑

k=0IB h

k(z)|φ|2L2

.

Hence, ∫Rd

|J h(z)φ|2L2ν(d z) ≤ Å

12

d∑i=1

|δh,iφ|2L2+N |φ|2L2

,

which, proves (2.2.62) and also combined with (2.2.63) and (2.2.64) proves (2.2.61).

The proof of the following theorem goes in the same way as the proof of Theorem

3.2 from [27], but we include it for the convenience of the reader.

Theorem 2.2.7. Let f i ∈ Hm for i ∈ 0, ...,d, g ∈ Hm(l2) and r ∈ Hm(ν). Suppose

that Assumption 2.2.1 (i) holds. For any h 6= 0, there exists a unique L2(Rd )−valued

49

solution of

duht (x) = (L h

t uht (x)+I huh

t (x)+δh,i f it (x))d t + (M h,%

t uht (x)+ g%t (x))d w%

t

+∫Rd

J h(z)uht−(x)+ rt (x, z)N (d z,d t ), (2.2.65)

for any H m−valued F0−measurable initial conditionψ. This solution is H m−valued

càdlàg process. Moreover, if Assumption 2.2.1 (ii) holds, then

E supt≤T

|uht |2H m +E

∫ T

0

d∑i=1

|δh,i uht |2H m ≤ N E |ψ|2H m

+N E∫ T

0

( d∑i=0

| f it |2H m +|g t |2H m (`2) +

∫Rd

|rt (z)|2H mν(d z))d t , (2.2.66)

where N depends only on d ,m,Å,Km ,T and ν.

Proof. Note that (2.2.65) is an SDE in L2(Rd ) with Lipschitz continuous coefficients,

and consequently there exists a unique solution. Similarly, it has a unique solution

with values in H m , and since H m ⊂ L2(Rd ), the first assertion of the theorem follows.

It is also easy to see that estimate (2.2.66) holds with a constant depending on h. By

Ito’s formula we have

d |uht |2L2

= Qt (uht )+|g t |2L2(`2) +

∫Rd

|rt (z)|2L2ν(d z)

+2(δh,i uht , f i

t )+2(bi%t δh,i uh

t , g%t )+2∫Rd

(J ht (z)uh

t ,rt (z))ν(d z)d t

+2(uht ,bi%δh,i uh

t + g%t )d w%t +2

∫Rd

(J h(z)uht−+ rt (z),uh

t−)N (d z,d t )

+∫Rd

|J h(z)uht−+ rt (z)|2L2

N (d z,d t ) (2.2.67)

Using Young’s inequality, (2.2.61) and (2.2.62) we obtain

E |uht |2L2

+ Å

8E

∫ t

0

∑i 6=0

|δh,i uhs |2L2

d s ≤ E |ψ|2L2

+N E∫ T

0

(|uh

s |2L2+

d∑i=0

| f it |2L2

+|g t |2L2(`2) +∫Rd

|rt (z)|2L2ν(d z)

)d t <∞. (2.2.68)

50

By Gronwall’s lemma we get

E∫ t

0

d∑i=0

|δh,i uhs |2L2

d s ≤ N E |ψ|2L2

+N E∫ T

0

( d∑i=0

| f it |2L2

+|g t |2L2(`2) +∫Rd

|rt (z)|2L2ν(d z)

)d t . (2.2.69)

Going back to (2.2.67) and applying Davis’ inequality (Chapter IV, Theorem 4.1 in

[56]) where appropriate, we get

E supt≤T

|ut |2L2≤ N ′A1 +N ′A2 +N E |ψ|2L2

+N E∫ T

0

( d∑i=0

| f it |2L2

+|g t |2L2(`2) +∫Rd

|rt (z)|2L2ν(d z)

)d t

+2∫ T

0

∫Rd

|J h(z)uht + rt (z)|2L2

ν(d z)d t

where

A1 = E(∫ T

0

∞∑%=1

(uht ,bi%δh,i uh

t + g%t )2d t)1/2

≤ (4N ′)−1E supt≤T

|uht |2L2

+N E∫ T

0(

d∑i=0

|δh,i uht |2L2

+|g t |2L2(`2))d t ,

A2 = E(∫ T

0

∫Rd

(J h(z)uht + rt (z),uh

t )2ν(d z)d t)1/2

≤ (4N ′)−1E supt≤T

|uht |2L2

+N E∫ T

0(

d∑i=0

|δh,i uht |2L2

+∫Rd

|rt (z)|2L2ν(d z))d t .

Using this, (2.2.62) and (2.2.69) we see that

E supt≤T

|ut |2L2+E

∫ T

0

d∑i=0

|δh,i uht |2L2

d t ≤ N E |ψ|2L2

+N E∫ T

0

( d∑i=0

| f it |2L2

+|g t |2L2(`2) +∫Rd

|rt (z)|2L2ν(d z)

)d t . (2.2.70)

This proves the theorem for m = 0. Now suppose we have (2.2.66) with n instead of

m, for some n ∈ 0, ...,m −1. Let α be a multi-index, with |α| = n +1. By differenti-

ating (2.2.65) and using the notation φ= ∂αφ, we have

duht (x) = (L h

t uht (x)+I huh

t (x)+δh,i f it (x))d t + (M h,%

t uht (x)+ g%t (x))d w%

t

+∫Rd

J h(z)uht−(x)+ rt (x, z)N (d z,d t ),

51

where

f i = f i , i 6= 0, f 0 = f 0 + ∑β<α

(α

β

)∂α−βai jδh,iδ−h, j∂

βuh ,

g% = g%+ ∑β<α

(α

β

)∂α−βbi%δh,i∂

βuh .

We proceed as before, and for |β| = n and j 6= 0 we use the estimate∫Rd

|uhs δh,iδ−h, j∂

βuhs |d x ≤ γ|∂ j∂

βδh,i uhs |2L2

+γ−1|uhs |2L2

≤ γ ∑|ζ|=n+1

|δh,i∂ζuh

s |2L2+γ−1|uh

s |2L2,

where γ> 0 is arbitrary, to obtain

E |uht |2L2

+E∫ t

0

d∑i=0

|δh,i uhs |2L2

d s ≤ N E |ψ|2L2

+(2d n)−1E∫ t

0

∑|ζ|=n+1i∈0,..,d

|δh,i∂ζuh

s |2L2d s

+N E∫ t

0

(|uh

s |2L2+

d∑i=0

| f it |2H n+1 +|g t |2H n+1(`2) +

∫Rd

|rt (z)|2H n+1ν(d z))d t . (2.2.71)

By writing the above relation for all multi-indices α, with |α| = n +1, and summing

them up, we see that the term with the factor (2d n)−1 can be dropped. After that, by

proceeding as before we obtain

E supt≤T

|uht |2L2

+E∫ T

0

d∑i=0

|δh,i uhs |2L2

d s ≤ N E |ψ|2L2

+N E∫ T

0

( d∑i=0

| f it |2H n+1 +|g t |2H n+1(`2) +

∫Rd

|rt (z)|2H n+1ν(d z))d t , (2.2.72)

which combined with the induction hypothesis, gives estimate (2.2.66) with n + 1

instead of m. This brings the proof to an end.

2.2.3 Proof of Theorem 2.2.2

Theorem 2.2.8. Let Assumptions 2.2.1 and 2.2.2 hold with m ≥ 2, and let u and uh

be the solutions of (2.2.35) and (2.2.52), respectively, with initial condition ψ. Then

E supt≤T

|ut −uht |2H m−2 +E

∫ T

0

d∑i=1

|δh,i ut −δh,i uht |2H m−2 d s ≤ N |h|2κ2

m , (2.2.73)

52

where N is a constant depending only on d ,m,Å,Km , and ν.

Proof. Set r h := uht −ut and set

F ht := (L h

t −Lt +I h −I )ut +d∑

i=1(δh,i −∂i ) f i

t ,

Gh,%t :=

(M

h,%t −M%

t

)ut ,

H ht (z) :=

(J h(z)−J (z)

)ut−.

By virtue of Assumption 2.2.1 (i), Lemmas 2.2.3, 2.2.4, 2.2.5, and estimate (2.2.40),

we have

E∫ T

0|Ft |2H m−2 d t ≤ N |h|2E

∫ T

0

(|ut |2H m+1 +

d∑i=1

| f it |H m

)d t ≤ N |h|2κ2

m

and

E∫ T

0

(|Gh

t |`2(H m−2) +∫Rd

|H ht (z)|2H m−2ν(d z)

)d t

≤ N |h|2E∫ T

0|ut |2H m d t ≤ N |h|2κ2

m ,

where N is a constant depending only on d ,m,Km , and ν. Therefore, F h ∈ Hm−2,

Gh ∈ Hm−2(`2), and H h ∈ Hm−2(ν). Note that r h is a H m−2-valued strongly càdlàg

process, satisfying the equation

dr ht =

((L h

t +I h)r ht +F h

t

)d t +

(M

h,%t r h

t +Gh,%t

)d w%

t

+∫Rd

(J h(z)r h

t−+H ht (z)

)N (d z,d t ), t ∈ (0,T ], (2.2.74)

with r h0 = 0. Hence, by Theorem 2.2.7,

E supt≤T

|r ht |2H m−2 +E

∫ T

0

d∑i=1

|δh,i r ht |2H m−2 d s ≤ N |h|2κ2

m , (2.2.75)

where N is a constant depending only on d ,m,Å,Km , and ν.

By virtue of Sobolev’s embedding theorem and (2.2.47), as in [27], we obtain

the following direct corollary of Theorem 2.2.8, with the convention that 1, ...,d0 =0, ...,0.

Corollary 2.2.9. Suppose the assumptions of Theorem 2.2.8 hold with m > n + 2+d/2, where n ≥ 0 is an integer. Then for all λ = (λ1, . . . ,λn) ∈ 1, . . . ,dn and δh,λ =

53

δh,λ1 · · ·δh,λn , we have

E supt≤T

supx∈Rd

|δh,λut (x)−δh,λuht (x)|2

+E supt≤T

|δh,λut −δh,λuht |2`2(Gh ) ≤ N |h|2κ2

m ,

where N is a constant depending only on d ,m,Å,Km ,ν and n .

Proof of Theorem 2.2.2. Let uh be the unique solution of (2.2.52) with initial condi-

tion ψ. By virtue of Theorems 2.2.1 and 2.2.7, u, uh are strongly càdlàg H m-valued

process. Notice that u, uh are càdlàg processes with values in Cb(Rd ). We only need

to show that almost surely uht (x) = uh

t (x), for all t ∈ [0,T ] and x ∈Gh , and then Theo-

rem 2.2.2 follows from Corollary 2.2.9. To this end, let I be the embedding operator

form H m to `2(Gh). By applying J to both sides of (2.2.65) we see that Juh satis-

fies (2.2.45) and therefore the result follows from the uniqueness of the solution of

(2.2.45).

