A PDE approach to large deviations in

Hilbert spaces

Andrzej Swiech∗

School of MathematicsGeorgia Institute of Technology

Atlanta, GA 30332, U.S.A.

Abstract

We introduce a PDE approach to the large deviation principle for Hilbert spacevalued diffusions. It can be applied to a large class of solutions of abstract stochas-tic evolution equations with small noise intensities and is adaptable to some specialequations, for instance to the 2-d stochastic Navier-Stokes equations. Our approachuses a lot of ideas from (and in significant part follows) the program recently de-veloped by Feng and Kurtz [19]. Moreover we present easy proofs of exponentialmoment estimates for solutions of stochastic PDE.

1 Introduction

In this paper we are interested in the large deviation principle for the family of solutions

of abstract stochastic evolution equations with small noise intensities of the form

dXn(s) = (−AXn(s) + b(s,Xn(s)))ds+ 1√nσ(s,Xn(s))Q

12dW (s) for s > t,

X(t) = x ∈ H.

(1.1)

Above H is a real separable Hilbert space, A is a linear, maximal monotone operator

in H, W is a cylindrical Wiener process in H with identity covariance operator, and Q

is a linear, bounded, nonnegative, self-adjoint operator of trace class on H. The form

of equation (1.1) is mainly motivated by semilinear stochastic PDE with multiplicative

noise, for instance stochastic reaction-diffusion systems. We will study the small noise

∗A. Swiech was supported by NSF grant DMS 0500270. The author also acknowledges the support ofCentro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy.

2000 Mathematics Subject Classification: 49L25, 35R15, 35K55, 60F10, 60H15, 60J60.Key words and phrases: large deviations, viscosity solutions, Hamilton-Jacobi-Bellman equations,

stochastic PDE, stochastic Navier-Stokes equations.

1

limit for the sequence of solutions Xn(·) as n→ +∞. Moreover we will also look at the

small noise limit for solutions of 2-d stochastic Navier-Stokes (SNS) equations. Our main

goal is to present a relatively easy to use set of mostly analytical tools for the problems

based on the theory of viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations

in infinite dimensional Hilbert spaces.

The use of viscosity solutions in large deviation type problems is not new in finite

dimensions (see for instance [20] and the references therein). In infinite dimensional spaces

such techniques were not available and large deviation results for abstract stochastic

evolution equations were off limits for the analytic PDE approach. However in recent years

the theory of viscosity solutions in infinite dimensions has undergone rapid development.

A few years ago Feng and Kurtz [19] proposed a new framework for large deviations

based on nonlinear semigroup techniques and viscosity solutions in abstract spaces. This

work has been a remarkable contribution, offering a fresh approach and a variety of tools.

Moreover their methods apply to general metric space valued Markov processes. However

because of this generality the techniques proposed in [19] and related papers [16, 17, 18]

may miss some “local” methods and it may not be very transparent how to apply them

in some particular cases. We think that a simpler and a more transparent technique can

be developed for Hilbert space valued diffusions. The reason is that Hilbert spaces admit

by now a rather well developed theory of second order HJB equations. This is a missing

ingredient in [19], understandably so as unfortunately no workable second order theory is

available at the moment even in Banach spaces. Without being too precise, in a nutshell,

to show large deviation principle one has to prove exponential tightness and the existence

of the so called Laplace limit. The expressions that appear in the Laplace limit at a single

time for the family of processes Xn can be shown to be viscosity solutions of second

order HJB equations. Therefore the key existence of the Laplace limit at a single time

(7.18) can be recast in terms of convergence of viscosity solutions of singularly perturbed

HJB equations and we can then use the whole existing machinery of viscosity solutions.

After this is done the rest of our analysis follows the program outlined in [19]. Overall we

propose a PDE based technique that, in the spirit, is a more direct generalization of the

finite dimensional method. This setup can accomodate a large class of abstract stochastic

evolution equations and can be adapted to other situations whenever a good theory of

second order equations exists. For instance we also show how these techniques can be

used to obtain the large deviation result for two-dimensional SNS equations. The obvious

drawback of our approach is that it is limited to Hilbert space valued diffusions. We also

mention that in a much simpler setting of risk sensitive control for bounded evolutions, a

singular limit for bounded second order equations in a Hilbert space was studied in [40].

There exist many results on large deviations for small noise limits for problems of type

2

(1.1). We refer the reader to [2, 4, 6, 7, 18, 19, 21, 29, 35, 37] and the book [12]. In particular

in [2, 4] large deviation estimates for the small noise limit for systems of stochastic reaction

diffusion equations with linearly growing diffusion coefficients are obtained. Our abstract

setup is similar to the one in [35], which too considers linearly growing non-additive

noise. Overall our assumptions are slightly different from these of the above mentioned

papers. The conditions we impose on the operator A are very weak. We do not assume

any coercivity of A and it does not have to come from a second order elliptic operator.

Because of this we have to look for a large deviation result in C([0,+∞);H−1), where

H−1 is an ambient space with a weaker topology. We also allow the diffusion coefficient

σ to have linear growth. In fact, as it is seen in Section 7, the linear growth case is just

an easy consequence of the bounded diffusion case. The main goal and contribution of

the paper is to present an analytic approach to large deviations however, to the best of

our knowledge, our method produces large deviation results in some cases which are not

currently available even though there may be big overlap with the existing literature. It

also gives an alternative to other approaches. Many extensions seem possible. For instance,

under suitable assumptions, it seems possible to allow A to depend on t if A(t) admit the

same operator B (see Section 2). Also if A had some coercivity property we could look for

a large deviation result in C([0,+∞);H) using a stronger definition of viscosity solution.

This is what in fact we do for SNS equations and so Section 9 shows how such a case

could be handled.

As far as SNS equations are concerned large deviation principle was obtained by dif-

ferent techniques in [5, 38]. In particular in [38] a large deviation result was shown for

a small noise limit with linearly growing multiplicative noise by the variational repre-

sentation method. We could reproduce a similar result by the techniques of this paper

however, since we would not prove anything new, we chose instead to present our method

in a simpler case of additive noise.

Finally we think that the exponential moment estimates of Proposition 6.1 and their

simple proofs may be interesting on their own.

We refer the reader to [14, 15, 19, 22, 42] for the general theory, references, and other

approaches to large deviations.

2 Notation and assumptions

We recall that H is a real separable Hilbert space with the inner product 〈·, ·〉 and norm

‖ · ‖, A is a linear, maximal monotone operator in H, W is a cylindrical Wiener process

in H defined on a probability space (Ω,F ,P) with a normal filtration Ft (see [12]), and

Q is a linear, bounded, nonnegative, self-adjoint, trace-class operator on H.

3

Throughout Sections 4-8, B will be a fixed linear, bounded, positive, self-adjoint

operator such that A∗B is bounded and

〈(A∗B + C0B)x, x〉 ≥ 0, for some C0 > 0 and all x ∈ H. (2.1)

Such an operator always exists. We refer the reader to [36] for this and to [10] for examples

of possible choices of B in some particular cases. We will also require that

B is compact. (2.2)

This assumption is not needed for the results of Section 5 and Theorem 7.3 however we

impose it throughout the paper to simplify the presentation since the proofs are then a

little less technical.

Operator B defines spaces Hα. For α < 0 we define Hα as the completion of H under

the norm ‖x‖α = ‖Bα/2x‖, and for α > 0, Hα = R(Bα/2) equipped with the norm ‖x‖α =

‖B−α/2x‖. They are Hilbert spaces with the inner product 〈x, y〉α = 〈B−α/2x,B−α/2x〉,Hα and H−α are dual to each other, and for every α, β ∈ R, Bα/2 is an isometry between

Hβ and Hβ+α.

We will denote the space of bounded, self-adjoint operators in H by S(H) and we

will write L(H) for the space of bounded, linear operators in H. Both spaces are equipped

with the operator norm (also denoted by ‖ · ‖).Throughout Sections 4-8 we will also assume that b : [0,+∞)×H → H, σ : [0,+∞)×

H → L(H) are continuous and such that for every 0 ≤ t ≤ T , x, y ∈ H,

‖b(t, x) − b(t, y)‖, ‖σ(t, x) − σ(t, y)‖ ≤ LT‖x− y‖−1 (2.3)

for some constant LT .

Let e1, e2, . . . , en, . . . be the basis of H composed of the eigenvectors of B. Given

N ≥ 1 let VN = spane1, e2, . . . , eN, and let PN be the orthogonal projection of H onto

VN . We denote QN = I − PN .

We will be using the function spaces

C2(H) = u : H → R : Du,D2u are continuous,

C1,2((0, T ) ×H) = u : (0, T ) ×H → R : ut, Du,D2u are continuous.

The symbols Du,D2u denote the Frechet derivatives of u. We will always identify H with

its dual space. With this identification we can interpret the Frechet derivatives Du(x) and

D2u(x) as respectively an element of H and a bounded, self-adjoint operator in H. For a

Hilbert space Z we denote

Cb(Z) = u : Z → R : u is bounded and continuous,

4

Lipb(Z) = u : Z → R : u is bounded and Lipschitz continuous.

We will sometimes write X(s; t, x) for solutions of (1.1) when the dependence on t and x

needs to be emphasized.

3 Large deviations

We recall here for readers’ convenience the definition of the large deviation principle and

exponential tightness (see [15]).

Let H be a separable metric space, and Yn be a sequence of random variables on a

probability space (Ω,F ,P) with values in H.

Definition 3.1. A function I : H → [0,+∞] with compact sublevel sets x : I(x) ≤ rfor every r < +∞ is called a rate function on H.

Definition 3.2. We say that the sequence Yn satisfies the large deviation principle on

H with rate function I if the following conditions hold:

• For every closed subset F of H

lim supn→∞

1

nlog P(Yn ∈ F) ≤ −I(F ) := − inf

x∈FI(x).

• For every open subset G of H

lim infn→∞

1

nlog P(Yn ∈ G) ≥ −I(G).

Definition 3.3. The sequence Yn is called exponentially tight if for every M ∈ (0,+∞)

there exists a compact set K ⊂ H such that

lim supn→∞

1

nlog P(Yn ∈ H \K) ≤ −M.

4 Viscosity solutions

The definition of viscosity solution which we employ in this paper is slightly weaker than

the one used in [39] and [30]. The reason for this is explained in Remark 4.3.

Definition 4.1. A function ψ is a test function if ψ = ϕ+ h(‖x‖), where:

(i) ϕ ∈ C1,2 ((0, T ) ×H), is weakly sequentially upper semicontinuous, ϕt, Dϕ,D2ϕ,

A∗Dϕ are locally uniformly continuous, they have at most polynomial growth on

[ε, T −ε]×H for every ε > 0, and ϕ is bounded on every set [ε, T −ε]×‖x‖−1 ≤ r.

5

(ii) h ∈ C2([0,+∞)) and is such that h′(0) = 0, h′(r) ≥ 0 for r ∈ (0,+∞).

We remark that even though ‖x‖ is not differentiable at 0, the function h(‖x‖) ∈C2(H) for a test function h as above.

We give below the definition of viscosity solution for a terminal value problem for

equation

ut − 〈Ax,Du〉 + F (t, x,Du,D2u) = 0 in (0, T ) ×H, (4.1)

where F : (0, T ) ×H ×H × S(H) → R.