54

Chapter 3

Stochastic Partial differential

equations

In this chapter we turn our attention to Stochastic PDEs. The equation under con-

sideration is

dut = (Lt ut +∂i f it + f 0

t )d t + (M kt ut + g k

t )d w kt , u0 =ψ, (3.0.1)

for (t , x) ∈ [0,T ]×Q, where the operators Lt , and M kt are given by

Lt u = ∂ j (ai jt ∂i u)+bi

t∂i u + ct u, M kt u =σi k

t ∂i ut +µkt u. (3.0.2)

Here, Q is a bounded Lipschitz domain in Rd and he summation for the parameters

i and j takes place over the set 1, ...,d, and for k over the positive integers.

Solvability of these type of equations in Sobolev spaces has been extensively

studied, mainly through a variational approach (L2-theory, see for example [43],

[57], and [54]) and an analytic approach (Lp -theory, see [40], [37]). By the results

in these two approaches, one can obtain regularity (integrability or/and differentia-

bility) of the solution under the assumption that the data of the equation are regular

enough. In this chapter we deal with another question. Under the minimal condi-

tions that guarantee the existence of a solution, can we derive some further infor-

mation for the solution? As we will see in the next sections, the answer is affirmative.

The results of this chapter are from [6] and [5], two joint works with Máté Gerenc-

sér.

In order to ease the notation followed in the previous chapter, for p,r, q ∈ [1,∞],

55

let us set

Lr,q := Lr ([0,T ];Lq (Q))

Lp := Lp (Ω,F0;Lp (Q))

Lp := Lp (Ω× [0,T ],P ;Lp (Q))

Lp (l2) := Lp (Ω× [0,T ],P ;Lp (Q; l2))

and also

| · |p = | · |Lp (Q), ‖ ·‖r,q := | · |Lr,q , ‖ ·‖r := ‖·‖r,r .

The assumptions posed are the following.

Assumption 3.0.3. i) The coefficients ai j , bi and c are real-valued P ×B(Q) mea-

surable functions on Ω× [0,T ]×Q and are bounded by a constant K ≥ 0, for any

i , j = 1, ...,d . The coefficients σi = (σi k )∞k=1 and µ = (µk )∞k=1 are l2-valued P ×Q-

measurable functions on Ω× [0,T ]×Q such that

∑i

∑k|σi k

t (x)|2 +∑k|µk

t (x)|2 ≤ K for all ω, t and x,

ii) f l , for l ∈ 0, ...,d, and g = (g k )∞k=1 are P ×B(Q)-measurable functions on Ω×[0,T ]×Q with values in R and l2, respectively, such that

E(d∑

l=0‖ f l‖2

2 +‖|g |l2‖22) <∞

iii) ψ is an F0-measurable random variable in L2(Q) such that E |ψ|22 <∞

Assumption 3.0.4 (Parabolicity). There exists a constantλ> 0 such that for allω, t , x

and for all ξ= (ξ1, ...ξd ) ∈Rd we have

ai jt (x)ξiξ j − 1

2σi k

t (x)σ j kt (x)ξiξ j ≥λ|ξ|2,

We will refer to the constants K ,T,λ,d and |Q|, where the latter is the Lebesgue

measure of Q, as structure constants.

The solution of equation (3.0.1) is again understood to be an L2(Q)-valued, Ft−adapted,

strongly continuous process (ut )t∈[0,T ], such that

i) ut ∈ H 10 (Q), for dP ×d t almost every (ω, t ) ∈Ω× [0,T ]

ii) E∫ T

0 (|ut |22 +|∇ut |22)d t <∞

56

iii) for all φ ∈C∞c (Q) we have with probability one

(ut ,φ) = (ψ,φ)+∫ t

0(ai j

s ∂i us + f js ,∂− jφ)+ (bi

s∂i us + csus + f 0s ,φ)d s

+∫ t

0(M k

s us + g ks ,φ)d w k

s ,

for all t ∈ [0,T ].

3.1 Global Boundedness

We are interested in boundedness properties of solutions of (3.0.1) under Assump-

tions 3.0.3 and 3.0.4. The corresponding problem in the deterministic case, has

been extensively studied. The first results for non-degenerate equations in diver-

gence form are due to [18] and [51] for the elliptic case and [53] for both elliptic and

parabolic equations. Later, the techniques of [51] were extended to the parabolic

case in [52]. The approach of [18] was also applied for parabolic equations (see for

example [48]). In these articles, boundedness is obtained as an intermediate step

in order to prove Hölder continuity and Harnack inequalities. Another proof of the

parabolic Harnack inequality was given in [16]. Hölder estimates and Harnack in-

equality were also obtained in [59] and [47], for elliptic and parabolic equations in

non-divergence form. More recently, these results were also proved for a wider class

of parabolic equations, including, for example, the p-Laplacian as the driving oper-

ator (see [14] and references therein).

Boundedness of solutions of SPDEs can be proved through embedding theo-

rems of Sobolev spaces. Such results can be obtained from Lp−theory, see e.g.

[40], for equations considered on the whole space. This approach, however, re-

quires some regularity of the coefficients. For SPDEs where these regularity as-

sumptions are dropped or weakened, the literature has been expanding recently.

In [55] a maximum principle is obtained for a class of backward SPDEs. Under

the additional assumption σ = 0, variants of the problem are treated in [9], [33],

and [36], with methods that strongly rely on the absence of derivatives of u in the

noise term. In [10], through the technique of Moser’s iteration, introduced in [51],

boundedness results are derived without posing regularity assumptions on the co-

efficients, for a class of quasilinear equations, by staying in the L2−framework. This

served as a main motivation to our work. However, in [10], it is assumed that there

exist constants λ > β > 0, such that for any ξ ∈ Rd , one has ai jξiξ j ≥ λ|ξ|2 and

(72+ 1/2)σi kσ j kξiξ j ≤ β|ξ|2, which seems a rather strong condition. In the work

presented here, only the classical stochastic parabolicity condition will be assumed

57

in order to get estimates for the uniform bound of the solution of equation (3.0.1).

We note that the results of this section can also be extended to quasilinear equa-

tions under suitable conditions. Having accessibility in mind, such generalizations

are not included here.

3.1.1 The global L∞-estimates

Notice that under Assumptions 3.0.3 and 3.0.4, by virtue of Theorem 2.1.1 for exam-

ple, equation (3.0.1) admits a unique solution u, and the following estimate holds

E sup0≤t≤T

|ut |22 ≤ N E(|ψ|22 +d∑

l=0‖ f l‖2

2 +‖|g |l2‖22), (3.1.3)

where N = N (d ,K ,λ,T ).

Let

Γd =

(r, q) ∈ (1,∞]2∣∣∣∣1

r+ d

2q< 1

.

The following is our main result.

Theorem 3.1.1. Suppose that Assumptions 3.0.3 and 3.0.4 hold, and let u be the

unique solution of equation (3.0.1). Then for any (r, q) ∈ Γd and η> 0,

E‖u‖η∞ ≤ N E(|ψ|η∞+‖ f 0‖ηr,q +d∑

i=1‖ f i‖η2r,2q +‖|g |l2‖η2r,2q ), (3.1.4)

where N = N (η,r, q,d ,K ,λ, |Q|,T ).

Remark 3.1.1. Notice that in particular we obtain

E‖u‖2∞ ≤ N E(|ψ|2∞+

d∑l=0

‖ f l‖2∞+‖|g |l2‖2

∞), (3.1.5)

and by interpolating between (3.1.3) and (3.1.5), for any p ≥ 2, one obtains

E sup0≤t≤T

|ut |2p ≤ N E(|ψ|2p +d∑

l=0‖ f l‖2

p +‖|g |l2‖2p )

where N can be chosen to be independent of p. In fact, such a uniform estimate for

the Lp -norms of the solutions is equivalent to (3.1.5).

Theorem 3.1.1 will be proved in Section 3.1.3. We will adapt the technique of

Moser from [51] and [52]. The strategy, in short, and for the moment ignoring the

contributions from the initial and free data, is the following: with a suitable in-

termediate norm [u]n we obtain estimates of the form E‖u‖ηrn+1,qn+1≤ N (n)E [u]ηn ,

58

E [u]ηn ≤ N (n)E‖u‖ηrn ,qn, with rn , qn ∞. The constants N (n) in these estimates

are controlled so that one can iterate this procedure, take limits, and finally obtain

estimates for the supremum norm.

3.1.2 Itô’s formula for the Lp-norm and energy-type estimates

In this section we gather some results that we will need for the proof of Theorem

3.1.1. First let us invoke (II.3.4) from [48].

Lemma 3.1.2. Suppose that v ∈ L2([0,T ], H 10 (Q))∩L∞([0,T ],L2(Q)). Let r, q ∈ (2,∞),

satisfying 1/r +d/2q = d/4. Then v belongs to Lr ([0,T ],Lq (Q)), and

(∫ T

0

(∫Q|vt |q d x

)r /q

d t

)2/r

≤ N

(sup

0≤t≤T

∫Q|vt |2d x +

∫ T

0

∫Q|∇vt |2d xd t

)

with N = N (d , |Q|,T ).

The right hand side of the inequality in the above lemma plays the role of the

“suitable norm" (for n = 2), which was discussed at the end of the previous section.

We are also going to use the following result (see Proposition IV.4.7 and Exercise

IV.4.31/1, [56]).

Proposition 3.1.3. Let X be a non-negative, adapted, right-continuous process, and

let A be a non-decreasing, continuous process such that

E(Xτ|F0) ≤ E(Aτ|F0)

for any bounded stopping time τ. Then for any σ ∈ (0,1)

E supt≤T

X σt ≤σ−σ(1−σ)−1E Aσ

T .

In order to obtain our estimates, we will need an Itô formula for |ut |pp . The differ-

ence between the next lemma and Lemma 8 in [10], is that we obtain supremum (in

time) estimates, that are essential for having (3.1.7) almost surely, for all t ∈ [0,T ].

Therefore, we give a whole proof for the sake of completeness.

Lemma 3.1.4. Suppose that u satisfies equation (3.0.1), f l ∈ Lp , for l ∈ 0, ...,d, g ∈Lp (l2), andψ ∈ Lp for some p ≥ 2. Then there exists a constant N = N (d ,K ,λ, p), such

that

E supt≤T

|ut |pp +E∫ T

0

∫Q|∇us |2|us |p−2d xd s ≤ N E(|ψ|pp +

d∑l=0

‖ f l‖pp +‖|g |l2‖p

p ). (3.1.6)

59

Moreover, almost surely

∫Q|ut |p d x =

∫Q|u0|p d x +p

∫ t

0

∫Q

(σi ks ∂i us +µk us + g k )us |us |p−2d xd w k

s

+∫ t

0

∫Q−p(p −1)ai j

s ∂i us |us |p−2∂ j us −p(p −1) f is ∂i us |us |p−2d xd s

+∫ t

0

∫Q

p(bis∂i us + csus + f 0

s )us |us |p−2d xd s

+ 1

2p(p −1)

∫ t

0

∫Q

∞∑k=1

|σi ks ∂i us +µk us + g k

s |2|us |p−2d xd s, (3.1.7)

for any t ≤ T .

Proof. Consider the functions

φn(r ) =

|r |p if |r | < n

np−2 p(p−1)2 (|r |−n)2 +pnp−1(|r |−n)+np if |r | ≥ n.

Then one can see that φn are twice continuously differentiable, and satisfy

|φn(x)| ≤ N |x|2, |φ′n(x)| ≤ N |x|, |φ′′

n(x)| ≤ N ,

where N depends only on p and n ∈ N. We also have that for any r ∈ R, φn(r ) →|r |p , φ′

n(r ) → p|r |p−2r , φ′′n(r ) → p(p −1)|r |p−2, as n →∞, and

φn(r ) ≤ N |r |p , φ′n(r ) ≤ N |r |p−1, φ′′

n(r ) ≤ N |r |p−2, (3.1.8)

where N depends only on p. Then for each n ∈Nwe have almost surely

∫Qφn(ut )d x =

∫Qφn(u0)d x +

∫ t

0

∫Q

(σi ks ∂i us +µk us + g k )φ′

n(us)d xd w ks

+∫ t

0

∫Q−ai j

s ∂i usφ′′n(us)∂ j us − f iφ′′

n(us)∂i usd xd s

+∫ t

0

∫Q

bis∂i usφ

′n(us)+ csusφ

′n(us)+ f 0

s φ′n(us)d xd s

+ 1

2

∫ t

0

∫Q

∞∑k=1

|σi ks ∂i us +µk us + g k

s |2φ′′n(us)d xd s, (3.1.9)

for any t ∈ [0,T ] (see for example, Section 3 in [42]). By Young’s inequality, and the

60

parabolicity condition we have for any ε> 0,∫Qφn(ut )d x ≤ m(n)

t +∫

Qφn(u0)d x

+∫ t

0

∫Q

(−λ|∇us |2 +ε|∇us |2 +Nd∑

i=1| f i

s |2)φ′′n(us)d xd s

+∫ t

0

∫Q

(ε|∇us |2 +N |us |2 +N∞∑

k=1|g k

s |2)φ′′n(us)d xd s

+∫ t

0

∫Q

(bis∂i us + csus + f 0

s )φ′n(us)d xd s, (3.1.10)

where N = N (d ,K ,ε), and m(n)t is the martingale from (3.1.9). One can check that

the following inequalities hold,

i) |rφ′n(r )| ≤ pφn(r )

ii) |r 2φ′′(r )| ≤ p(p −1)φn(r )

iii) |φ′n(r )|2 ≤ 4p φ′′

n(r )φn(r )

iv) [φ′′n(r )]p/(p−2) ≤ [p(p −1)]p/(p−2)φn(r ),

which combined with Young’s inequality imply,

i) ∂i usφ′n(us) ≤ εφ′′

n(us)|∂i us |2 +Nφn(us)

ii) |usφ′n(us)| ≤ pφn(us)

iii) | f 0s φ

′n(us)| ≤ | f 0

s ||φ′′n(us)|1/2|φn(us)|1/2 ≤ N | f 0

s |p +Nφn(us)

iv) |us |2φ′′n(us) ≤ Nφn(us)

v)∑

k |g ks |2φ′′

n(us) ≤ Nφn(us)+N(∑

k |g ks |2

)p/2

vi)∑d

i=1 | f is |2φ′′

n(us) ≤ Nφn(us)+N∑d

i=1 | f is |p ,

where N depends only on p and ε.

By choosing ε sufficiently small, and taking expectations we obtain

E∫

Qφn(ut )d x +E I A

∫ t

0

∫Q|∇us |2φ′′

n(us)d xd s ≤ N EKt +N∫ t

0E

∫Qφn(us)d xd s,

where N = N (d , p,K ,λ) and

Kt = |ψ|pp +∫ t

0

d∑l=0

| f ls |pp +|gs |pp d s.

61

By Gronwall’s lemma we get

E∫

Qφn(ut )d x +E

∫ t

0

∫Q|∇us |2φ′′

n(us)d xd s ≤ N EKt

for any t ∈ [0,T ], with N = N (T,d , p,K ,λ). Going back to (3.1.10), using the same

estimates, and the above relation, by taking suprema up to T we have

E supt≤T

∫Qφn(ut )d x ≤ N E I AKt +E sup

t≤T|m(n)

t |.

≤ N EKT +N E

(∫ T

0

∑k

(∫Q|σi k∂i us +µk us + g k

s ||φ′′n(us)φn(us)|1/2d x

)2

d s

)1/2

≤ N EKT +N E

(∫ T

0

∫Q

(|∇us |2 +|us |2 +∞∑

k=1|g k

s |2)φ′′n(us)d x

∫Qφn(us)d xd s

)1/2

≤ N EKT + 1

2E sup

t≤T

∫Qφn(ut )d x <∞,

where N = N (T,d , p,K ,λ). Hence,

E supt≤T

∫Qφn(ut )d x +E

∫ T

0

∫Q|∇us |2φ′′

n(us)d xd s ≤ N EKT ,

and by Fatou’s lemma we get (3.1.6). For (3.1.7), we go back to (3.1.9), and by letting

a subsequence n(k) → ∞ and using the dominated convergence theorem, we see

that each term converges to the corresponding one in (3.1.7) almost surely, for all

t ≤ T . This finishes the proof.

Corollary 3.1.5. Let γ > 1 and denote κ = 4γ/(γ− 1). Suppose furthermore that

r,r ′, q, q ′ ∈ (1,∞), satisfying 1/r +2/r ′ = 1 and 1/q+2/q ′ = 1. Suppose that u satisfies

the conditions of Lemma 3.1.4 for any p ∈ 2γn ,n ∈N. Then, for any p ∈ 2γn ,n ∈N,

almost surely, for all t ≤ T

∫Q|ut |p d x + p2

4

∫ t

0

∫Q|∇ut |2|ut |p−2d xd s ≤ N ′mt

+N

[|ψ|pp +pκ‖u‖p

r ′p/2,q ′p/2 +p−p (‖ f 0‖pr,q +

d∑i=1

‖ f i‖p2r,2q +‖|g |l2‖p

2r,2q )

], (3.1.11)

where mt is the martingale from (3.1.7), and N , N ′ are constants depending only on

K ,d ,T,λ, |Q|,r, q.

Proof. By Lemma 3.1.4, the parabolicity condition, and Young’s inequality we have

62

∫Q|ut |p d x + p2

4

∫ t

0

∫Q|∇us |2|us |p−2d xd s ≤ N ′mt +N1

(∫Q|ψ|p d x

+∫ t

0

[∫Q

p2|us |p +p| f 0s ||us |p−1 +p2

d∑i=1

| f is |2|us |p−2 +p2|gs |2l2

|us |p−2d x

]d s

).

Then by Hölder’s inequality we have∫ t

0

∫Q| f 0

s ||us |p−1d xd s ≤ ‖ f 0‖r,q‖u‖p−1q ′(p−1)/2,r ′(p−1)/2,

and by Young’s inequality we obtain

p‖ f 0‖r,q‖u‖p−1q ′(p−1)/2,r ′(p−1)/2 ≤ p−p‖ f 0‖p

r,q +pκ‖u‖pr ′(p−1)/2,q ′(p−1)/2

≤ p−p‖ f 0‖pr,q +N2pκ‖u‖p

r ′p/2,q ′p/2.

Similarly, for n ≥ 1,

p2∫ t

0

∫Q| f i

s |2|us |p−2d xd s ≤ p2‖ f i‖22r,2q‖u‖p−2

r ′(p−2)/2,q ′(p−2)/2

≤ p−p‖ f i‖p2r,2q +pκ‖u‖p

r ′(p−2)/2,q ′(p−2)/2

≤ p−p‖ f i‖p2r,2q +N3pκ‖u‖p

r ′p/2,q ′p/2.

The same holds for g in place of f i . The case n = 0 can be covered separately with

another constant N4, and then N can be chosen to be maxN1(N2 +N3), N4. This

finishes the proof.

Lemma 3.1.6. Suppose that u satisfies equation (3.0.1), f l ∈ Lp , for l ∈ 0, ...,d, g ∈Lp (l2), and ψ ∈ Lp for some p ≥ 2. Then for any 0 < η< p, and for any ε> 0,

E

(supt≤T

|ut |pp + p2

4E

∫ T

0

∫Q|∇us |2|us |p−2d xd s

)η/p

≤ εE‖u‖η∞+N (ε, p)E

[|ψ|ηp +‖ f 0‖η1 +

d∑i=1

‖ f i‖η2 +‖|g |l2‖η2]

where N (ε, p) is a constant depending only on ε,η,K ,d ,T,λ, |Q|, and p.

Proof. As in the proof of corollary 3.1.5, for any F0−measurable set B , we have al-

most surely

IB

∫Q|ut |p d x + p2

4IB

∫ t

0

∫Q|∇us |2|us |p−2d xd s ≤ N ′IB mt +N1IB

(∫Q|ψ|p d x

63

+∫ t

0

[∫Q

p2|us |p +p| f 0s ||us |p−1 +p2

d∑i=1

| f is |2|us |p−2 +p2|gs |2l2

|us |p−2d x

]d s

),

(3.1.12)

for any t ∈ [0,T ]. The above relation, by virtue of Gronwal’s lemma implies that for

any stopping time τ≤ T

supt≤T

E IB

∫Q|ut∧τ|p d x +E IB

∫ τ

0

∫Q|∇us |2|us |p−2d xd s ≤ N E IBVτ, (3.1.13)

where

Vt :=∫

Q|ψ|p d x +

∫ t

0

∫Q| f 0

s ||us |p−1 +d∑

i=1| f i

s |2|us |p−2 +|gs |2l2|us |p−2d xd s.

Going back to (3.1.12), and taking suprema up to τ and expectations, and having in

mind (3.1.13), gives

E supt≤τ

IB

∫Q|ut |p d x ≤ N E sup

t≤τIB |mt |+N E IBVτ.

By the Burkholder-Gundy-Davis inequality and (3.1.13) we have

E supt≤τ

IB |mt | ≤ N E IB

(∫ τ

0

(∫Q|ut |p−2 (|∇ut |+ |ut |+ |g |l2

)d x

)2

d t

)1/2

≤ N E IB

(∫ τ

0

∫Q|ut |p d x

∫Q

(|∇ut |2 +|ut |2 +|g |2l2)|u|p−2d xd t

)1/2

≤ 1

2E sup

t≤τIB

∫Q|ut |p d x +N E IBVτ.