Definition 4.2. A weakly sequentially upper semicontinuous function u : (0, T )×H → R

is a viscosity subsolution of (4.1) if whenever u− ϕ− h(‖ · ‖) has a local maximum at a

point (t, x) for test functions ϕ, h(‖y‖) then

ψt(t, x) − 〈x,A∗Dϕ(t, x)〉 + F (t, x,Dψ(t, x), D2ψ(t, x)) ≥ 0,

where ψ(s, y) = ϕ(s, y) + h(‖y‖).A weakly sequentially lower semicontinuous function u : (0, T )×H → R is a viscosity

supersolution of (4.1) if whenever u + ϕ + h(‖ · ‖) has a local minimum at a point (t, x)

for test functions ϕ, h(‖y‖) then

ψt(t, x) + 〈x,A∗Dϕ(t, x)〉 + F (t, x,Dψ(t, x), D2ψ(t, x)) ≤ 0,

where ψ(s, y) = −ϕ(s, y) − h(‖y‖).A viscosity solution of (4.1) is a function which is both a viscosity subsolution and a

viscosity supersolution.

Remark 4.3. The maxima and minima in Definition 4.2 can be assumed to be global

and strict. We refer the reader to [31], Lemma 3.6. We use a slightly smaller class of

test functions than in [30] since we only deal with bounded solutions here. The additional

requirement that the test function ϕ be bounded on every set [ε, T − ε] × ‖x‖−1 ≤ ris needed to guarantee that we can always take the test functions ϕ and h(‖y‖) in the

definition of viscosity solution to be bounded which helps when dealing with expressions of

exponential type in the proof of Theorem 7.1. To see this suppose that u− h(‖ · ‖)−ϕ has

a (strict and global) maximum at (t0, x0) where u is bounded. Without loss of generality

we can assume that ϕ is bounded on every set (0, T ) × ‖x‖−1 ≤ r. Set K = ‖x0‖−1 + 1

and let r > 0 be such that ‖x‖ ≤ r ⊂ ‖x‖−1 ≤ K. Let η : IR → IR be a smooth

nonincreasing function such that η(s) = 1 if s < K and η(s) = 0 if s > 2K. Set

ϕ(s, y) = η(‖y‖−1)ϕ(s, y). Then ϕ is a good test function and

‖ϕ‖∞ ≤ sup(s,y)∈(0,T )×‖x‖−1≤2K

|ϕ(s, y)| =: C.

6

We now define a function h to be a smooth nondecreasing function such that h(s) = h(s)

if s ≤ ‖x0‖ and h(s) = 2‖u‖∞ + 2C + h(r) for s > r. The function h is bounded and it

follows from the construction that u− h(‖ · ‖) − ϕ has a (strict and global) maximum at

(t0, x0).

For large deviation applications we will have to deal with HJB equations of the form

ut+1

2ntr(σ(t, x)Qσ(t, x)∗D2u

)− 1

2‖(σ(t, x)Q

12 )∗Dun‖2 + 〈−Ax+ b(t, x), Du〉 = 0. (4.2)

This will be explained in Section 7. When we set n = +∞ in (4.2) we formally have a

first order equation. We will refer to it as (4.2)∞.

5 Comparison principles

In this section we establish the main tools which will be needed in Section 7 to show the

existence of Laplace limit. First we show a comparison result for equation (4.2). We need

an additional assumption

supt∈[0,T ],x∈H

‖σ(t, x)‖ ≤M (5.1)

for some constant M .

Theorem 5.1. Let (2.1)-(5.1) hold. Let g ∈ Lipb(H−1). Let u be a viscosity subsolution

of (4.2), and v be a viscosity supersolution of (4.2) such that u,−v are weakly sequentially

upper semicontinuous on (0, T ] ×H,

u,−v ≤ C for some C > 0 (5.2)

and

limt→T

(u(t, x) − g(x))+ + (v(t, x) − g(x))− = 0,

uniformly on bounded sets. (5.3)

Assume in addition that either ψ = u or ψ = v satisfies

|ψ(t, x) − ψ(t, y)| ≤ L‖x− y‖−1 (5.4)

for some L ≥ 0 and all t ∈ (0, T ), x, y ∈ H. Then u ≤ v.

Proof. The proof is similar to a typical comparison proof.

Define uµ(t, x) = u(t, x) − µt

and vµ(s, y) = v(s, y) + µs, If u 6≤ v then for some γ > 0

we have for sufficiently small µ > 0

0 < γ < mε := limR→+∞

limη→0

supuµ(t, x)−vµ(s, y)− 1

2ε‖x−y‖2

−1 : ‖x‖, ‖y‖ ≤ R, |t−s| < η.

7

Denote

mε,δ := limη→0

supuµ(t, x) − vµ(t, y) − 1

2ε‖x− y‖2

−1 − δ(‖x‖2 + ‖y‖2) : |t− s| < η,

mε,δ,β := supuµ(t, x) − vµ(s, y) − 1

2ε‖x− y‖2

−1 − δ(‖x‖2 + ‖y‖2) − 1

2β(t− s)2.

As usual we have the following convergencies:

mε = limδ→0

mε,δ, (5.5)

mε,δ = limβ→0

mε,δ,β. (5.6)

For ε, δ, β > 0 we now consider the function

Φ(t, s, x, y) = uµ(t, x) − vµ(s, y)−‖x− y‖2

−1

2ε− (t− s)2

2β− δ(‖x‖2 + ‖y‖2).

Since Φ is weakly sequentially upper semicontinuous, it has a global maximum over (0, T ]×H (which can be assumed to be strict) at some points t, s, x, y, where 0 < t, s, and x, y

are bounded independently of ε for a fixed δ. Using (5.5)-(5.6), we have (see [27]) that

lim supδ→0

lim supβ→0

δ(‖x‖2 + ‖y‖2) = 0 for fixed ε, (5.7)

lim supβ→0

(t− s)2

2β= 0 for fixed ε, δ. (5.8)

It then follows from (5.7), (5.8), the fact that g ∈ Lipb(H−1), and (5.3) that for small µ

and ε, δ, β, we have 0 < t, s < T .

We can assume without loss of generality that v satisfies (5.4). Then, since Φ(t, s, x, x) ≤Φ(t, s, x, y), we obtain

‖x− y‖2−1

2ε≤ L‖x− y‖−1 + δ‖x‖2

which, in light of (5.7), implies

lim supδ→0

lim supβ→0

‖x− y‖−1

ε≤ 2L. (5.9)

We now define

u1(t, x) = uµ(t, x) − 〈BQN(x− y), x〉ε

− ‖QN(x− x)‖2−1

ε

+‖QN (x− y)‖2

−1

2ε− δ‖x‖2

and

v1(s, y) = vµ(s, y) − 〈BQN(x− y), y〉ε

+‖QN(y − y)‖2

−1

ε+ δ‖y‖2.

8

Then

u1(t, x) − v1(s, y) −1

2ε‖PN(x− y)‖2

−1 −1

2β|t− s|2

has a strict global maximum at (t, s, x, y). Now similarly to the proof of Theorem 4.4 of

[31] (see also Theorem 2.1 of [9], Lemma 6.4 of [28] Lemma 3.1 of [30], Theorem 3.8 of

[23], [32], [39] and Theorem 9.3 here) we obtain test functions of type (i), ϕk, ψk with

uniformly continuous derivatives such that

u1(t, x) − ϕk(t, x)

has a global maximum at some point (tk, xk),

v1(s, y) + ψk(s, y)

has a global minimum at some point (sk, yk), and

(tk, xk, u1(tk, xk),

∂ϕk∂t

(tk, xk), Dϕk(tk, xk), D2ϕk(tk, xk)

)→

(t, x, u1(t, x),

t− s

β,BPN (x− y)

ε,XN

), (5.10)

(sk, yk, v1(sk, yk),

∂ψk∂t

(sk, yk), Dψk(sk, yk), D2ψk(sk, yk)

)→

(s, y, v1(s, y),

s− t

β,BPN(y − x)

ε, YN

), (5.11)

M

ε

(I 00 I

)≤(XN 00 YN

)≤ 2

ε

(B −B−B B

), (5.12)

where the above convergencies are in R ×H × R × R ×H2 × L(H) and M is a constant

depending only on ‖B‖. (Most of the arguments are the same and in fact much simpler

since B is compact, PN has a special form here, and u and −v are weakly sequentially

upper semicontinuous.)

Therefore using the definition of viscosity solution and letting k → ∞ we obtain

− µ

t2+t− s

β+

1

2ntr

(σ(t, x)Qσ(t, x)∗(XN +

2BQN

ε+ 2δI)

)

− 1

2‖(σ(t, x)Q

12 )∗(

B(x− y)

ε+ 2δx)‖2 − 〈x, A

∗B(x− y)

ε〉

+ 〈b(t, x), B(x− y)

ε+ 2δx〉 ≥ 0

(5.13)

and

µ

s2+t− s

β− 1

2ntr

(σ(s, y)Qσ(s, y)∗(YN +

2BQN

ε+ 2δI)

)

− 1

2‖(σ(s, y)Q

12 )∗(

B(x− y)

ε− 2δy)‖2 − 〈y, A

∗B(x− y)

ε〉

+ 〈b(s, y), B(x− y)

ε− 2δy〉 ≤ 0.

(5.14)

9

Combining (5.13) and (5.14), using the assumptions, (5.7), (5.8) and (5.9) we thus

have

2µ

T 2≤ σ1(N) + σ2(β) + σ3(δ) + C1

‖x− y‖2−1

ε+ C2

‖x− y‖3−1

ε2, (5.15)

where σ1(N) → 0 as N → ∞ for ε, δ, β fixed, σ2(β) → 0 as β → 0 for ε, δ fixed,

σ3(δ) → 0 as δ → 0 for ε fixed, and C1, C2 are absolute constants. It now remains to send

N → ∞, β → 0, δ → 0, ε → 0 in this order and use (5.9) to obtain a contradiction µ ≤ 0

which completes the proof.

Theorem 5.2. Let (2.1)-(5.1) hold. Let g ∈ Lipb(H−1). Let un be a bounded viscosity

solution of (4.2), and v be a bounded viscosity solution of (4.2)∞ such that

limt→T

|un(t, x) − g(x)| + |v(t, x) − g(x)| = 0, uniformly on bounded sets (5.16)

and

|v(t, x) − v(t, y)| ≤ L‖x− y‖−1 (5.17)

for some L ≥ 0 and all t ∈ (0, T ], x, y ∈ H. Then there exists a constant C independent

of n such that

‖un − v‖∞ ≤ C√n. (5.18)

Proof. Let C1, C2 be the constants from (5.15) and C3 be such that

1

2|tr (σ(t, x)Qσ∗(t, x)B) | ≤ C3 on (0, T ) ×H.

We set

vn = v +1√n

(T − t)(4C1L2 + 8C2L

3 + C3) +2L2

√n.

Then vn is a viscosity supersolution of

(vn)t +1

2ntr(σ(t, x)Qσ(t, x)∗D2vn

)− 1

2‖(σ(t, x)Q

12 )∗Dvn‖2

+ 〈−Ax + b(t, x), Dvn〉 = − 1√n

(4C1L2 + 8C2L

3 + C3).(5.19)

The proof then follows the proof of Theorem 5.1 applied to un and vn, however we now

fix the constant ε = 1/√n. If un is not less than or equal to vn then exactly as before we

obtain that for small µ, δ, β the function

Φ(t, s, x, y) = (un)µ(t, x) − (vn)

µ(s, y) −√n

2‖x− y‖2

−1 −(t− s)2

2β

− δ(‖x‖2 + ‖y‖2)

(5.20)

has a strict global maximum over (0, T ] ×H at some points t, s, x, y, where 0 < t, s < T .

Moreover (5.7),(5.8), and (5.9) (with ε = 1/√n) are satisfied. The rest of the proof is

10

much easier as there is no need now to produce the test functions ϕk and ψk. We use

(5.20) and the definition of viscosity solution to obtain

µ

t2+t− s

β+

1

2ntr(σ(t, x)Qσ∗(t, x)(

√nB + 2δI)

)

− 1

2‖(σ(t, x)Q

12 )∗(

√nB(x− y) + 2δx)‖2 − 〈x,

√nA∗B(x− y)〉

+ 〈b(t, x),√nB(x− y) + 2δx〉 ≥ 0

(5.21)

and

− µ

s2+t− s

β− 1

2‖(σ(s, y)Q

12 )∗(

√nB(x− y) − 2δy)‖2

− 〈y,√nA∗B(x− y)〉 + 〈b(s, y),

√nB(x− y) − 2δy〉

≤ − 1√n

(4C1L2 + 8C2L

3 + C3).

(5.22)

Combining (5.21) and (5.22) and using (5.9) we thus obtain

2µ

T 2≤ σ2(β) + σ3(δ) +

C3√n

+√nC1‖x− y‖2

−1 + nC2‖x− y‖3−1 −

1√n

(4C1L2 + 8C2L

3 + C3)

≤ σ2(β) + σ3(δ)

(5.23)

which gives a contradiction after we send β → 0 and δ → 0.

Therefore we have obtained that un ≤ v + 1√n(T − t)(4C1L

2 + 8C2L3 + C3) + 2L2

√n.

Applying a similar argument to functions vn = v− 1√n(T − t)(4C1L

2 + 8C2L3 +C3)− 2L2

√n

and un produces v− 1√n(T − t)(4C1L

2 +8C2L3 +C3)− 2L2

√n≤ un which, together with the

previous estimate, yields (5.18).

6 Exponential moment estimates

In order to use Theorem 5.2 we need a layer of approximations. For r > 0 we denote by

P rH−1

the projection onto ‖x‖−1 ≤ r in H−1 and then define

σr(t, x) := σ(t, P rH−1

x).

It is easy to see that the σr satisfy (2.3) with the same constants LT and (5.1) on [0, T ]

with some constants Mr. We will denote by Xrn(s) or Xr

n(s; t, x) the solutions of (1.1) with

σ replaced by σr. We will also use X∞n (s) to denote the solution Xn(s) of (1.1).

Exponential moment estimates and various continuity estimates for the processes X rn

will be essential in later sections. Exponential moment estimates are interesting on their

11

own so we decided to state a general result as we could not find estimates (6.3) and (6.4)

in such forms under our assumptions in the literature. We point out that (6.4) is stronger

than what is needed to obtain exponential tightness of the family Xn. We refer the reader

to [12], Chapter 7 for the definition of mild solution.

Proposition 6.1. Let 0 ≤ t ≤ T .

(i) Let b : [0, T ] ×H → H, σ : [0, T ] ×H → L(H) be continuous and such that for every

0 ≤ s ≤ T , x, y ∈ H,

‖b(s, x) − b(s, y)‖, ‖σ(s, x) − σ(s, y)‖ ≤ L‖x− y‖ (6.1)

for some constant L. Assume moreover that

‖σ(s, x)‖ ≤ L(1 + ‖x‖γ) (6.2)

for some γ ∈ [0, 1]. Then there exists a unique mild solution of (1.1). The solution has

trajectories a.s. in C([t, T ];H) and there exist constants c1 > 0, c2 > 0 (depending only

on T, L,Q, γ, with c2 depending also on ‖x‖) such that

IE(

supt≤s≤T

enc1‖Xn(s)‖2(1−γ)) ≤ enc2 if γ ∈ [0, 1) (6.3)

and

IE(

supt≤s≤T

enc1(log(1+‖Xn(s)‖))2) ≤ enc2 if γ = 1. (6.4)

(ii) Let (2.3) be satisfied. If Xn(s), and Yn(s) are solutions of (1.1) with initial conditions

x and y respectively then

IE‖Xn(s) − Yn(s)‖2−1 ≤ C1(T )‖x− y‖2

−1, (6.5)

IE‖Xn(s) − x‖2−1 ≤ C2(‖x‖, T )(s− t), (6.6)

and

IE‖Xn(s) − x‖2 ≤ ωx(s− t) (6.7)

for some modulus ωx.

Proof. (i) The existence, uniqueness, and continuity of trajectories of mild solutions are

standard, see for instance [12], Chapter 7. It remains to show (6.3) and (6.4). To make

things simple we begin with the case γ = 0 since the proof is then particularly transparent.

We will first prove (6.3) for the processes Xmn , where Xm

n is the solution of (1.1) with

A replaced by its Yosida approximation Am. Denote by τk the minimum of T and the exit

12

time of Xmn from the set ‖z‖ ≤ k. Let β > 0 and let α > 0 be a number to be specified

later. Since the Xmn are strong solutions we can apply Ito’s formula to get

enβe−α(s∧τk)(1+‖Xm

n (s)‖2) = enβe−α(s∧τk)(1+‖Xm

n (s∧τk)‖2)

= enβe−αt(1+‖x‖2) −

∫ s∧τk

t

αnβe−αr(1 + ‖Xmn (r)‖2)enβe

−αr(1+‖Xmn (r)‖2)dr

+

∫ s∧τk

t

2nβe−αrenβe−αr(1+‖Xm

n (r)‖2)〈−AmXmn (r) + b(r,Xm

n (r)), Xmn (r)〉dr

+

∫ s∧τk

t

2√nβe−αrenβe

−αr(1+‖Xmn (r)‖2)〈Xm

n (r), σ(r,Xmn (r))Q

12dW (r)〉

+

∫ s∧τk

t

enβe−αr(1+‖Xm

n (r)‖2)tr(σ(r,Xm

n (r))Qσ(r,Xmn (r))∗

[2nβ2e−2αrXm

n (r) ⊗Xmn (r) + βe−αrI

] )dr

≤ enβe−αt(1+‖x‖2) +

∫ s∧τk

t

enβe−αr(1+‖Xm

n (r)‖2)(−α + C(β))nβe−αr(1 + ‖Xmn (r)‖2)dr

+

∫ s

t

2√nβe−αrenβe

−αr(1+‖Xmn (r)‖2)1[t,τk]〈Xm

n (r), σ(r,Xmn (r))Q

12dW (r)〉

(6.8)

for some absolute constant C(β), nondecreasing in β and also depending on various pa-

rameters of b, σ, Q. Therefore choosing α = C(β) + 1 we obtain

enβe−α(s∧τk)(1+‖Xm

n (s∧τk)‖2) +

∫ s∧τk

t

enβe−αr(1+‖Xm

n (r)‖2)nβe−αr(1 + ‖Xmn (r)‖2)dr

≤ enβe−αt(1+‖x‖2) +

∫ s

t

2√nβe−αrenβe

−αr(1+‖Xmn (r)‖2)1[t,τk ]〈Xm

n (r), σ(r,Xmn (r))Q

12dW (r)〉.

(6.9)

Taking expectation in (6.9), we get

IEenβe−α(s∧τk)(1+‖Xm

n (s∧τk)‖2) + IE

∫ s∧τk

t

enβe−αr(1+‖Xm

n (r)‖2)nβe−αr(1 + ‖Xmn (r)‖2)dr

≤ enβe−αt(1+‖x‖2).

(6.10)

We now take α which satisfies (6.9) and (6.10) for β = 2. This α also satisfies these

inequalities for β = 1. It then follows from (6.9) applied with this α and β = 1 that

supt≤u≤s

ene−α(u∧τk)(1+‖Xm

n (u∧τk)‖2) ≤ ene−αt(1+‖x‖2)

+ supt≤u≤s

∣∣∣∣∫ u

t

2√nβe−αrene

−αr(1+‖Xmn (r)‖2)1[t,τk]〈Xm

n (r), σ(r,Xmn (r))Q

12dW (r)〉

∣∣∣∣

13

and therefore using Burkholder-Davis-Gundy inequality (see for instance [13]) and (6.10)

with β = 2 we can estimate

IE supt≤u≤s

ene−α(u∧τk)(1+‖Xm

n (u∧τk)‖2) ≤ ene−αt(1+‖x‖2)

+

(IE sup

t≤u≤s

∣∣∣∣∫ u

t

2√nβe−αrene

−αr(1+‖Xmn (r)‖2)1[t,τk]〈Xm

n (r), σ(r,Xmn (r))Q

12dW (r)〉

∣∣∣∣2) 1

2

≤ ene−αt(1+‖x‖2) +

(IE

∫ s

t

C1ne−2αre2ne

−αr(1+‖Xmn (r)‖2)‖Xm

n (r)‖21[0,τk]dr

) 12

≤ ene−αt(1+‖x‖2) + C2e

ne−αt(1+‖x‖2) ≤ C3en(1+‖x‖2).

(6.11)

We thus have obtained

IE supt≤s≤T

ene−αT ‖Xm

n (s∧τk)‖2 ≤ C3en(1+‖x‖2). (6.12)

Letting k → +∞ in (6.12) gives

IE supt≤s≤T

ene−αT ‖Xm

n (s)‖2 ≤ C3en(1+‖x‖2).

It now remains to use that

limm→∞

IE(

supt≤s≤T

‖Xmn (s) −Xn(s)‖2

)= 0.

(see [12], page 197) to conclude the proof.

For 0 < γ ≤ 1 we repeat the same arguments applied to the function

enβe−αsh(‖x‖),

for properly chosen α, where h is a smooth, symmetric, nondecreasing function such that

h(0) = 1 and

h(r) = r2(1−γ) for big r if 0 < γ < 1,

and

h(r) = (log r)2 for big r if γ = 1.

We leave the details to the readers.

(ii) Estimates (6.5)-(6.7) are standard and their proofs can be found for instance in [30],

pages 23-25.

Corollary 6.2. Let 0 ≤ t ≤ T, r ∈ (0,+∞], and let (2.3) be satisfied. Then

IE(

supt≤s≤T

enc1(log(1+‖Xrn(s)‖))2) ≤ enc2 , (6.13)

where c1, c2 are from Proposition 6.1 and are independent of r. Moreover estimates (6.5)

-(6.7) are satisfied for Xrn(s) uniformly in r.

14

7 Laplace limit at a single time T

The main goal of this section is to show that the functions urn defined by (7.1) converge

as n→ +∞.

Theorem 7.1. Let 0 < r < +∞, let (2.1)-(2.3) hold and let g ∈ Lipb(H−1). There exist

a constant C1 and, for every R > 0, a constant C2 = C2(R) (both possibly depending on

n) such that the function

urn(t, x) = − 1

nlog IE

(e−ng(X

rn(T ))

)(7.1)

satisfies

|urn(t, x)−urn(s, y)| ≤ C1‖x−y‖−1 +C2(max‖x‖, ‖y‖)|t−s| 12 for x, y ∈ H, t, s ∈ [0, T ].

(7.2)

The function urn is the unique, bounded viscosity solutions of the HJB equation

(urn)t +12n

tr (σr(t, x)Qσr(t, x)∗D2urn) − 1

2‖(σr(t, x)Q

12 )∗Durn‖2

+〈−Ax + b(t, x), Durn〉 = 0,

urn(T, x) = g(x) in (0, T ) ×H

(7.3)

satisfying (5.3).

Proof. Estimate (7.2) follows directly from Corollary 6.2 and (6.5), (6.6), and the Markov

property of the process Xrn. In particular urn satisfies (5.3). The proof that urn is a viscosity

solution of (7.3) is technical but rather standard. It is similar to the proof of Theorem

4.1 in [30] or Theorem 6.3 in [25]. We will sketch its main points, explaining only the

subsolution property as the supersolution part is similar.