Hence,

E supt≤τ

IB

∫Q|ut |p d x ≤ N E IBVτ,

which combined with (3.1.13), by virtue of Lemma 3.1.3 gives

E

(supt≤T

|ut |pp + p2

4E

∫ T

0

∫Q|∇us |2|us |p−2d xd s

)η/p

≤ N EVη/p

T

≤ N E

[|ψ|pp +‖u‖p−1

∞ ‖ f 0‖1 +‖u‖p−2∞

(d∑

i=1‖ f i‖2

2 +‖|g |l2‖22

)]η/p

≤ εE‖u‖η∞+N E

[|ψ|ηp +‖ f 0‖η1 +

d∑i=1

‖ f i‖η2 +‖|g |l2‖η2]

,

which brings the proof to an end.

64

3.1.3 Proof of Theorem 3.1.1

Proof. Throughout the proof, the constants N in our calculations will be allowed to

depend on η,r, q as well as on the structure constants. Notice that we may, and we

will assume that r, q <∞. Without loss of generality we assume that the right hand

side in (3.1.4) is finite. Also, in the first part of the proof we make the assumption

that ψ, f l , l = 0, . . . ,d , and g are bounded by a constant M . in particular, by (3.1.6),

u ∈ Lη(Ω,Lr,q ) for any η,r, q .

Let us introduce the notation

Mr,q,p (t ) = ‖1[0,t ] f 0‖pr,q +

d∑i=1

‖1[0,t ] f i‖p2r,2q +‖1[0,t ]|g |l2‖p

2r,2q .

Since (r, q) ∈ Γd , if we define r ′ and q ′ by 1/r +2/r ′ = 1, 1/q +2/q ′ = 1, we have

d

4< 1

r ′ +d

2q ′ =: γd

4

for some γ> 1. Then r = γr ′ and q = γq ′ satisfy

1

r+ d

2q= d

4.

By applying Lemma 3.1.2 to r , q , and v = |v |p/2, we have, for any p ≥ 2

E

|ψ|η∞∨(∫ T

0

(∫Q|vt |qp/2d x

)r /q

d t

)2η/r p

≤[

E |ψ|η∞∨Nη/p(

sup0≤t≤T

∫Q|vt |p d x + p2

4

∫ T

0

∫Q|∇vt |2|vt |p−2d xd t

)η/p]. (3.1.14)

To estimate the right-hand side above, first notice that, if p = 2γn for some n, then

by taking supremum in (3.1.11), we have for any stopping time τ≤ T , and any F0−measurable set B ,

IB sup0≤s≤τ

∫Q|vs |p d x

≤ N IB

(|ψ|p∞+pκ‖1[0,τ]v‖p

r ′p/2,q ′p/2 +p−pMr,q,p (τ))+N ′IB sup

0≤s≤τ|ms |, (3.1.15)

By the Davis inequality we can write

E IB sup0≤s≤τ

|ms | ≤ N E IB

(∫ τ

0

∑k

(∫Q

p(σi ks ∂i vs +µk vs + g k )vs |vs |p−2d x

)2

d s

)1/2

65

≤ N E IB

(sup

0≤s≤τ

∫Q|vs |p d x

)1/2(∫ τ

0

∫Q

p2∑k|σi k

s ∂i vs +µk vs + g k |2|vs |p−2d xd s

)1/2

.

Applying Young’s inequality and recalling the already seen estimates in the proof of

Corollary 3.1.5 (i) for the second term yields

E IB sup0≤s≤τ

|ms | ≤ εE IB sup0≤s≤τ

∫Q|vs |p d x

+N

εE IB

(p2

∫ τ

0

∫Q|∇vs |2|vs |p−2d xd s +pκ‖1[0,τ]v‖p

r ′p/2,q ′p/2 +p−p‖1[0,τ]|g |l2‖p2r,2q

)for any ε > 0. With the appropriate choice of ε, combining this with (3.1.15) and

using (3.1.11) once again, now without taking supremum, we get

E IB

(sup

0≤s≤τ

∫Q|vs |p d x + p2

4

∫ τ

0

∫Q|∇vs |2|vs |p−2d xd s

)

≤ N E IB

(|ψ|p∞+p2

∫ τ

0

∫Q|∇vs |2|vs |p−2d xd s +pκ‖1[0,τ]v‖p

r ′p/2,q ′p/2 +p−pMr,q,p (τ))

≤ N E IB

(|ψ|p∞+pκ‖1[0,τ]v‖p

r ′p/2,q ′p/2 +p−pMr,q,p (τ))+N ′E IB mτ,

and the last expectation vanishes. Now consider

X t = |ψ|p∞∨(

sup0≤s≤t

∫Q|vs |p d x + p2

4

∫ t

0

∫Q|∇vs |2|vs |p−2d xd s

)and

At =C pκ(|ψ|p∞∨‖1[0,t ]v‖p

r ′p/2,q ′p/2 +p−pMr,q,p (t ))

for a large enough, but fixed C . The argument above gives that

E IB Xτ ≤ E IB

(|ψ|p∞+ sup

0≤s≤τ

∫Q|vs |p d x + p2

4

∫ τ

0

∫Q|∇vs |2|vs |p−2d xd s

)≤ N E IB

(|ψ|p∞+pκ‖1[0,τ]v‖p

r ′p/2,q ′p/2 +p−pMr,q,p (τ))≤ E IB Aτ.

Therefore the condition of Proposition 3.1.3 is satisfied, and thus for η< p we obtain

E

(|ψ|p∞∨

(sup

0≤t≤T

∫Q|vt |p d x + p2

4

∫ T

0

∫Q|∇vt |2|vt |p−2d xd t

))η/p

66

≤ (N pκ+1)η/p p

p −ηE(|ψ|p∞∨‖v‖p

r ′p/2,q ′p/2 +p−pMr,q,p (T ))η/p

≤ (N pκ+1)η/p p

p −ηE(|ψ|η∞∨‖v‖ηr ′p/2,q ′p/2 +p−ηMr,q,η(T )

). (3.1.16)

Let us choose p = pn = 2γn for n ≥ 0, and use the notation cn = (N pκ+1n )η/pn pn

pn−η .

Upon combining (3.1.14) and (3.1.16), for pn > ηwe can write the following inequal-

ity, reminiscent of Moser’s iteration:

E |ψ|η∞∨‖v‖ηr ′pn+1/2,q ′pn+1/2 ≤ cnE[|ψ|η∞∨‖v‖ηr ′pn /2,q ′pn /2 +N p−η

n Mr,q,η(T )]

.

(3.1.17)

Consider the minimal n0 = n0(d ,η) such that pn0 > 2η. Taking any integer m ≥ n0

we have

m∏n=n0

cn ≤m∏

n=n0

(Nγκ+1)ηn/2γne2η/2γn

= exp

[log(Nγκ+1)

m∑n=n0

ηn

2γn+

m∑n=n0

η

γn

]≤ N0,

where N0 does not depend on m. Also,

Nm∑

n=n0

p−ηn ≤ N1,

where N1 does not depend on m. Therefore, by iterating (3.1.17) we get

liminfm→∞ E |ψ|η∞∨‖v‖ηr ′pm /2,q ′pm /2 ≤N0N1EMr,q,η(T )

+N0E |ψ|η∞∨‖v‖ηr ′(pn0+1)/2,q ′(pn0+1)/2,

and thus by Fatou’s lemma

E‖v‖η∞ ≤ N E(|ψ|η∞∨‖v‖ηr ′(pn0+1)/2,q ′(pn0+1)/2 +Mr,q,η(T )), (3.1.18)

in particular, the left-hand side is finite. By Lemma 3.1.6 we get

E

(|ψ|p∞∨

(sup

0≤t≤T

∫Q|vt |p d x + p2

4

∫ T

0

∫Q|∇vt |2|vt |p−2d xd t

))η/p

≤ εE‖v‖η∞+N (ε, p)E(|ψ|η∞+M1,1,η(T )

)(3.1.19)

for any ε> 0. Combining (3.1.14) and (3.1.19) for p = pn0 gives

E |ψ|η∞∨‖v‖ηr ′(pn0+1)/2,q ′(pn0+1)/2 = E |ψ|η∞∨‖v‖ηr pn0 /2,q ′pn0 /2

67

≤ εE‖v‖η∞+N (ε, pn0 )E(|ψ|η∞+M1,1,η(T )

). (3.1.20)

Choosing ε sufficiently small, plugging (3.1.20) into (3.1.18), and rearranging yields

the desired inequality

E‖v‖η∞ ≤ N E(|ψ|η∞+Mr,q,η(T )). (3.1.21)

As for the general case, set

ψ(n) = (−n)∨ (ψ∧n), f l ,(n) = (−n)∨ ( f l ∧n), g k,(n) = (−n/k)∨ (g k ∧ (n/k)),

define M (n)r,q,p correspondingly, and let vn be the solution of the corresponding equa-

tion. This new data is now bounded by a constant, so the previous argument applies,

and thus

E‖vn‖η∞ ≤ N E(|ψ(n)|η∞+M (n)r,q,η(T ) ≤ N E(|ψ|η∞+Mr,q,η(T )).

Since vn → v in L2, for a subsequence k(n), vk(n) → v for almost every ω, t , x. In

particular, almost surely ‖v‖∞ ≤ liminfn→∞ ‖vk(n)‖∞, and by Fatou’s lemma

E‖v‖η∞ ≤ liminfn→∞ E‖vk(n)‖η∞ ≤ N E(|ψ|η∞+Mr,q,η(T )).

3.2 Application to semilinear equations

In this section, we will use the uniform norm estimates obtained in the previous

section, to construct solutions for the following equation

dut = (Lt ut + ft (ut ))d t + (M kt ut + g k

t )d w kt ,u0 =ψ (3.2.22)

for (t , x) ∈ [0,T ]×Q, where f is a real function defined on Ω× [0,T ]×Q ×R and is

P ×B(Rd )×B(R)−measurable.

Assumption 3.2.1. The function f satisfies the following

i) for all r,r ′ ∈R and for all (ω, t , x) we have

(r − r ′)( ft (x,r )− ft (x,r ′)) ≤ K |r − r ′|2

ii) For all (ω, t , x), ft (x,r ) is continuous in r

68

iii) for all N > 0, there exists a function hN ∈ L2 with E‖hN‖∞ <∞, such that for

any (ω, t , x)

| ft (x,r )| ≤ |hNt (x)|,

whenever |r | ≤ N .

iv) E |ψ|∞+E‖|g |l2‖∞ <∞

Definition 3.2.1. An B-solution of equation (3.2.22) is an Ft−adapted, strongly

continuous process (ut )t∈[0,T ] with values in L2(Q) such that

i) ut ∈ H 10 (Q), for dP ×d t almost every (ω, t ) ∈Ω× [0,T ]

ii)∫ T

0 |ut |2H 10 (Q)

d t <∞ (a.s.)

iii) almost surely, u is essentially bounded in (t , x)

iv) for all φ ∈C∞c (Q) we have with probability one

(ut ,φ) = (ψ,φ)+∫ t

0(ai j

s ∂i us ,∂− jφ)+ (bis∂i us + csus ,φ)+ ( fs(us),φ)d s

+∫ t

0(M k

s us + g ks ,φ)d w k

s ,

for all t ∈ [0,T ].

Notice that by Assumption 3.2.1 iii), and (iii) from Definition 3.2.1, the term∫ t0 ( fs(us),φ)d s is meaningful.

Theorem 3.2.1. Under Assumptions 3.0.3, 3.0.4, and 3.2.1, there exists a unique B-

solution of equation (3.2.22).

Remark 3.2.1. From now on we can and we will assume that the function f is de-

creasing in r or else, by virtue of Assumption 3.2.1, we can replace ft (x,r ) by ft (x,r ) :=ft (x,r )−K r and ct (x) with ct (x) := ct (x)+K .

Proof of Theorem 3.2.1. We truncate the function f by setting

f n,mt (x,r ) =

ft (x,m) if r > m

ft (x,r ) if −n ≤ r ≤ m

ft (x,−n) if r <−n,

for n, m ∈Nwe consider the equation

dun,mt = (Lt un,m

t + f n,mt (un,m

t ))d t + (M kt un,m

t + g kt )d w k

t ,

un,m0 =ψ (3.2.23)

69

We first fix m ∈ N. One can easily check that under Assumptions 3.0.3, 3.0.4 and

3.2.1, by virtue of Theorem 2.1.1, equation (3.2.23) has a unique solution (in the

sense of definition 2.1.1) (un,mt )t∈[0,T ]. We also have that for n′ ≥ n, f n′,m ≥ f n,m . By

Theorem 2.1.2 we get that almost surely, for all t ∈ [0,T ]

un′,mt (x) ≥ un,m

t (x), for almost every x. (3.2.24)

We define now the stopping time

τR,m := inft ≥ 0 :∫

Q(u1,m

t +R)2− d x > 0∧T.

We claim that for each R ∈ N, there exists a set ΩR of full probability, such that for

each ω ∈ΩR , and for all n ≥ R we have that

un,mt = uR,m

t , for t ∈ [0,τR,m]. (3.2.25)

Notice that by (3.2.24) and the definition of τR,m , for all n ≥ R

f n,mt (x,un,m

t (x)) = f R,mt (x,un,m

t (x)), for t ∈ [0,τR,m].

This means that for all n ≥ R the processes un,mt satisfies

d vt = (Lt vt + f R,mt (vt ))d t + (M k

t vt + g kt )d w k

t ,

v0 =ψ, (3.2.26)

on [0,τR,m]. The uniqueness of the L2−solution of the above equation shows (3.2.25).

Notice that by Assumption 3.2.1 (iii) and (iv), Theorem 3.1.1 guarantees that u1,m is

almost surely essentially bounded in (t , x). Therefore, for almost everyω ∈Ω, τR,m =T for all R large enough. On the set Ω := ∩R∈NΩR we define u∞,m

t = limn→∞ un,mt ,

where the limit is in the sense of L2(Q). Since for each ω ∈ Ω, we have u∞,mt = un,m

t

for all t ≤ τR,m , and for any n ≥ R, it follows that the process (u∞,mt )t∈[0,T ] is an

adapted continuous L2(Q)−valued process such that

i) u∞,mt ∈ H 1

0 (Q), for dP ×d t almost every (ω, t ) ∈Ω× [0,T ]

ii)∫ T

0 |u∞,mt |2

H 10 (Q)

d t <∞(a.s.)

iii) u∞,mt is almost surely essentially bounded in (t , x)

iv) for all φ ∈C∞c (Q) we have with probability one

(u∞,mt ,φ) =

∫ t

0(ai j

s ∂i u∞,ms ,∂− jφ)+ (bi

s∂i u∞,ms + csum

s ,φ)+ ( f ms (u∞,m

s ),φ)d s

70

+∫ t

0(σi k

s ∂i u∞,ms +νk

s u∞,ms + g k

s ,φ)d w ks + (ψ,φ),

for all t ∈ [0,T ], where

f mt (x,r ) =

ft (x,m) if r > m

ft (x,r ) if r ≤ m.

Now we will let m →∞. Let us define the stopping time

τR := inft ≥ 0 :∫

Q(u∞,1

t −R)2+d x > 0∧T.

As before we claim that for any R > 0, there exists a set Ω′R of full probability, such

that for any ω ∈Ω′R and any m,m′ ≥ R,

u∞,m′t = u∞,m

t on [0,τR ]. (3.2.27)

To show this it suffices to show that for each R ∈N, almost surely, for all m ≥ R, we

have un,mt = un,R

t on [0,τR ] for all n ∈N. To show this we set

τRn := inft ≥ 0 :

∫Q

(un,1t −R)2

+d x > 0∧T.

For all m ≥ R we have that the processes un,mt satisfy the equation

d vt = (Lt vt + f n,Rt (vt ))d t + (M k

t vt + g kt d w k

t ,

v0(x) =ψ(x), (3.2.28)

for t ≤ τRn . It follows that almost surely, un,m

t = un,Rt for t ≤ τR

n , for all n. We just

note here that by the comparison principle again, we have τR ≤ τRn and this shows

(3.2.27). Also for almost every ω ∈Ω, we have τR = T for R large enough. Hence we

can define ut = limm→∞ u∞,mt , and then one can easily see that ut has the desired

properties.

For the uniqueness, let u(1) and u(2) be B-solutions of (3.2.22). Then one can

define the stopping time

τN = inft ≥ 0 :∫

Q(|u(1)

t |−N )2+d x ∨

∫Q

(|u(2)t |−N )2

+d x > 0,

to see that for t ≤ τN , the two solutions satisfy equation (3.2.23) with n = m = N ,

and the claim follows, since τN = T almost surely, for large enough N .

71

3.3 Weak Harnack inequality

Harnack inequalities, introduced by [32], provide a comparison of values at dif-

ferent points of nonnegative functions which satisfy a partial differential equation

(PDE). This type of inequalities have a vast number of applications, in particular,

they played a significant role in the study of PDEs with discontinuous coefficients

in divergence form. This is the celebrated De Giorgi-Nash-Moser theory ([18], [53],

[51]), in which Hölder continuity of the solutions is established. Later, by using a

weaker version of Harnack’s inequality, a simpler proof in the parabolic case was

given in [39]. Harnack inequality and Hölder estimate for equations in non diver-

gence form, also known as the Krylov-Safonov estimate, was proved in [47] and [59].

Since then, similar results have been proved for more general equations, including

for example integro-differential operators of Lévy type (see [2]) and singular equa-

tions (see [14] and references therein).

It is well known (see e.g. [43], [40]) that the stochastic partial differential equa-

tions (SPDEs)

dut =Lt ut d t +M kt ut d w k

t , (3.3.29)

are in many ways the natural stochastic extensions of parabolic equations dut =Lt ut d t . It is therefore also natural to ask whether the above mentioned results,

fundamental in deterministic PDE theory, have stochastic counterparts.

Here, following the approach of [39], we derive a stochastic version of the afore-

mentioned weak Harnack inequality in Theorem 3.3.1. Here “weak” stands for that

in order to estimate the minimum of a nonnegative solution u, not only the max-

imum of u is required to be bounded from below by 1, but u itself on at least half

of the domain. In the deterministic case such an inequality yields the Hölder conti-

nuity of the solutions almost immediately. This last step appears to be more elusive

in the stochastic setting, but a weaker type of continuity is nevertheless obtained

in Theorem 3.3.2. We note that Harnack inequalities for solutions of SPDEs - not

to be confused with Harnack inequalities for the transition semigroup of SPDEs, for

which we refer the reader to [61] and the references therein - have not been previ-

ously established even for equations with smooth coefficients. We also remark that

the supremum estimates needed in the proofs are local, as opposed to the global

ones established in the previous Section. The latter is proved through a stochastic

Moser’s iteration, while here, for the sake of novelty we use a stochastic version of

De Giorgi’s iteration.

Let us introduce some notation used in this section. For R ≥ 0, let BR = x ∈ Rd :

|x| < R, GR = [4−R2,4]×BR , and G :=G2. We will also assume that for simplicity that

72

T = 4 and Q = B2. Subsets of Rd+1 of the formJ×(BR +x), where J is a closed interval

in [0,4] and x ∈ Rd , will be referred to as cylinders. For p ∈ [1,∞] and a subset A of

Rd or Rd+1, the norm in Lp (A) will be denoted by | · |p,A and ‖ ·‖p,A, respectively. For

a real function defined on a set A ⊂Rd we write oscAu := supA u − infA u. By inf,sup

we always mean essential ones.

3.3.1 Formulation and main results

Notice that neither boundary nor initial condition is posed on (3.3.29). We will de-

note by H the set of all strongly continuous L2(B2)-valued predictable processes

u = (ut )t∈[0,4] on Ω× [0,4] such that ut ∈ H 1(B2) for dP ⊗d t-almost every (ω, t ) ∈Ω× [0,4], and

E supt≤4

|ut |22 +E∫ 4

0

∫B2

|∇ut |2d xd t <∞.

Definition 3.3.1. We will say that u satisfies (or is a solution of) (3.3.29), if u ∈ H

and for each φ ∈C∞c (B2), with probability one,

(ut ,φ) = (u0,φ)+∫ t

0(ai j

t ∂i ut ,∂− jφ)d t +∫ t

0(σi k

t ∂i ut ,φ)d w kt ,

for all t ∈ [0,4].

Denote by Λ the set of functions v on [0,4]×B2 such that v ≥ 0 and

|x ∈ B2|v0(x) ≥ 1| ≥ 1

2|B2|.

Let us recall the Harnack inequality that is essentially proved in [39] : If u is a solu-

tion of du = ∂i (ai j∂ j u)d t and u ∈Λ, then

infG1

u ≥ h

with h = h(d ,λ,K ) > 0. In the stochastic case clearly it can not be expected that such

a lower estimate holds uniformly in ω. It does hold, however, with h above replaced

with a strictly positive random variable, this is the assertion of our main theorem.

Theorem 3.3.1. Let u be a solution of (3.3.29) such that on an event A ∈ F , u ∈ Λ.

Then for any N > 0 there exists a set D ∈F , with P (D) ≤C N−δ, such that on A∩Dc ,

inf(t ,x)∈G1

ut (x) ≥ e−N .

where C and δ, are positive constants depending only on d ,λ and K .

73

Later on we will refer to the quantity e−N above as the lower bound correspond-

ing to the probability C N−δ. With the help of Theorem 3.3.1, we obtain the following

continuity result.

Theorem 3.3.2. Let u be a solution of (3.3.29) and (t0, x0) ∈ (0,4)×B2. Then u is

almost surely continuous at (t0, x0).