Suppose that urn−h(‖ ·‖)−ϕ has a local maximum at (t, x) (which we can assume to

be strict and global). Since urn is bounded by Remark 4.3 we can also assume that h, h′, h′′

and ϕ are bounded and uniformly continuous. Denote ψ(s, y) = h(‖y‖) + ϕ(s, y). Then

urn(t + ε,Xrn(t+ ε)) − ψ(t+ ε,Xr

n(t + ε)) ≤ urn(t, x) − ψ(t, x).

Therefore, setting vrn = e−nurn we have

vrn(t+ ε,Xn(t + ε))

vrn(t, x)≥ e−nψ(t+ε,Xr

n(t+ε))enψ(t,x).

Taking the expectation of both sides of the above inequality and using the Markov prop-

erty of Xrn(s) yields

e−nψ(t,x) ≥ IEe−nψ(t+ε,Xrn(t+ε)).

15

Therefore, applying Ito’s formula and using the monotonicity of A (see Lemma 4.3 of

[30]), we obtain

0 ≥ IE1

ε

e−nψ(t+ε,Xr

n(t+ε)) − e−nψ(t,x)

≥ IE1

ε

∫ t+ε

t

−e−nψ(τ,Xrn(τ))nϕt(τ,X

rn(τ))dτ

−IE1

ε

∫ t+ε

t

ne−nψ(τ,Xrn(τ))〈Dψ(τ,Xr

n(τ)), b(τ,Xrn(τ))〉dτ

+IE1

ε

∫ t+ε

t

ne−nψ(τ,Xrn(τ))〈Xr

n(τ), A∗Dϕ(τ,Xr

n(τ))〉dτ

+IE1

2nε

∫ t+ε

t

e−n(ψ(τ,Xrn(τ))tr

[σr(τ,X

rn(τ))Qσr(τ,X

rn(τ))

∗(

n2Dψ(τ,Xrn(τ)) ⊗Dψ(τ,Xr

n(τ)) − nD2ψ(τ,Xrn(τ))

)]dτ. (7.4)

We now need to pass to the limit as ε → 0 in (7.4). This is very technical however it

is not really difficult using (6.7) and the properties of test functions. (The reader can

consult the proofs of Theorem 4.1 of [30], Theorem 6.3 of [25] and Theorem 5.4 of [26] for

such arguments.) Sending ε→ 0 in (7.4) and denoting ψ(s, y) = h(‖y‖) +ϕ(s, y) we then

obtain

0 ≥ −ne−nψ(t,x)

(ψt(t, x) − 〈x,A∗Dϕ(t, x)〉 + 〈b(t, x), Dψ(t, x)〉

+1

2ntr[σr(t, x)Qσr(t, x)

∗ (−nDψ(t, x) ⊗Dψ(t, x) +D2ψ(t, x))])

(7.5)

which establishes that urn is a viscosity subsolution of (7.3). Since urn is Lipschitz continuous

in x the uniqueness follows from Theorem 5.1 (once we also establish that urn is a viscosity

supersolution).

If we pass to the limit as n→ +∞ in (7.3) we obtain the first order equation

(vr)t − 12‖(σr(t, x)Q

12 )∗Dvr‖2 + 〈−Ax + b(t, x), Dvr〉 = 0,

vr(T, x) = g(x) in (0, T ) ×H

(7.6)

Since

−1

2‖(σr(t, x)Q

12 )∗p‖2 = inf

z∈H〈σr(t, x)Q

12 z, p〉 +

1

2‖z‖2

we can interpret this equation as the dynamic programming equation associated with an

optimal control problem. The control problem is the following.

For 0 ≤ t ≤ T, x ∈ H and z(·) ∈ L2(t, T ;H), the state equation is

ddsXr(s) = −AXr(s) + b(s,Xr(s)) + σr(s,X

r(s))Q12 z(s) for s > t,

Xr(t) = x,

(7.7)

16

and we want to minimize the cost functional

J(t, x; z(·)) =

∫ T

t

1

2‖z(s)‖2ds+ g(Xr(T )) (7.8)

over all z(·) ∈ L2(t, T ;H). However, since g is bounded we can restrict the minimization

to the functions z(·) such that

z(·) ∈ Mt := z(·) ∈ L2(t, T ;H) :

∫ T

t

‖z(s)‖2ds ≤ K = 2‖g‖∞. (7.9)

Therefore if vr is the value function for the problem then

vr(t, x) = infz(·)∈L2(t,T ;H)

J(t, x; z(·)) = infz(·)∈Mt

J(t, x; z(·)). (7.10)

We first collect several properties of solutions of (7.7).

Lemma 7.2. Let r ∈ (0,+∞] and let (2.3) be satisfied. Let let 0 ≤ t ≤ T and z(·) ∈L2(t, T ;H). Then:

(i) There exists a unique mild solution X r(s) of (7.7) in C([t, T ];H). Moreover there

exists a constant C1 = C1(R, T, ‖z(·)‖L2(t,T ;H)) (independent of r) such that if ‖x‖ ≤ R

then

supt≤s≤T

‖Xr(s)‖ ≤ C1. (7.11)

(ii) There exists a constant C2 = C2(T, ‖z(·)‖L2(t,T ;H)), such that if Xr(s), and Y r(s) are

solutions of (7.7) with initial conditions x and y respectively then

‖Xr(s) − Y r(s)‖−1 ≤ C2‖x− y‖−1, (7.12)

(iii) There exists a constant C3 = C3(R, ‖z(·)‖L2(t,T ;H), T ), such that if ‖x‖ ≤ R then

‖Xr(s) − x‖−1 ≤ C3(s− t)12 , (7.13)

and for every x ∈ H there exists a modulus ωx, depending on ‖z(·)‖L2(t,T ;H), T such that

‖Xr(s) − x‖ ≤ ωx(s− t). (7.14)

Proof. The existence and uniqueness of a mild solution is standard, see for instance [33],

Chapter 2, Proposition 5.3. The proofs of the estimates are also standard, similar to and

much easier than the proofs of the corresponding properties of solutions of the stochastic

differential equation (1.1) given in Proposition 6.1. Some estimates in a slightly different

case are shown in [33], Chapters 2 and 6, and [11]. Here we will only indicate how to

obtain (7.12).

17

To do this, since

d

ds‖Xr(s) − Y r(s)‖2

−1 = −2〈A∗B(Xr(s) − Y r(s)), Xr(s) − Y r(s)〉

+ 2〈B(Xr(s) − Y r(s)), b(s,Xr(s)) − b(s, Y r(s)) + (σr(s,Xr(s)) − σr(s, Y

r(s)))Q12 z(s)〉

we have

‖Xr(s) − Y r(s)‖2−1 ≤ ‖x− y‖2

−1 + C

∫ s

t

‖Xr(τ) − Y r(τ)‖2−1(1 + ‖z(τ)‖)dτ

and then Gronwall’s inequality yields

‖Xr(s) − Y r(s)‖2−1 ≤ ‖x− y‖2

−1eC

R s

t(1+‖z(τ)‖)dτ ≤ C2‖x− y‖2

−1.

We can now identify vr with the solution of the HJB equation (7.6).

Theorem 7.3. Let r ∈ (0,+∞], let (2.1)-(2.3) hold and let g ∈ Lipb(H−1). There exist

a constant D1 and, for every R > 0, a constant D2 = D2(R) such that the value function

vr defined by (7.10) satisfies

|vr(t, x)−vr(s, y)| ≤ D1‖x−y‖−1 +D2(max‖x‖, ‖y‖)|t−s| 12 for x, y ∈ H, t, s ∈ [0, T ].

(7.15)

If r ∈ (0,+∞), the function vr is the unique, bounded viscosity solution of the HJB

equation (7.6) satisfying (5.3).

Proof. The Lipschitz continuity in x is a direct consequence of (7.12) and the fact that

g ∈ Lipb(H−1). To show the continuity in time let x ∈ H and s < t. For ε = |s − t| 12 let

zε(·) ∈Mt be such that

vr(t, x) ≥ J(t, x; zε(·)) − ε.

Extending zε(·) by 0 to [s, T ] we can assume that zε(·) ∈Ms. Therefore

vr(s, x) − vr(t, x) ≤ J(s, x; zε(·)) − J(t, x; zε(·)) + ε

≤ g(Xr(T ; s, x)) − g(Xr(T ; t, x)) + ε ≤ C2C3L|s− t| 12 + |s− t| 12 ,

where we have used (7.12), (7.13), and L is the Lipschitz constant of g. On the other hand

if if now zε(·) ∈Ms is such that

vr(s, x) ≥ J(s, x; zε(·)) − ε

then zε(·) ∈Mt and we again have

vr(s, x) − vr(t, x) ≥ J(s, x; zε(·)) − ε− J(t, x; zε(·))≥ g(Xr(T ; s, x)) − g(Xr(T ; t, x)) − ε ≥ C2C3L|s− t| 12 + |s− t| 12 .

18

This completes the proof of (7.15). In particular vr satisfies (5.3).

The proof that vr is a viscosity solution of (7.6) follows well known arguments (see for

instance the proof of Theorem 4.1 in [11] and the proof of Theorem 3.2 in [33], Chapter

6). However the proof of the supersolution property has a small technical difference due

to the unboundedness of the controls z(·). We explain it below.

First we notice that the dynamic programming principle holds, i.e. if 0 ≤ t < t+ ε ≤T, x ∈ H then

vr(t, x) = infz(·)∈L2(t,T ;H)

∫ t+ε

t

‖z(s)‖2ds+ vr(t+ ε,Xr(t + ε))

.

Let now vr + h(‖x‖) + ϕ have a local minimum at (t, x). By the dynamic programming

principle for every 0 < ε < T − t there exists a control zε(·) such that.

vr(t, x) ≥∫ t+ε

t

‖zε(s)‖2ds+ vr(t+ ε,Xrε (t+ ε)) − ε2

Denote ψ(s, y) = −ϕ(s, y)− h(‖y‖). A version of Lemma 4.3 of [30] which we used in the

stochastic case is also true here. The formula for ϕ is proved in [33], Chapter 2, Proposition

5.5 (see also [11], page 437) and it can be easily shown using Yosida approximations,

similarly to the proof of Proposition 5.5 of [33], Chapter 2 (see also [33], Chapter 2,

Lemma 3.4 and [11], page 437) that, denoting h(x) := h(‖x‖),

h(Xrε (t+ ε)) ≤ h(x) +

∫ t+ε

t

〈b(s,Xrε (s)) + σr(s,X

rε (s))Q

12 zε(s), Dh(X

rε (s))〉ds.

Using this and Proposition 5.5 of [33], Chapter 2 we have

ε ≥ 1

ε(vr(t + ε,Xr

ε (t + ε)) − vr(t, x)) +1

2ε

∫ t+ε

t

‖zε(s)‖2ds

≥ 1

ε(−ϕ(t+ ε,Xr

ε (t+ ε)) + ϕ(t, x) − h(‖Xrε (t + ε)‖) + h(‖x‖)) +

1

2ε

∫ t+ε

t

‖zε(s)‖2ds

≥ 1

ε

∫ t+ε

t

[− ϕt(s,X

rε (s)) + 〈Xr

ε (s), A∗Dϕ(s,Xr

ε (s))〉

+〈b(s,Xrε (s)) + σr(s,X

rε (s))Q

12 zε(s), Dψ(s,Xr

ε (s))〉 +1

2‖zε(s)‖2

]ds

≥ 1

ε

∫ t+ε

t

[− ϕt(s,X

rε (s)) + 〈Xr

ε (s), A∗Dϕ(s,Xr

ε (s))〉

+〈b(s,Xrε (s)), Dψ(s,Xr

ε (s))〉 −1

2‖(σr(s,Xr

ε (s))Q12 )∗Dψ(s,Xr

ε (s))‖2

]ds

(7.16)

Using (7.14) and the fact that ‖zε(·)‖L2(t,t+ε;H) are bounded we can now easily pass to the

limit as ε→ 0 in (7.16) to obtain

0 ≥ ψt(t, x) + 〈x,A∗Dϕ(t, x)〉 + 〈b(t, x), Dψ(t, x)〉 − 1

2‖(σr(t, x)Q

12 )∗Dψ(t, x)‖2

19

which shows that vr is a supersolution. The uniqueness follows from Theorem 5.1 since

vr is Lipschitz continuous in x.