Remark 3.3.1. Notice that Theorems 3.3.1 and 3.3.2 remain valid if L u and M k u

are replaced by L u = ∂i (ai j (u)∂ j u), and M k u =σi k (u)∂i u where the function

(αi j (·))di , j=1 = (2αi j (·)−σi k (·)σ j k (·))d

i , j=1

takes values in the set (βi j )di , j=1 : ∀z ∈ Rd ,λ−1|z|2 ≥ βi j zi z j ≥ λ|z|2 for some λ> 0.

Also, we only consider equations with higher order terms, in the same spirit as in

[52], since the lower order terms with bounded and measurable coefficients can be

easily treated.

3.3.2 Martingale growth estimates and Itô’s formula

The first three lemmas might be considered standard in the context of stochastic

processes and parabolic PDEs, respectively. For the sake of completeness we pro-

vide short proofs.

Lemma 3.3.3. Let T > 0 and let (mt )t∈[0,T ] be a continuous local martingale. Then

for any α> 0, and Å≥ 2

P ( inft∈[0,T ]

mt ≥−α, supt∈[0,T ]

mt ≥ Åα) ≤C

√logÅ

Å≤CÅ−1/2

with an absolute constant C .

Proof. Without loss of generality, we can assume that our probability space can sup-

port a Wiener process B for which B⟨m⟩t = mt . Then for any α,β≥ 0,

P ( inft∈[0,T ]

mt ≥−α,⟨m⟩T ≥β) ≤ P ( infs∈[0,β]

Bs ≥−α).

Using the well-known distribution of the minimum of B ,

P ( infs∈[0,β]

Bs ≥−α) =√

2

πβ

∫ 0

−αe−x2/2βd x ≤C

α√β

.

74

On the other hand, for any β,γ

P (⟨m⟩T ≤β, supt∈[0,T ]

mt ≥ γ) ≤ P ( sups∈[0,β]

Bs ≥ γ),

and

P ( sups∈[0,β]

Bs ≥ γ) ≤√

2

πβ

β

γ

∫ ∞

γ

x

βe−x2/2βd x =C

√β

γ2e−γ2/2β. (3.3.30)

Substituting β= (Åα)2/logÅ2 and γ=αÅ yields the first inequality, while the second

one is obvious.

Lemma 3.3.4. For any c > 0, there exists N0(c) > 0, such that for any continuous local

martingale mt , and for any N ≥ N0,

P

(supt≥0

(mt − c⟨m⟩t ) > N

)≤Ce−N c/4,

with an absolute constant C .

Proof. Again, we can assume that our probability space can support a Wiener pro-

cess B for which B⟨m⟩t = mt . Then for any β> 0

P

(supt≥0

(mt − c⟨m⟩t ) > N

)≤ P

(sups≥0

(Bs − cs) > N

)

≤ P ( sups∈[0,β]

Bs > N )+∞∑

i=1P ( sup

s∈[0,(i+1)β]Bs > i cβ).

By (3.3.30),

P

(supt≥0

(mt − c⟨m⟩t ) > N

)≤C

√β

N 2e−N 2/2β+

∞∑i=1

C

√(i +1)

c2i 2βe−c2i 2β/2(i+1).

Choosing β= N /c yields the claim.

Lemma 3.3.5. Suppose that u ∈ L2([0,4], H 1(B2))∩L∞([0,4],L2(B2)). Let J ⊂ [0,4] be

a subinterval, Q = Bρ for some 0 < ρ < 2, ϕ ∈C∞c (Q), and α>β≥ 0. Then

‖(u −α)+ϕ‖22,J×Q ≤C (|ϕ|2∞+|∇ϕ|2∞)

[‖(u −β)+‖2,J×Q

α−β] 4

d+2

×[

supt∈J

|(u −α)+|22,Q +‖Iu>α∇u‖22,J×Q

],

with C =C (d).

75

Proof. By Hölder’s inequality,

‖(u −α)+ϕ‖22,J×Q ≤ ‖(u −α)+ϕ‖2

2(d+2)/d ,J×Q‖Iu>α‖4/d+22,J×Q .

Noticing that

Iu>α ≤ (u −β)+

α−β ,

and using the embedding inequality

‖v‖22(d+2)/d ,G ≤C (d)

(sup

t∈[0,4]|v |22,B2

+‖∇v‖22,G

)

for v ∈ L2([0,4], H 10 (B2))∩L∞([0,4],L2(B2)) (see, e.g. Lemma 3.2, [52]), applied to the

function (u −α)+I Jϕ, we get the required inequality.

Finally, let us formulate the version of Itô’s formula we will use later. We denote

by D the set of twice continuously differentiable functions f from R to R, such that

f ′′ is bounded. Notice that if f ∈ D, then there exists a constant K such that for all

r ∈R| f (r )| ≤ K (1+|r |2), | f ′(r )| ≤ K (1+|r |).

We denote by C the set of twice continuously differentiable functions f from R to

R, such that both f ′ and f f ′′ are bounded.

Lemma 3.3.6. Let u satisfy (3.3.29), and let g ∈ D, ϕ ∈ C∞c (B2), and ψ ∈ C∞[0,4].

Then almost surely,∫B2

ϕ2ψ2t g (ut )d x =

∫B2

ϕ2ψ20g (u0)d x +

∫ t

0

∫B2

2ψsψ′sϕ

2g (us)d xd s

−∫ t

0

∫B2

2ψ2sϕ∂ jϕg ′(us)ai j

s ∂i usd xd s +∫ t

0

∫B2

ψ2sϕ

2g ′(us)σi k∂i usd w ks

−∫ t

0

∫B2

ψ2sϕ

2g ′′(us)[ai js ∂i us∂ j us − 1

2σi kσ j k∂i us∂ j us]d xd s, (3.3.31)

for all t ∈ [0,4].

Proof. Letκbe nonnegative a C∞ function onRd , bounded by 1, supported on |x| <1, and having unit integral. We denote κε(x) = ε−dκ(x/ε), for ε > 0 and for v ∈L2(B2) we write

vε(x) = (v)ε(x) =∫

B2

κε(x − y)v(y)d y, for x ∈Rd .

Let us choose ε > 0 small enough such that ϕ is supported in B2−ε. Then for

76

x ∈ B2−ε we have

uεt (x) = uε

0(x)+∫ t

0(ai j

s ∂ j us ,∂iκε(x −·))d t +∫ t

0(σi k

s ∂i us)ε(x)d w ks .

Then one can write Itô’s formula for the processes ϕ2(x)ψ2t g (uε

t (x)) for x ∈ B2, use

Fubini and stochastic Fubini theorems (for the latter, see [46]), and integrate by

parts to obtain that almost surely,∫B2

ϕ2ψ2t g (uε

t )d x =∫

B2

ϕ2ψ0g (uε0)d x +

∫ t

0

∫B2

2ψsψ′sϕ

2g (uεs )d xd s

−∫ t

0

∫B2

2ψ2sϕ∂ jϕg ′(uε

s )(ai js ∂i us)εd xd s +

∫ t

0

∫B2

ψ2sϕ

2g ′(uεs )(σi k

s ∂i us)εd xd w ks

−∫ t

0

∫B2

ψ2sϕ

2g ′′(uεs )[(ai j

s ∂ j us)ε∂i uεs −

1

2(σi k

s ∂i us)ε(σ j ks ∂ j us)ε]d xd s,

for all t ∈ [0,4]. Then for fixed t one lets ε→ 0 to obtain that (3.3.31) holds almost

surely, and the result follows since both sides of (3.3.31) are continuous in t .

Lemma 3.3.7. Let u satisfy (3.3.29) , and let g ∈C , ϕ ∈C∞c (B2), andψ ∈C∞[0,4]. Set

vt = (g (ut ))+. Then v ∈H , and almost surely,

|ϕψt vt |22 = |ϕψ0v0|22 +∫ t

0

∫B2

2ψ2sϕ

2vsσi k∂i vsd xd w k

s +∫ t

0

∫B2

2ψsψ′sϕ

2v2s d xd s

−∫ t

0

∫B2

ψ2sϕ

2[2ai js ∂i vs∂ j vs −σi kσ j k∂i vs∂ j vs]d xd s

−∫ t

0

∫B2

ψ2sϕ

2vs g ′′(us)[2ai js ∂i us∂ j us −σi kσ j k∂i us∂ j us]d xd s

−∫ t

0

∫B2

4ψ2sϕ∂ jϕvs ai j

s ∂i vsd xd s, (3.3.32)

for all t ∈ [0,4].

Proof. Since g has bounded first derivative, it follows easily that v ∈ H . We intro-

duce now the functions αδ(r ), βδ(r ) and γδ(r ) on R, for δ> 0, given by

αδ(r ) =

2 if r > δ

2rδ if 0 ≤ r ≤ δ0 if r < 0,

βδ(r ) =∫ r

0aδ(s)d s, γδ(r ) =

∫ r

0βδ(s)d s.

For all r ∈ R we have αδ(r ) → 2Ir>0, βδ(r ) → 2r+ and γδ(r ) → (r+)2 as δ→ 0. Also,

77

for all r,r1,r2 and δ, the following inequalities hold

|αδ(r )| ≤ 2, |βδ(r )| ≤ 2|r |, |γδ(r )| ≤ r 2.

It follow then that since g ∈ C , the function ζδ(r ) := γδ(g (r )) lies in D. Hence, by

virtue of Lemma 3.3.6 one can write Itô’s formula for |ψϕζδ(ut )|22, i.e. (3.3.31) with

g , g ′ and g ′′ replaced by γδ(g ),βδ(g )g ′ and αδ(g )|g ′|2 +βδ(g )g ′′ respectively. Then

we let δ→ 0 to obtain (3.3.32).

3.3.3 Local supremum estimates

While supremum estimate have been obtained in Section 3.1.1, there are two dif-

ferences to that in the following version. First, the estimate presented here is local,

and therefore no initial or boundary condition needs to be specified. Second, we

need the estimates to hold not only for the solutions, but for certain functions of

solutions as well.

Theorem 3.3.8. Let f ∈C such that f f ′′ ≥ 0, and let u be a solution of (3.3.29). Then

there exist positive constantsδ,C ,C , depending only on d ,λ,K , such that for anyα> 0

and κ≥ 1

(i) P (‖ f (u)+‖2∞,G1

≥ Cκα,‖ f (u)+‖22,G3/2

≤α) ≤Cκ−δ,

(ii) P (‖ f (u)‖2∞,G1

≥ Cκα,‖ f (u)‖22,G3/2

≤α) ≤Cκ−δ.

Proof. First we prove (i ). It is easy to see that it suffices to show the existence of

γ,δ1,δ2,C > 0 such that

P (‖ f (u)+‖2∞,G1

≥ 1,‖ f (u)+‖22,G3/2

≤ κ−δ1 γ) ≤Cκ−δ2 , (3.3.33)

since by plugging in (3.3.33) f = (γ/κδ1α)1/2 in place of f , we obtain the desired

inequality with δ= δ2/δ1 and C = 1/γ.

To this end, take r ∈ [0,4], ρ ∈ [1,2], ψ ∈ C∞([0,4]) with ψ = 0 on [0,r ], and ϕ ∈C∞

c (Bρ). For j = 0,1, . . . let g j (u) := f (u)− (1−2− j ), v j = (g j (u))+, and let us apply

Lemma 3.3.7 with g j+1. Using the parabolicity condition and Young’s inequality, as

well as the nonnegativity of v j+1(g j+1)′′, we get for any ε> 0

∫Bρϕ2ψ2

t |v j+1t |2d x

≤ m j+1t +

∫ t

r

∫Bρ

2ψsψ′sϕ

2|v j+1s |2d xd s −

∫ t

r

∫Bρλψ2

sϕ2|∇v j+1

s |2d xd s

78

+∫ t

r

∫Bρ

[εψ2sϕ

2K 2|∇v j+1s |2 +16/εψ2

s |∇ϕ|2|v j+1s |2]d xd s

almost surely for all for t ∈ [r,4], where

m j+1t =

∫ t

r

∫Bρ

2ϕ2ψ2s v j+1

s M k v j+1s d xd w k

s .

Choosing ε sufficiently small, we arrive at∫Bρϕ2ψ2

t |v j+1t |2d x +

∫ t

r

∫Bρϕ2ψ2|∇v j+1

s |2d xd s

≤C ′m j+1t +C

∫ t

r

∫Bρϕ2ψsψ

′s |v j+1

s |2 +|∇ϕ|2ψ2s |v j+1

s |2d xd s. (3.3.34)

Now let us choose r = r j = 3−(5/4)2− j and ρ = ρ j = 1+(1/2)2− j , that is, [r0,4]×Bρ0 =G3/2. Also we introduce the notation F j = [r j ,4]×Bρ j . Furthermore, choose ψ=ψ j

and ϕ=ϕ j such that

(i) 0 ≤ψ j ≤ 1, ψ j |[0,r j ] = 0, ψ j |[r j+1,4] = 1;

(ii) 0 ≤ϕ j ≤ 1, ϕ j ∈C∞0 (Bρ j ), ϕ j |Bρ j+1

= 1;

(iii) |∂tψj |+ |∇ϕ j |2 <C 4 j .

Then by running t over [r j ,4], by (3.3.34) we obtain

supt∈[r j+1,4]

|v j+1t |22,Bρ j+1

+‖∇v j+1‖22,F j+1

≤C 4 j‖v j+1‖22,F j

+C supt∈[r j ,4]

m j+1t . (3.3.35)

Notice that, since the left-hand side of (3.3.34) is nonnegative, running t over I j

gives

inft∈[r j ,4]

m j+1t ≥−C 4 j‖v j+1‖2

2,F j. (3.3.36)

Applying Lemma 3.3.5 with α= 1−2−( j+1), β= 1−2− j , and ϕ=ϕ j+1, we get

‖v j+1‖2,F j+2 ≤ ‖ϕ j+1v j+1‖2,F j+1

≤C j‖v j‖4/d+22,F j+1

[sup

t∈[r j+1,4]|v j+1

t |22,Bρ j+1+‖∇v j+1‖2,F j+1

].

Combining this with (3.3.35) yields

‖v j+1‖22,F j+2

≤C j‖v j‖4/d+22,F j+1

[‖v j+1‖2

2,F j+ sup

t∈[r j ,4]m j+1

t

].

79

Since for j > i , we have v j ≤ v i and F j ⊂ Fi , we obtain for V j = ‖v j‖22,F j

V j+2 ≤C j V 2/(d+2)j

[4 j V j + sup

t∈[r j ,4]m j+1

t

].

Let γ0,γ ∈ (0,1) and suppose that V j ≤ γ0γj on a set Ω j ⊂Ω. By (3.3.36) we have

inft∈[r j ,4]

m j+1t ≥−C 4 j‖v j+1‖2

2,F j≥−C 4 j V j ,

and Lemma 3.3.3 can be applied with α = C 4 jγ0γj , and Å replaced by κ4 j . That is

we obtain a subset Ω j+2 ⊂Ω j such that P (Ω j \Ω j+2) ≤Cκ−1/22− j and on Ω j+2

supt∈[r j ,4]

m j+1t ≤ κC 16 jγ0γ

j .

Consequently, on Ω j+2,

V j+2 ≤C jγ2/(d+2)0 γ2 j /(d+2)γ0γ

j (1+κ) ≤ γ0γj+2,

provided that

γ=C−(d+2)/2, γ0 ≤ (γ2/(1+κ))(d+2)/2.

Proceeding iteratively, we can conclude that on ∩ j≥0Ω2 j , V j → 0, and therefore

‖ f (u)+‖2∞,G1

≤ 1,

and moreover,

P(Ω0 \∩ j≥0Ω2 j

)= ∑j≥0

P (Ω2 j \Ω2 j+2) ≤ 2Cκ−1/2.

This proves (3.3.33), sinceΩ0 = ‖ f (u)+‖22,G3/2

=V0 ≤ γ0 = κ−(d+2)/2γ with a constant

γ= γ(d ,λ,K ).

For part (i i ), we have

P (‖ f (u)‖2∞,G1

≥ Cκα,‖ f (u)‖22,G3/2

≤α)

≤ P (‖ f (u)+‖2∞,G1

≥ Cκα,‖ f (u)+‖22,G3/2

≤α)

+P (‖ f (u)−‖2∞,G1

≥ Cκα,‖ f (u)−‖22,G3/2

≤α),

which by virtue of (i ) and the fact that − f satisfies the conditions of the lemma,

finishes the proof.

80

Let us consider the case when the initial value is 0. Note that in this case in the

proof of Theorem 3.3.8 the time-cutoff function ψ can be omitted. Doing so and

repeating the same steps afterwards, we get the following.

Theorem 3.3.9. Let f ∈C such that f f ′′ ≥ 0, u be a solution of (3.3.29) on [s,r ]×B2,

where 0 ≤ s < r ≤ 4, and suppose that f (v)(s, ·) ≡ 0. Then there exist positive constants

δ,C ,C , depending only on d ,λ,K , such that for any α> 0 and κ≥ 1

(i) P (‖ f (u)+‖2∞,[s,r ]×B1

≥ Cκα,‖ f (u)+‖22,[s,r ]×B2

≤α) ≤Cκ−δ,

(ii) P (‖ f (u)‖2∞,[s,r ]×B1

≥ Cκα,‖ f (u)‖22,[s,r ]×B2

≤α) ≤Cκ−δ.

Recall that from [57] it is known that solutions of (3.3.29) with 0 boundary and

Lp initial conditions are weakly continuous in Lp for any p ∈ (0,∞). A simple con-

sequence of Theorem 3.3.8 is that in fact strong continuity holds, away from t = 0.

Corollary 3.3.10. Let u be a solution of (3.3.29) and p ∈ (0,∞). Then

(a) (ut )t∈[3,4] is strongly continuous in Lp (B1);

(b) If furthermore u|∂B2 = 0, then (ut )t∈[3,4] is strongly continuous in Lp (B2).

Proof. (a) First notice that the supremum in time can be taken to be real (and not

only essential) supremum: the function |(u − ‖u‖∞,G1 )+|2,B1 is 0 for almost all t ,

hence by the continuity of u in L2 it is 0 for all t , and therefore, for all t , almost

every x, ut (x) ≤ ‖u‖∞,G1 . Now fix t ∈ [3,4], and take a sequence tn → t . Then

utn → ut in L2(B2), hence for a subsequence tnk , for almost every x. On the other

hand, |utnkIB1 | ≤ ‖u‖∞,G1 < ∞, therefore by the dominated convergence theorem,

utnk→ ut in Lp .

For part (b), notice that when u ∈ H 10 (B2) for almost all ω, t , then in the spe-

cial case f (r ) = r the space-cutoff function ϕ in the proof of Theorem 3.3.8 can be

omitted. We then obtain that ‖u‖∞,[3,4]×B2 <∞ with probability 1, and by the same

argument as above we get the claim.

3.3.4 Proof of Theorem 3.3.1

Before turning to the proof we need one more lemma, which can be considered as a

weak version of Theorem 3.3.1.

Lemma 3.3.11. Let u be a solution of (3.3.29), such that on A ∈ F , u ∈Λ. Then for

any N > 0, there exists a set D1 ∈F , with P (D1) ≤Ce−cN , such that on A∩Dc1, for all

t ∈ [0,4],

|(x ∈ Bρ|v(t , x) ≥ e−N | ≥ 1

8|Bρ|,

81

where ρ is defined by

|Bρ| = 3

4|B2|,

and the constants c,C > 0, depend only on d ,λ,K .

Proof. Clearly it is sufficient to prove the statement for N > N0 for some N0. Intro-

duce the functions

fh(x) =

ah x +bh if x <−h/2

log+ 1x+h if x ≥−h/2,

for h > 0 where ah and bh is chosen such that fh and f ′h are continuous. Let κ be

nonnegative a C∞ function on R, bounded by 1, supported on |x| < 1, and having

unit integral. Denote κh(x) = h−1κ(x/h) and

Fh = fh ∗κh/4.

We claim that Fh has the following properties:

(i) Fh(x) = 0 for x ≥ 1;

(ii) Fh(x) ≤ log(2/h) for x ≥ 0;

(iii) Fh(x) ≥ log(1/2h) for x ≤ h/2;

(iv) Fh ∈D and F ′′h (x) ≥ (F ′

h(x))2 for x ≥ 0.

The first three properties are obvious, while for the last one notice that Fh has bounded

second derivative, f ′′h (x) ≥ ( f ′

h(x))2 for x ≥−h/2, and therefore, for x ≥ 0

(F ′h(x))2 =

(∫f ′

h(x − z)κ1/2h/4(z)κ1/2

h/4(z)d z

)2

≤∫

( f ′h(x − z))2κh/4(z)d z

≤∫

f ′′h (x − z)κh/4(z)d z = F ′′

h (x).

Let us denote v = Fh(u). Applying Lemma 3.3.6 and using the parabolicity condi-

tion, we get∫B2

ϕ2vt d x −∫

B2

ϕ2v0 d x ≤∫ t

0

∫B2

Cϕ∇ϕ∇v − (λ/2)ϕ2F ′′h (u)(∇u)2 d x d s

+∫ t

0

∫B2

ϕ2M k v d x d w ks (3.3.37)

82

for any ϕ ∈ C∞c . Let us denote the stochastic integral above by mt , and notice that

provided |ϕ| ≤ 1,

⟨m⟩t ≤C∫ t

0

∫B2

ϕ2(∇v)2 d x d s.