Combining Theorems 5.2, 7.1 and 7.3 we have therefore proved the following result.

Corollary 7.4. Let r ∈ (0,+∞], let (2.1)-(2.3) hold and let g ∈ Lipb(H−1). Then

limn→+∞

‖urn − vr‖∞ = 0. (7.17)

We can now establish the existence of Laplace limit at a single time for solutions of

the original equations. For a function f we denote

Vn(t)f(x) =1

nlog IE

(e−nf(Xn(t)) : Xn(0) = x

),

where Xn is the solution of (1.1) on [0,+∞).

Lemma 7.5. Let t ≥ 0, let (2.1)-(2.3) hold and let f ∈ Lipb(H−1). Let Xn(t) =

Xn(t; 0, x). Then the Laplace limit

V (t)f(x) := limn→∞

Vn(t)f(x) = −v+∞(0, x) (7.18)

exists and the convergence is uniform on bounded subsets of H, where v+∞ is defined with

terminal time T = t and g = f . In particular V (t)f ∈ Lipb(H−1).

Proof. First of all we notice from (7.11) that obviously for every M > 0, v+∞(0, x) =

vr(0, x) for ‖x‖ ≤M if r ≥ r0 for some r0 = r0(M). Therefore, in light of (7.17) (applied

with T = t) it is enough to show that for every M > 0 there exists r > r0 such that

limn→∞

sup‖x‖≤M

|Vn(t)f(x) + urn(0, x)| = 0. (7.19)

Let ‖x‖ ≤M . Estimate (6.4) yields that if Ωn1 = ω : sup0≤s≤t ‖Xn(s)‖ > r then

ec1n(log r)2P(Ωn

1 ) ≤ ec2n

for some constants c1, c2 > 0, where c2 also depends on M . Therefore there exists r > r0,

depending on M , such that

P(Ωn1 ) ≤ e−3‖f‖∞n. (7.20)

Denote Ωn2 = Ω \ Ωn

1 . On Ωn2 we have Xn = Xr

n, where Xrn(·) = Xr

n(·; 0, x). Denote

An = IE[e−nf(Xn(t))].

Then

IE[e−nf(Xrn(t))] = An + IE[e−nf(Xr

n(t))χΩn1] − IE[e−nf(Xn(t))χΩn

1].

20

Therefore

− 1

nlog(An + IE[e−nf(Xr

n(t))χΩn1]) ≤ urn(0, x)

≤ − 1

nlog(An − IE[e−nf(Xn(t))χΩn

1])

This implies that

−Vn(t)f(x) − 1

nlog(1 +

IE[e−nf(Xrn(t))χΩn

1]

An) ≤ urn(0, x)

≤ −Vn(t)f(x) − 1

nlog(1 −

IE[e−nf(Xn(t))χΩn1]

An)

However (7.20) yields that

IE[e−nf(Xrn(t))χΩn

1]

An,IE[e−nf(Xn(t))χΩn

1]

An≤ e−2‖f‖∞n

e−‖f‖∞n= e−‖f‖∞n.

Therefore, using the fact that −2s ≤ log(1 − s), and log(1 + s) ≤ s for small s > 0, we

obtain

−Vn(t)f(x) − 1

ne−n‖f‖∞ ≤ urn(0, x) ≤ −Vn(t)f(x) +

2

ne−n‖f‖∞ .

This proves the claim.

However we need to extend this result to a larger class of functions f .

Lemma 7.6. Let (2.1)-(2.3) be satisfied. Let f, fm be weakly sequentially continuous on

H and such that ‖fm‖∞ ≤ M for m ≥ 1, and let fm → f uniformly on bounded subsets

of H. Then for every R > 0, ε > 0 there exists m0 such that if m,n ≥ m0 then

sup‖x‖≤R

|Vn(t)fm(x) − Vn(t)f(x)| ≤ ε. (7.21)

Proof. We argue similarly as in the proof of the previous lemma. Let ‖x‖ ≤ R and let

R1 > R. Denote

Ωn = ω : ‖Xn(t)‖ > R1.

Again, by (6.4), we can choose R1 such that

P(Ωn) ≤ e−3Mn. (7.22)

Let m1 be such that for m ≥ m1

sup‖x‖≤R1

|fm(x) − f(x)| ≤ ε

2. (7.23)

21

We have

IEe−nfm(Xn(t)) = IEe−nf(Xn(t)) + IE((e−nfm(Xn(t)) − e−nf(Xn(t)))χΩ\Ωn

)

+ IE((e−nfm(Xn(t)) − e−nf(Xn(t)))χΩn

).

Therefore, using (7.22) and (7.23), we have

IEe−nf(Xn(t)) + IEe−nf(Xn(t))(e−nε2 − 1) − 2e−2Mn

≤ IEe−nfm(Xn(t))

≤ IEe−nf(Xn(t)) + IEe−nf(Xn(t))(enε2 − 1) + 2e−2Mn

and so

1

nlog IEe−nf(Xn(t)) − ε

2+

1

nlog

(1 − 2e−2Mn

IEe−nf(Xn(t))e−nε2

)

≤ 1

nlog IEe−nfm(Xn(t))

≤ 1

nlog IEe−nf(Xn(t)) +

ε

2+

1

nlog

(1 +

2e−2Mn

IEe−nf(Xn(t))enε2

).

Using (7.22) again and the inequalities −2s ≤ log(1 − s), and log(1 + s) ≤ s for small

s > 0, we obtain for ε < M, m ≥ m1, and n big enough that

1

nlog IEe−nf(Xn(t)) − ε

2− 4

ne−Mne

nε2

≤ 1

nlog IEe−nfm(Xn(t))

≤ 1

nlog IEe−nf(Xn(t)) +

ε

2+

2

ne−Mne−

nε2

which proves the claim.

We remark that since B is compact, weak sequential continuity on H is equivalent

to uniform continuity in the ‖ · ‖−1 norm on bounded subsets of H.

Proposition 7.7. Let (2.1)-(2.3) be satisfied and let f be bounded and weakly sequentially

continuous on H. Then:

(i) For every n the function Vn(t)f is weakly sequentially continuous on H.

(ii) For every x ∈ H

V (t)f(x) = limn→+∞

Vn(t)f(x)

exists and is uniform on bounded subsets of H. In particular V (t)f(x) is weakly sequen-

tially continuous on H.

22

(iii) If fn are weakly sequentially continuous on H, such that ‖fn‖∞ ≤ M for n ≥ 1 and

fn → f uniformly on bounded subsets of H then

limn→+∞

Vn(t)fn(x) = V (t)f(x) (7.24)

uniformly on bounded subsets of H.

Proof. (i) We fix R > 0 and take x, y ∈ H such that ‖x‖, ‖y‖ ≤ R. Let Xn, and Yn be

solutions of (1.1) such that Xn(0) = x and Yn(0) = y. It follows for instance from seconm

moment estimates (or from (6.4)) that

P(max(‖Xn(t)‖, ‖Yn(t)‖) > m) ≤ C(R, T )1

m2. (7.25)

Let ρm be the modulus of continuity of e−nf in the ‖ · ‖−1 norm on the set ‖z‖ ≤ m.Then, using (7.25), (6.5) and the concavity of ρm, we obtain

∣∣IEe−nf(Xn(t)) − IEe−nf(Yn(t))∣∣ ≤ 2en‖f‖∞

m2+ IEρm(‖Xn(t) − Yn(t)‖−1)

≤ 2en‖f‖∞

m2+ ρm(IE‖Xn(t) − Yn(t)‖−1) ≤

2en‖f‖∞

m2+ ρm(C‖x− y‖−1)

which shows the weak sequential continuity of enVn(t)f and hence the weak sequential

continuity of Vn(t)f .

(ii) We notice that we can find functions fm ∈ Lipb(H−1) which converge to f uniformly

on bounded subsets of H. To do this for every m ≥ 1 we first define f : H−1 → R by

f(x) =

f(x) if ‖x‖ ≤ m,

−‖f‖∞ otherwise.

Then f is upper semi-continuous on H−1 and if we take for δ > 0 its δ sup-convolution

fδ(x) = supy∈H−1

f(y) − ‖x− y‖2

−1

δ

then fδ ∈ Lipb(H−1) and it is easy to see, since the set ‖x‖ ≤ m is compact in H−1,

that fδ → f = f uniformly on ‖x‖ ≤ m. Therefore for some small δ = δ(m), denoting

fm := fδ(m), we can have sup‖x‖≤m |fm(x) − f(x)| ≤ 1/m.

Having defined the functions fm it follows from Lemma 7.6 that for ε, R > 0 there

exists n0 such that if m,n ≥ m0 then (7.21) is satisfied. Therefore if n,m1, m2 ≥ m0 we

have

sup‖x‖≤R

|Vn(t)fm1(x) − Vn(t)fm2(x)| ≤ 2ε. (7.26)

23

However fm1 , fm2 ∈ Lipb(H−1) so using Corollary 7.5 we can pass to the limit as n→ +∞in (7.26) to obtain

sup‖x‖≤R

|V (t)fm1(x) − V (t)fm2(x)| ≤ 2ε (7.27)

for all m1, m2 ≥ n0. Therefore limm→+∞ V (t)fm(x) exists uniformly on bounded subsets

of H (which is also clear from the representation formula for V (t)fm(x)). Again, using

the convergence of Vn(t)fm0 as n→ +∞ provided by Corollary 7.5, we obtain from (7.21)

that for n ≥ n1 = n1(m0)

sup‖x‖≤R

|V (t)fm0(x) − Vn(t)f(x)| ≤ 2ε. (7.28)

Combining (7.27) and (7.28) we finally have that for n ≥ n1

sup‖x‖≤R

| limm→+∞

V (t)fm(x) − Vn(t)f(x)| ≤ 4ε, (7.29)

which shows that

V (t)f(x) = limn→+∞

Vn(t)f(x) = limm→+∞

V (t)fm(x) (7.30)

and the convergence is uniform on bounded sets of H. The weak sequential continuity of

V (t)f follows from (7.30) and part (i).

(iii) Convergence (7.24) follows directly from part (ii) and Lemma 7.6 since

|Vn(t)fn(x) − V (t)f(x)| ≤ |Vn(t)fn(x) − Vn(t)f(x)| + |Vn(t)f(x) − V (t)f(x)|.

It is also easy to see from the above that if f is bounded and weakly sequentially

continuous on H then V (t)f(x) has a representation

V (t)f(x) = − infz(·)∈L2(0,t;H)

∫ t

0

1

2‖z(s)‖2ds+ f(X(t; 0, x))

. (7.31)

8 Large deviation principle

In this section we will show how to use the program developed in [19] to prove that the

sequence Xn(·) satisfies large deviation principle in C([0,+∞);H−1) (equipped with

the topology of local uniform convergence), where Xn(s) satisfies (1.1) on [0,+∞) and

Xn(0) = x ∈ H.

Following [19] we denote by DH−1 [0,+∞) the space of H−1-valued right continuous

with left limit functions on [0,+∞) equipped with Skorohod topology.

24

We will first show that Xn(·) satisfies large deviation principle in DH−1 [0,+∞).

According to Corollary 4.29 of [19] in order to show that Xn(·) satisfies the large

deviation principle in DH−1[0,+∞) it is enough to prove that the sequence Xn(·) is

exponentially tight in DH−1[0,+∞) and that for each 0 ≤ t1 ≤ ... ≤ tm and f1, ..., fm ∈Cb(H−1) the limit

limn→+∞

1

nlog IE

[e−n(f1(Xn(t1))+...+fm(Xn(tm)))

](8.1)

exists. The existence of this limit is now a rather easy consequence of Proposition 7.7

and the Markov property of the processes Xn(s). (Recall again that functions in Cb(H−1)

are weakly sequentially continuous on H.) Indeed using the Markov property of Xn(·) we

have

IE[e−n(f1(Xn(t1))+...+fm(Xn(tm)))

]

= IE[IE[e−n(f1(Xn(t1))+...+fm(Xn(tm)))|Ftm−1

]]

= IE[e−n(f1(Xn(t1))+...−Vn(tm−tm−1)fm(Xn(tm−1)))

]

= ... = IE[e−n(f1−Vn(t2−t1)(f2−...−Vn(tm−tm−1)fm)...)(Xn(t1))

]

= enVn(t1)(f1−Vn(t2−t1)(f2−...−Vn(tm−tm−1)fm)...)(x).

By Proposition 7.7 we know that the functions Vn(tm−tm−1)fm are uniformly bounded,

weakly sequentially continuous and they converge uniformly on bounded subsets of H to

V (tm − tm−1)fm, and then Vn(tm−1 − tm−2)(fm−1 − Vn(tm − tm−1)fm) are also uniformly

bounded, weakly sequentially continuous and converge uniformly on bounded subsets of

H to V (tm−1 − tm−2)(fm−1 − V (tm − tm−1)fm) and so on. Continuing this process we

therefore we obtain

limn→+∞

1

nlog IE

[e−n(f1(Xn(t1))+...+fm(Xn(tm)))

]

= V (t1)(f1 − V (t2 − t1)(f2 − ...− V (tm − tm−1)fm)...)(x). (8.2)

It remains to show that the sequence Xn(·) is exponentially tight in DH−1 [0,+∞).

This will be accomplished with the help of the following theorem.

Theorem 8.1. ([19], Theorem 4.4) Let Z be a complete, separable metric space. The

sequence of Z-valued processes Xn(·) is exponentially tight in DZ[0,+∞) if and only if:

(i) (exponential compact containment) for every T,M > 0 there exists a compact set

KM,T ⊂ Z such that

lim supn→+∞

1

nlog P(there exists 0 ≤ t ≤ T such that Xn(t) 6∈ KM,T) ≤ −M ; (8.3)

25

(ii) there exists a family of functions A ⊂ C(Z) that is closed under additions and

isolates points in Z such that for each f ∈ A, f(Xn) is exponentially tight in

DR[0,+∞).

We refer the reader to Definition 3.18 in [19] for the definition of a family isolating

points. Condition (i) is called the exponential compact containment condition.

We need to find a family satisfying Theorem 8.1(ii) with Z = H−1.

Lemma 8.2. Let

A = m∑

i=1

fi(‖x− xi‖−1) : m ∈ N, xi ∈ H, fi ∈ C2([0,+∞)),

f ′i(0) = 0, fi, f

′i , f

′′i are bounded

.

Then A is closed under addition and isolates points in H−1. Moreover if f ∈ A then for

every r > 0 there exists C(f, r) ≥ 0 such that

sup‖x‖≤r

‖A∗Df(x)‖ ≤ C(f, r). (8.4)

Proof. The fact that A isolates points in H−1 is an easy consequence of the definition of

A and the definition of a family isolating points. As regards (8.4) we have

sup‖x‖≤r

‖A∗Df(x)‖ ≤ sup‖x‖≤r

m∑

i=1

‖A∗B(x− xi)‖f ′i(‖x− xi‖−1)

‖x− xi‖−1

≤ C(f, r).

We can now prove that Xn(·) is exponentially tight in DH−1 [0,+∞).

Theorem 8.3. Let (2.1)-(2.3) be satisfied. Then the sequence Xn(·) is exponentially

tight in DH−1 [0,+∞).

Proof. The fact that the sequence satisfies the exponential compact containment con-

dition (8.3) follows from (6.4). Indeed taking KM,T = ‖x‖ ≤ r we have that KM,T is

compact in H−1 and moreover by (6.4)

1

nlog P(there exists 0 ≤ t ≤ T such that Xn(t) 6∈ KM,T)

≤ 1

nlog(en(c2−c1(log r)2)

)= c2 − c1(log r)2 ≤ −M

if r = e√

(M+c2)/c1 .

Therefore it remains to show that condition (ii) of Theorem 8.1 holds for the family

A of Lemma 8.2. Let f ∈ A. According to [19], Theorem 4.1 it is enough to show that for

26

T, s > 0, λ ∈ R there exist random variables γn(s, λ, T ), nondecreasing in s, such that for

0 ≤ t ≤ t+ s ≤ T

IE[enλ(f(Xn(t+s))−f(Xn(t)))|Ft] ≤ IE[eγn(s,λ,T )|Ft], (8.5)

and

lims→0

lim supn→+∞

1

nlog IE[eγn(s,λ,T )] = 0. (8.6)

Let r > 0 be big enough so that if

Ωn1 = ω : there exists 0 ≤ t ≤ T such that ‖Xn(t)‖ > r

then

P(Ωn1 ) ≤ e−2n|λ|‖f‖∞ . (8.7)

Denote Ωn2 = Ω \ Ωn

1 . Then, applying Ito’s formula and Lemma 4.3 of [30] as we did in

the proof of Theorem 7.1, and using (8.4), we obtain

IE[enλ(f(Xn(t+s))−f(Xn(t)))|Ft]

≤ IE[enλ(f(Xn(t+s))−f(Xn(t)))χΩn2|Ft] + IE[e2n|λ|‖f‖∞χΩn

1|Ft]

≤ IE

[exp

(nλ

(∫ t+s

t

(〈A∗Df(Xn(τ)), Xn(τ)〉 + 〈Df(Xn(τ)), b(τ,Xn(τ))〉)dτ

+

∫ t+s

t

(1

2ntr(σ(τ,Xn(τ))Qσ(τ,Xn(τ))

∗D2f(Xn(τ)))

+λ

2‖(σ(τ,Xn(τ))Q

12 )∗Df(Xn(τ))‖2

−λ2‖(σ(τ,Xn(τ))Q

12 )∗Df(Xn(τ))‖2

)dτ

+

∫ t+s

t

1√n〈Df(Xn(τ)), σ(τ,Xn(τ))Q

12dW (τ)〉

))χΩn

2|Ft

]

+IE[e2n|λ|‖f‖∞χΩn1|Ft]

≤ IE

[(exp

(∫ t+s

t

√nλ〈Df(Xn(τ)), σ(τ,Xn(τ))Q

12dW (τ)〉

−∫ t+s

t

nλ2

2‖(σ(τ,Xn(τ))Q

12 )∗Df(Xn(τ))‖2dτ

)

+enC(f,r,λ)s

)χΩn

2|Ft

]+ IE[e2n|λ|‖f‖∞χΩn

1|Ft]

≤ IE

[exp

(∫ t+s

t

√nλ〈Df(Xn(τ)), σ(τ,Xn(τ))Q

12dW (τ)〉

−∫ t+s

t

nλ2

2‖(σ(τ,Xn(τ))Q

12 )∗Df(Xn(τ))‖2dτ

)|Ft

]

+IE[enC(f,r,λ)s + e2n|λ|‖f‖∞χΩn1|Ft] (8.8)

27

for some positive constant C(f, r, λ). However it is easy to see using Ito’s formula that

the process

E(u) = exp

(∫ u

t

√nλ〈Df(Xn(τ)), σ(τ,Xn(τ))Q

12dW (τ)〉

−∫ u

t

nλ2

2‖(σ(τ,Xn(τ))Q

12 )∗Df(Xn(τ))‖2dτ

)

is the solution of the stochastic differential equation

dE(u) =√nλE(u)〈Df(Xn(u)), σ(τ,Xn(u))Q

12dW (u)〉

with E(t) = 1, i.e.

E(u) = 1 +

∫ u

t

√nλE(τ)〈Df(Xn(τ)), σ(τ,Xn(τ))Q

12dW (τ)〉.

In particular it is a martingale with respect to the filtration Fu, so IE[E(t + s)|Ft] = 1.

Therefore we can take

γn(s, λ, T ) = log(enC(f,r,λ)s + e2n|λ|‖f‖∞χΩn

1

).

Then, by (8.7),

1

nlog IE[eγn(s,λ,T )] ≤ 1

nlog(enC(f,r,λ)s + 1) ≤ C(f, r, λ)s+

1

ne−nC(f,r,λ)s

and so (8.6) follows and the proof is complete.

To conclude that Xn(·) satisfies large deviation principle in C([0,+∞);H−1) we

recall that the processes Xn(·) are a.s. in C([0,+∞);H−1). Therefore the sequence Xn(·)is so called C-exponentially tight in DH−1 [0,+∞) (see [19], Definition 4.12). This, together

with the fact that Xn(·) satisfies large deviation principle in DH−1[0,+∞) implies that

it satisfies large deviation principle in C([0,+∞);H−1) (see [19], Section 4.4). We also

mention that using (7.31) and (8.2) one can give an explicit representation formula for

the rate function (see Corollary 4.29 of [19]). We have thus obtained the following result.

Theorem 8.4. Let (2.1)-(2.3) be satisfied. Then the sequence Xn(·) = Xn(·; 0, x) of

solutions of (1.1) satisfies the large deviation principle in C([0,+∞);H−1).

9 Large deviations for stochastic Navier-Stokes equa-

tions

In this section we will show how the techniques developed in this paper can be adapted

to deal with large deviation principle for solutions of two-dimensional stochastic Navier-

Stokes (SNS) equations with small noise intensities. Unfortunately, because of the un-

bounded nonlinearity, SNS equations do not fall into the case (1.1) discussed before.

28

In particular, viscosity solutions of the corresponding HJB equations must be defined

differently now. However the basic strategy still works. As we have mentioned in the in-

troduction we could show the large deviation principle for solutions of SNS equations with

nonlinear and Lipschitz continuous diffusion coefficient, however since this result has been

already obtained in [38] we will consider instead the case with a simple linear noise to

minimize the technicalities and explain better the main difficulties. The SNS equations

we are interested in can be described in the following way.

Let U = [0, L] × [0, L], and let ν > 0. We define the spaces

V =

x ∈ H1

p

(U ; R2

), div x = 0,

∫

U

x = 0

,

H = the closure of V in L2(U ; R2

),

where for an integer k ≥ 1, Hkp (U ; R2) is the space of R

2 valued functions x that are in

Hkloc (R2; R2) and such that x(y + Lei) = x(y) for every y ∈ R

2 and i = 1, 2. We will

denote by 〈·, ·〉, and ‖ · ‖ respectively the inner product and the norm in L2 (U ; R2). The

space H inherits the same inner product and norm. Let PH be the orthogonal projection

in L2 (U ; R2) onto H. Define Ax = −PH∆x, with the domain D (A) = H2p (U ; R2) ∩ V,

and we denote B(x,y) = PH [(x · ∇)y].