Let c be such that cC ≤ λ/4. From Lemma 3.3.4, there exists a set D1 with P (D1) ≤Ce−N c/4, such that on Dc

1 we have

∫B2

ϕ2vt d x −∫

B2

ϕ2v0 d x

≤ N +∫ t

0

∫B2

Cϕ∇ϕ∇v − (λ/2)ϕ2F ′′h (u)(∇u)2 + cCϕ2(∇v)2 d x d s. (3.3.38)

On A∩Dc1, by the property (iv) above, we have F ′′

h (u)(∇u)2 ≥ (∇v)2, and therefore∫B2

ϕ2vt d x ≤ N +C∫

B2

|∇ϕ|2 d x +∫

B2

ϕ2v0 d x. (3.3.39)

Let us denote

Ot (h) = x ∈ Bρ : u(t , x) ≥ h.

Choosingϕ to be 1 on Bρ, by properties (i), (ii), and (iii) of Fh and (3.3.39), on A∩Dc1,

for all t ∈ [0,4]

|Bρ \Ot (h/2)| log(1/2h) ≤C +N + 1

2log(2/h)|B2| =C +N + 2

3log(2/h)|Bρ|.

Hence

|Ot (h/2)| ≥ |Bρ|− C +N

log(1/2h)− 2

3

log(2/h)

log(1/2h)|Bρ|,

and choosing N0 = C and h = 2e−C ′N for a sufficiently large C ′ finishes the proof of

the lemma.

Proof of Theorem 3.3.1 By Lemma 3.3.11, there exists a set D1 with P (D1) ≤Ce−cN

such that on A∩Dc1 we have

|(x ∈ Bρ|v(t , x) ≥ e−N | ≥ 1

8|Bρ|, (3.3.40)

for all t ∈ [0,4]. Let us denote h := e−N . For 0 < ε≤ h/2, we introduce the function

fε(x) =

aεx +bε if x <−ε/2

log+ hx+ε if x ≥−ε/2,

where aε and bε is chosen such that fε and f ′ε are continuous. Let κ be a nonnegative

C∞ function on R, bounded by 1, supported on |x| < 1, and having unit integral.

83

Denote κε(x) = ε−1κ(x/ε) and

Fε = fε∗κε/4.

Similarly to Fh in the proof of Lemma 3.3.11, Fε has the following properties:

(i) Fε(x) = 0 for x ≥ h;

(ii) Fε(x) ≤ log(2h/ε) for x ≥ 0;

(iii) Fε(x) ≥ log(h/(x +ε))−1 for x ≥ 0;

(iv) Fε ∈D and F ′′ε (x) ≥ (F ′

ε(x))2 for x ≥ 0.

Let us denote v = Fε(u). Similarly to (3.3.38), there exists a set D2 with P (D2) ≤Ce−N c , such that on Dc

2 we have

∫B2

ϕ2vt d x −∫

B2

ϕ2v2 d x

≤ N +∫ t

0

∫B2

Cϕ∇ϕ∇v − (λ/2)ϕ2F ′′ε (u)(∇u)2 + (λ/4)ϕ2(∇v)2 d x d s.

On A∩Dc2, by property (iv), we have,

∫ 4

0

∫B2

ϕ2|∇vt |2 d xd t ≤C (N +∫

B2

|∇ϕ|2 d x +∫

B2

ϕ2v2 d x). (3.3.41)

By choosing ϕ ∈C∞c (B2) with 0 ≤ϕ≤ 1 and ϕ= 1 on Bρ we get,

∫ 4

0

∫Bρ

|∇vt |2 d xd t ≤C (N +∫

B2

|∇ϕ|2 d x +∫

B2

ϕ2v2 d x).

Hence, by property (ii),

∫ 4

0

∫Bρ

|∇vt |2 d xd t ≤C N +C +C log2h

ε. (3.3.42)

Using property (i), by a version of Poincaré’s inequality (see, e.g., Lemma II.5.1, [48])

we get for all t ∫Bρ

|vt |2d x ≤Cρ2(d+1)

|Ot (h)|2∫

Bρ|∇vt |2d x,

which, by virtue of (3.3.40) and (3.3.42) implies

∫ 4

0

∫Bρ

|vt |2d x ≤C +C N +C log2h

ε.

84

on A∩Dc1 ∩Dc

2. By Theorem 3.3.8 and noting that G3/2 ⊂ [0,4]×Bρ we get that there

exists a set D3 ∈F with P (D3) ≤C N−δ, such that on A∩Dc1 ∩Dc

2 ∩Dc3 we have

sup(t ,x)∈G1

vt (x) ≤ [N (C +C N +C log2h

ε)]1/2.

By applying property (iii), we get

sup(t ,x)∈G1

logh

ut (x)+ε ≤ [N (C +C N +C log2h

ε)]1/2 +1,

and therefore,

inf(t ,x)∈G1

ut (x) ≥ he−[N (C+C N+C log2h−C logε)]1/2−1 −ε.

Letting ε= e−c ′N with a sufficiently large c ′, it is easy to see that the right-hand side

above is bounded from below by ε, finishing the proof.

3.3.5 Proof of Theorem 3.3.2

Proof. Consider the parabolic transformations Pα,t ′,x ′ :

t →α2t + t ′,

x →αx +x ′.

It is easy to see that if v is a solution of (3.3.29) on a cylinder V , then v P−1α,t ′,x ′ is

also solution of (3.3.29), on the cylinder Pα,t ′,x ′V , with another sequence of Wiener

martingales on another filtration, and with different coefficients that still satisfy As-

sumptions 3.0.3-3.0.4 with the same bounds. To ease notation, for a cylinder V let

PV denote the unique parabolic transformation that maps V to G , if such exists.

Also, for an interval [s,r ] ⊂ [0,4] let P[s,r ] =P2/p

r−s,−4s/(r−s),0. That is, P[s,r ][s,r ]×B1 = [0,4]×B2/

pr−s , which, when r − s ≤ 1, contains G .

Without loss of generality x0 = 0 can and will be assumed, as will the almost sure

boundedness of u on G , since these can be achieved with appropriate parabolic

transformations, using the boundedness obtained on sub-cylinders in 3.3.8. Also

let us fix a probability δ > 0, denote the corresponding lower bound 3ε2 obtained

from the Harnack inequality, and take an arbitrary 0 < ε1 < ε2/2.

Apply Theorem 3.3.9 (i ) twice, with the function f (r ) = r , with the interval [t0 −4s, t0 + s], and with solutions v = u−supt0−4s×B2

u and v =−u+ inft0−4s×B2 u. Also

85

notice that (for both choices of v)

‖v+‖22,[t0−4s,t0+s]×B2

≤C s‖u‖2∞,G → 0

as s → 0 for almost every ω, and thus in probability as well (recall that the fact that

the functions v are well-defined and that the above - seemingly trivial - inequality

holds, is justified in the proof of Corollary 3.3.10). In other words,

P (‖v+‖22,[t0−4s,t0+s]×B2

>α)

can be made arbitrarily small by choosing s sufficiently small. Therefore, we obtain

an s > 0 and an event Ω0, with P (Ω0) > 1−δ, such that on Ω0,

sup[t0−4s,t0+s]×B1

u − supt0−4s×B2

u < ε21/6

inf[t0−4s,t0+s]×B1

u − inft0−4s×B2

u >−ε21/6.

Let us rescale u at the starting time:

u′±(t , x) =±

(2

u(t , x)− supt0−4s×B2u

supt0−4s×B2u − inft0−4s×B2 u

+1

),

that is, supB2u′±(t0−4s, ·) = 1, infB2 u′

±(t0−4s, ·) =−1. Now we can writeΩ0 =ΩA∪ΩB ,

where

• On ΩA, osct0−4s×B2 u < ε1/3, and therefore, osc[t0−4s,t0+s]×B1 u < ε1/3+2ε21/6 <

ε1;

• On ΩB , |u′±| < 1+2(ε2

1/6)/(ε1/3) = 1+ε1, on [t0 −4s, t0 + s]×B1.

Notice that in the eventΩB , on the cylinder [t0−4s, t0+s]×B1, the functions u′±/(1+

ε1)+1 take values between 0 and 2. Therefore one of (u′±/(1+ε1)+1)P−1

[t0−4s,t0+s]

∣∣∣G

(see Figure 3.1), denoted for the moment by u′′, satisfies the conditions of Theorem

3.3.1 with A =ΩB .

We obtain that on an event Ω′B

infG1

u′′ > 3ε2,

and thus

oscV u < (2−3ε2)(1+ε1)

2osct0−4s×B2 u < (1−ε2)osct0−4s×B2 u,

86

Figure 3.1:

where V =P−1[t0−4s,t0+s]G1. Moreover, P (ΩB \Ω′

B ) < δ. Also, notice that (t0,0) ∈V. Let

us denote Ω1 =ΩA ∪Ω′B . We have shown the following lemma:

Lemma 3.3.12. Let δ> 0 and let 3ε2 be the lower bound corresponding to the proba-

bility δ obtained from the Harnack inequality. For any u that is a solution of (3.3.29)

on G, t0 > 0, and for any sufficiently small ε1 > 0 there exists an s > 0 and an eventΩ1

such that

(i) P (Ω1) > 1−2δ;

(ii) On Ω1, at least one of the following is satisfied:

(a) oscV u < ε1;

(b) oscV u < (1−ε2)oscG u,

where V =P−1[t0−4s,t0+s](G1).

Now take u = u(0) and t0 = t (0)0 from the statement of the theorem and a sequence

(ε(n)1 )∞n=0 ↓ 0, and for n ≥ 0 proceed inductively as follows:

• Apply Lemma 3.3.12 with u(n), t (n)0 , and ε(n)

1 , and take the resulting Ω(n)1 and

V (n);

• Let u(n+1) = u(n) P−1V (n) and (t (n+1)

0 ,0) =PV (n) (t (n)0 ,0).

87

On limsupn→∞Ω(n)1 the function u is continuous at the point (t0,0). Indeed,

the sequence of cylinders V (0),P−1V (0)V

(1),P−1V (0)P

−1V (1)V

(2), . . . contain (t0,0), and the

oscillation of u on these cylinders tends to 0. However, P (limsupn→∞Ω(n)1 ) ≥ 1−2δ,

and since δ can be chosen arbitrarily small, u is continuous at (t0,0) with probability

1, and the proof is finished.

Remark 3.3.2. It is natural to attempt to modify the above argument to bound expec-

tations and higher moments of the oscillations, in the hope to apply Kolmogorov’s

continuity criterion and obtain Hölder estimates. The main obstacle appears to be

to establish a uniform integrability property to a family of (normalized) oscillations.

Indeed, the present Harnack inequality can bring down the oscillation by a given

factor outside of a small event, and therefore one would like to exclude the possi-

bility that the majority of the oscillation’s mass is concentrated on that exceptional

event.

88

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