Let t ≥ 0. The abstract SNS equations with small noise intensities we will be con-

cerned with describe the evolution of the velocity vector field Xn : [t,+∞) → H that

satisfies the Ito type equation

dXn(s) = (−νAXn(s) − B (Xn(s),Xn(s)) + f(s)) ds+ 1√nQ

12dW(s) for s > t,

Xn(t) = x ∈ V,(9.1)

where f : [0,+∞) → V, W is a cylindrical Wiener process in H with identity covariance

operator defined on a probability space (Ω,F ,P) with a normal filtration Ft, and Q :

H → H is a linear, bounded, operator that is self-adjoint, Q ≥ 0, and

tr(Q) < +∞.

We will show that the sequence Xn(·) satisfies large deviation principle in C([0,+∞);H)

if we take t = 0.

9.1 Spaces, operators, and basic estimates

For γ > 0 we denote by Vγ the domain of Aγ2 , D(A

γ2 ), equipped with the norm

‖x‖γ = ‖A γ

2 x‖.

29

The space V1 coincides with V. The operator B defines a trilinear form b by setting

b(x,y, z) = 〈B(x,y), z〉.

It satisfies the estimate

|b(x,y, z)| ≤ C‖x‖γ1‖y‖1+γ2‖z‖γ3 (9.2)

where γi ≥ 0 and γ1 + γ2 + γ3 ≥ 1 if γi 6= 1, i = 1, 2, 3, and γ1 + γ2 + γ3 > 1 if γi = 1, for

some i (see for instance [41]). We also have the orthogonality relations

b (x,y, z) = −b (x, z,y) , b (x,y,y) = 0 for x,y, z ∈ V (9.3)

and, because of the periodic boundary conditions (see [41]),

b (x,x,Ax) = 0 for x ∈ V2. (9.4)

In particular if x,y ∈ V then (9.2) and an interpolation inequality yield

|b (x,y,x)| ≤ C ‖x‖ ‖x‖1 ‖y‖1 . (9.5)

Also, if x ∈ V, y ∈ V2, z ∈ H then

|b (x,y, z)| ≤ C ‖x‖1 ‖y‖2 ‖z‖ . (9.6)

We denote Q1 = A12 QA

12 . We will require throughout the rest of the paper that

tr(Q1) < +∞. (9.7)

We will use truncated operators B. Following [1, 34], for r > 0 we introduce operators

Br defined by

Br(x) = Br(x,x) :=

B(x,x) if ‖x‖1 ≤ r,

r2

‖x‖21B(x,x) if ‖x‖1 > r.

(9.8)

It is a rather standard calculation to show that for every r there exists a constant C =

C(ν, r) such that

〈(νAx + Br(x)) − (νAy + Br(y)),x − y〉 + C‖x − y‖2 ≥ ν

2‖x − y‖2

1 (9.9)

for all x,y ∈ V2. We will denote B+∞ := B.

30

9.2 Viscosity solutions

Since we will only be concerned with bounded solutions we will use a slight variation of

the definition of viscosity solutions given in [25].

Definition 9.1. A function ψ is a test function if ψ = ϕ± h(‖x‖1), where

(i) ϕ ∈ C1,2 ((0, T ) × H) and is such that ϕ, ϕt, Dϕ,D2ϕ are locally uniformly contin-

uous.

(ii) h ∈ C2([0,+∞)) and is such that h′(0) = 0, h′′(0) > 0, h′(r) > 0 for r ∈ (0,+∞).

We remark that even though ‖x‖1 is not differentiable at 0, the function h(‖x‖1) ∈C2(V). Since the function h(x) = h(‖x‖1) is not Frechet differentiable in H, following

[25] we define

Dh(x) =h′(‖x‖1)

‖x‖1Ax,

D2h(x) = h′(‖x‖1)

(A

‖x‖1

− Ax ⊗ Ax

‖x‖31

)+ h′′(‖x‖1)

Ax ⊗ Ax

‖x‖21

and in what follows for a function ψ = ϕ± h we will write

Dψ = Dϕ±Dh, D2ψ = D2ϕ±D2h

even though this is a slight abuse of notation.

We will be dealing with terminal value problems for the HJB equations of the type

ut +k

2tr(QD2u) − 〈νAx + Br(x,x), Du〉 + F (t,x, Du) = 0 in (0, T ) × V, (9.10)

where F : [0, T ] × V × H → R and k ≥ 0.

Definition 9.2. A weakly sequentially upper semicontinuous function u : (0, T )×V → R

is a viscosity subsolution of (9.10) if whenever u− ϕ− h(‖ · ‖1) has a local maximum at

a point (t,x) for test functions ϕ, h(‖y‖1) then x ∈ V2 and

ψt(t,x) +k

2tr(QD2ψ(t,x))− 〈νAx +Br(x,x), Dψ(t,x)〉+F (t,x, Dψ(t,x)) ≥ 0, (9.11)

where ψ(s,y) = ϕ(s,y) + h(‖y‖1).

A weakly sequentially lower semicontinuous function u : (0, T )×V → R is a viscosity

supersolution of (9.10) if whenever u−ϕ+ h(‖ · ‖1) has a local minimum at a point (t,x)

for test functions ϕ, h(‖y‖1) then x ∈ V2 and

ψt(t,x) +k

2tr(QD2ψ(t,x))− 〈νAx +Br(x,x), Dψ(t,x)〉+F (t,x, Dψ(t,x)) ≤ 0, (9.12)

where ψ(s,y) = ϕ(s,y) − h(‖y‖1).

A viscosity solution of (9.10) is a function which is both a viscosity subsolution and

a viscosity supersolution.

31

We remark that maxima and minima in the above definition can be assumed to

be strict and global and, for bounded sub- and supersolutions, ϕ, ϕt, Dϕ,D2ϕ may be

assumed to be bounded and uniformly continuous on (0, T ) ×H.

9.3 Comparison principles

We first prove a comparison theorem for terminal value problems

(un)t +12n

tr (QD2un) − 12‖Q 1

2Dun‖2 + 〈−νAx − Br(x) + f(t), Dun〉 = 0,

un(T, x) = g(x) in (0, T ) × V.

(9.13)

Theorem 9.3. Let tr(Q1) < +∞, f : [0, T ] → V be continuous, r ∈ (0,+∞), and let

g ∈ Lipb(H). Let u be a viscosity subsolution of (9.13), and v be a viscosity supersolution

of (9.13) where n ≥ 1 or n = +∞. Let u,−v be weakly sequentially upper semicontinuous

on (0, T ] ×H,

u,−v ≤ C for some C > 0 (9.14)

and

limt→T

(u(t,x) − g(x))+ + (v(t,x) − g(x))− = 0,

uniformly on bounded sets of V. (9.15)

Assume in addition that either ψ = u or ψ = v satisfies

|ψ(t,x) − ψ(t,y)| ≤ L‖x − y‖ (9.16)

for some L ≥ 0 and all t ∈ (0, T ),x,y ∈ V. Then u ≤ v.

Proof. The proof follows with some modifications the lines of the proof of Theorem 5.2

in [25] and the proof of Theorem 5.1 in this paper. We sketch it here for the sake of

completeness. We will only deal with the case n < +∞ as the proof for the first order

equation is essentially the same but much easier. For µ > 0 we define

uµ(t,x) = u(t,x) − µ

t, vµ(t,x) = v(t,x) +

µ

t.

If u 6≤ v then arguing exactly as in the proof of Theorem 5.1 with 12ε‖x − y‖2

−1

replaced by 12ε‖x − y‖2 and δ(‖x‖2 + ‖y‖2) replaced by δ(‖x‖2

1 + ‖y‖21) we have that for

sufficiently small µ, ε, δ, β > 0 the function

Φ(t, s,x,y) = uµ(t,x) − vµ(s,y) − ‖x−y‖2

2ε− δ(‖x‖2

1 + ‖y‖21) −

(t− s)2

2β

32

is weakly sequentially upper-semicontinuous on (0, T ]×H and so it has a global maximum

at some points t, s, x, y, where 0 < t, s, and x, y are bounded for a fixed δ, and

lim supδ→0

lim supδ→0

δ(‖x‖21 + ‖y‖2

1) = 0 for fixed ε, (9.17)

and

lim supβ→0

(t− s)2

2β= 0 for fixed δ, ε. (9.18)

We can assume the maxima to be strict and it follows from the definition of solution

that x, y ∈ V2. It then follows from (9.17), (9.18), the fact that g ∈ Lipb(H) and (9.15)

that for small µ, ε, δ, β > 0 we have t, s < Tδ. Moreover, exactly as in (5.9) we obtain

lim supδ→0

lim supβ→0

‖x − y‖ε

≤ 2L. (9.19)

Let now H1 ⊂ H2 ⊂ ... be finite dimensional subspaces of H generated by eigenvectors

of A such that⋃∞N=1 HN = H. Given N ∈ N, N > 1, denote by PN the orthogonal

projection onto HN , let QN = I − PN , and H⊥N = QNH. We then have an orthogonal

decomposition H = HN ×H⊥N and we will denote by xN an element of HN and by x⊥

N an

element of H⊥N . For x ∈ H we will write x = (PNx,QNx) = (xN ,x

⊥N).

We now fix N ∈ N. Then we have

‖QN(x − y)‖2 ≤ 2〈QN(x − y),x − y〉 − ‖QN(x − y)‖2

+2‖QN(x − x)‖2 + 2‖QN(y − y)‖2

with equality if x = x,y = y. We define

u1(t,x) = uµ(t,x) − 〈x,QN(x − y)〉ε

+‖QN (x − y)‖2

2ε− ‖QN(x − x)‖2

ε− δ‖x‖2

1

and

v1(s,y) = vµ(s,y) − 〈y,QN(x − y)〉ε

+‖QN(y − y)‖2

ε+ δ‖y‖2

1.

We remark again that u1 and v1 are respectively weakly sequentially upper- and lower-

semicontinuous on (0, T ] × H.

We now have that the function

Φ(t, s,x,y) = u1(t,x) − v1(s,y) − ‖PN(x − y)‖2

2ε− (t− s)2

2β(9.20)

always satisfies Φ ≤ Φ and attains a strict global maximum at t, s, x, y, where Φ(t, s, x, y) =

Φ(t, s, x, y).

33

Define, for xN , yN ∈ HN , the functions

u1(t,xN) = supx⊥

N∈H⊥

N

u1(t,xN ,x⊥N), v1(s,yN) = inf

y⊥

N∈H⊥

N

v1(s,yN ,y⊥N).

Since u1 and −v1 are weakly sequentially upper-semicontinuous on (0, T ]×H, u1 and −v1

are upper semicontinuous on (0, T ] × HN (see [9]). Moreover it follows that

u1(t,PN x) = u1(t, x), v1(s,PN y) = v1(s, y). (9.21)

and it is easy to see that the function

u1(t,xN ) − v1(s,yN) − ‖xN − yN‖2

2ε− (t− s)2

2β

= supx⊥

N, y⊥

N∈H⊥

N

Φ(t, s, (xN ,x

⊥N), (yN ,y

⊥N)),

attains a strict global maximum over (0, T ] × (0, T ] × HN × HN at (t, s, xN , yN). By

the finite dimensional maximum principle (see [8]) for every n ∈ lN there exist points

tn, sn ∈ (0, T ) and xnN , ynN ∈ HN such that

tn → t, sn → s, xnN → xN , ynN → yN as n→ ∞ (9.22)

u1(tn,xnN) → u1(t, xN), v1(s

n,ynN) → v1(s, yN), as n→ ∞ (9.23)

and there exist functions ϕn, ψn ∈ C1,2((0, T )×HN) with uniformly continuous derivatives

such that u1 − ϕn, and −v1 + ψn have strict, global maxima at (tn,xnN), and (sn,ynN)

respectively, and

(ϕn)t(tn,xnN) → t− s

β, Dϕn(t

n,xnN) → 1

ε(xN − xN ), (9.24)

(ψn)t(sn,ynN) → t− s

β, Dψn(s

n,ynN) → 1

ε(xN − xN), (9.25)

D2ϕn(tn,xnN) → XN , D2ψn(s

n,ynN) → YN , where XN ≤ YN . (9.26)

We now consider the function

u1(t,x) − v1(s,y) − ϕn(t,PNx) + ψn(s,PNy). (9.27)

It attains its global maximum (which we can assume to be strict) at some point (tn, sn,

xn, yn). Repeating now the arguments of [23, page 409] (see also [9]) it is not difficult to

show that

u1(tn, xn) → u1(t, x), v1(s

n, yn) → u1(s, y) (9.28)

34

and

tn = tn, sn = sn, xnN = xnN , ynN = ynN , (xn, yn) → (x, y) in H × H (9.29)

as n→ ∞, and moreover that

‖xn‖1 → ‖x‖1, ‖yn‖1 → ‖y‖1.

But this then obviously implies that

xn → x, yn → y in V. (9.30)

Therefore using the definition of viscosity solution and letting k → ∞ we obtain

− µ

T 2+tk − sk

β+

1

2ntr

(Q(D2ϕ(tk, xk) +

2QN

ε) + 2δQ1

)

− 1

2‖Q 1

2 (Dϕk(tk, xk) +

QN(x − y)

ε+

2QN(xk − x)

ε+ 2δAxk)‖2

+ 〈−νAxk + f(tk), Dϕk(tk, xk) +

QN(x − y)

ε+

2QN(xk − x)

ε+ 2δAxk〉

− 〈Br(xk), Dϕk(tk, xk) +

QN(x − y)

ε+

2QN(xn − x)

ε〉 ≥ 0

We remark that we have used (9.4) to get 〈Br(xk),Axk) = 0. We now want to pass

to the limit as k → ∞.

Now, using the boundedness of ‖xk‖1, boundedness of Q12A

12 and A

12 PN in H (which,

together with (9.24), implies in particular that the ‖Dϕn(tn, xk)‖1 are bounded), and the

continuity of 〈Br(x),y)〉 on V × V we obtain from (9.31) that the ‖Axk‖ are bounded

and therefore that xk x in V2. Thus, using (9.22), (9.24), (9.29), (9.30), and (9.6) we

can pass to the lim sup as k → ∞ in (9.31) to get

t− s

β+

1

2ntr

(Q(XN +

2QN

ε) + 2δQ1

)− 1

2‖Q 1

2 (x − y

ε+ 2δAx)‖2

+〈−νAx,x − y

ε〉 − 〈Br(x),

x − y

ε〉 + 〈f(t), x − y

ε+ 2δAx〉 ≥ µ

T 2. (9.31)

Similarly we obtain

t− s

β+

1

2ntr

(Q(YN − 2QN

ε) − 2δQ1

)− 1

2‖Q 1

2 (x − y

ε− 2δAx)‖2

+〈−νAy,x − y

ε〉 − 〈Br(y),

x − y

ε〉 + 〈f(s), x − y

ε− 2δAy〉 ≤ − µ

T 2. (9.32)

Since

2δ(‖Q 12 Ax‖ + ‖Q 1

2Ax‖) ≤ D1δ(‖x‖1 + ‖y‖1) ≤ D2

√δ,

35

using (9.19) we have that

1

2|‖Q 1

2x − y

ε+ 2δQ

12Ax‖2 − ‖Q 1

2x − y

ε− 2δQ

12Ay‖2| ≤ D3

√δ

Moreover

|〈f(t), 2δAx〉| + |〈f(s), 2δAy〉| ≤ 2Rδ(‖x‖1 + ‖y‖1) ≤ D4

√δ.

Therefore, combining (9.31) and (9.32) and using the above, XN ≤ YN , (9.18) and (9.19)

we obtain

〈(νAx + Br(x)) − (νAy + Br(y)),x − y

ε〉 −D5

√δ − σ1(N) − σ2(β) ≤ −2µ

T,

where σ1(N) → 0 as N → ∞ for ε, δ, β fixed, and σ2(β) → 0 as β → 0 for ε, δ fixed. It

now follows from (9.9) that

−C(ν, r)‖x − y‖2

ε−D5

√δ − σ1(N) − σ2(β) ≤ −2µ

T.

It then remains to pass to the lim inf in the above inequality as β → 0, δ → 0, and ε→ 0

and invoke (9.19) to obtain a contradiction. This completes the proof.

We remark that Theorem 9.3 holds without the assumption that either u or v is

Lipschitz continuous, with a very similar proof. However the next theorem needs the

Lipschitz continuity condition and its proof follows in large parts the proof of Theorem

9.3 in the Lipschitz case. Therefore, in order not to repeat the proof again we stated

Theorem 9.3 in a weaker form. We remark that we do not know of Theorem 9.3 is true

for r = +∞. This is why we have to go through this layer of approximations.

Theorem 9.4. Let tr(Q1) < +∞, f : [0, T ] → V be continuous, r ∈ (0,+∞),and let

g ∈ Lipb(H). Let un be a bounded viscosity solution of (9.13) for n < +∞, and v be a

viscosity solution of (9.13) for n = +∞ such that

limt→T

|un(t,x) − g(x)| + |v(t,x) − g(x)| = 0, uniformly on bounded sets of V (9.33)

and

|v(t,x) − v(t,y)| ≤ L‖x − y‖ (9.34)

for some L ≥ 0 and all t ∈ (0, T ],x,y ∈ V. Then there exists a constant C independent

of n such that

‖un − v‖∞ ≤ C√n. (9.35)

Proof. Let C(ν, r) be the constant from (9.9) and C1 be such that

1

2tr(Q) ≤ C1.

36

We set

vn = v +1√n

(T − t)(4C(ν, r)L2 + C1) +2L2

√n.

Then vn is a viscosity supersolution of

(vn)t −1

2‖Q 1

2Dvn‖2

+ 〈−νAx + Br(x) + f(t), Dvn〉 = − 1√n

(4C(ν, r)L2 + C1).(9.36)

We then combine the proofs of Theorem 5.2 and 9.3. If un 6≤ vn then we obtain that for

small µ, δ, β the function

(un)µ(t,x) − (vn)

µ(s,y) −√n

2‖x − y‖2 − (t− s)2

2β− δ(‖x‖2

1 − ‖y‖21) (9.37)

has a strict global maximum over (0, T ]×H at some points t, s, x, y, where 0 < t, s < T .

Moreover (9.17), (9.18), and (9.19) are satisfied. We then use (9.37) and the definition of

viscosity solution to obtain

− µ

t2+t− s

β+

1

2ntr(√

nQ + 2δQ1))

− 1

2‖Q 1

2 (√n(x − y) + 2δAx)‖2 + 〈−νAx + f(t),

√n(x − y) + 2δAx〉

− 〈Br(x),√n(x − y)〉 ≥ 0

(9.38)

and

µ

s2+t− s

β− 1

2‖Q 1

2 (√n(x − y) − 2δAy)‖2

+ 〈−νAy + f(s),√n(x − y) − 2δAy〉 − 〈Br(y),

√n(x − y)

≤ − 1√n

(4C(ν, r)L2 + C1).

(9.39)

Combining (9.38) and (9.39) and using (9.9) and (9.19) (where ε = 1/√n) we thus obtain

as in the proof of Theorem 9.3 that

2µ

T 2≤ σ1(β) + σ2(δ) +

C1√n

+√nC(ν, r)‖x − y‖2 − 1√

n(4C(ν, r)L2 + C1)

≤ +σ1(β) + σ2(δ)

(9.40)

which gives a contradiction after we send β → 0 and δ → 0.

Therefore un ≤ v + 1√n(T − t)(4C(ν, r)L2 + C1) + 2L2

√n. As before, applying a similar

argument to functions vn = v − 1√n(T − t)(4C(ν, r)L2 + C1) − 2L2

√n

and un produces

v − 1√n(T − t)(4C(ν, r)L2 + C1) − 2L2

√n≤ un which, together with the previous estimate,

yields (9.35).

37

9.4 Stochastic Navier-Stokes equations

Proposition 9.5. Let r ∈ (0,+∞] and 0 ≤ t ≤ T . Let tr(Q1) < +∞, and let f : [0, T ] →V be continuous and such that

‖f(t)‖1 ≤ R for all t ∈ [0, T ].

Then:

(i) For every x ∈ V there exists a unique strong solution X(·) = X(·; t,x) of

dXn(s) = (−νAXn(s) − Br (Xn(s)) + f(s)) ds+ 1√nQ

12dW(s) for s > t,

Xn(t) = x

(9.41)

such that

IE‖X(s)‖21 + νIE

∫ s

t

‖X(τ)‖22dτ ≤ ‖x‖2

1 + Cν(R2 +

1

ntr(Q1))(s− t). (9.42)

The solution has trajectories a.s. in C([t, T ];V).

(ii) If r < +∞ there exists a constant C = C(ν, r, T ) such that for every initial condi-

tions x,y ∈ V

IE‖Xn(s) − Yn(s)‖2 ≤ C(ν, r, T )‖x − y‖2, (9.43)

where Xn(·) = Xn(·; t,x),Yn(·) = Yn(·; t,y) are solutions of (9.41) with initial

conditions x and y respectively.

(iii) There exists a constant C = C(ν, R,M, tr(Q)) such that for every initial condition

x ∈ V, ‖x‖1 ≤M,

IE‖Xn(s) − x‖2 ≤ C(ν, R,M, tr (Q))(s− t). (9.44)

Moreover there exists a modulus ωx such that

IE‖Xn(s) − x‖21 ≤ ωx(s− t). (9.45)

Above, Xn(·) = Xn(·; t,x) is the solution of (9.41).

(iv) There exists an absolute constant C > 0 such that if x ∈ V then

IE

(supt≤s≤T

enc1‖Xn(s)‖21

)≤ enc2, (9.46)

where

c1 =νC

tr(Q1), (9.47)

c2 = c2 (‖x‖1, ν, T, R, tr(Q1)) .

38

Proof. The existence and uniqueness of variational and strong solutions if r = +∞ is

standard ([43], [34]). The estimates (9.44)-(9.45) for r = +∞ are explained and proved in

[25]. Since the truncated operators Br satisfy the same (and even better) estimates as the

one for B, uniformly in r, the existence of solutions of (9.41) for r < +∞ can be proved

by the same techniques as for r = +∞, for instance by the method of [34]. Estimates

(9.44)-(9.45) when r < +∞ are also proved in exactly the same way. Estimate (9.43) is

shown by noticing that the process Z = Xn − Yn satisfies the equation

d

dsZ(s) = −νAZ(s) − Br(Xn(s)) + Br(Yn(s))

and therefore, using (9.9) we obtain

‖Z(s)‖2 = ‖x − y‖2 −∫ s

t

〈νAZ(τ) + Br(Xn(τ)) − Br(Yn(τ))〉dτ

≤ ‖x − y‖2 + C(ν, r)

∫ s

t

‖Z(s)‖2dτ.

The application of Gronwall’s inequality now yields (9.43).

It remains to prove (iv). We will do it only for r = +∞ as the estimate for the case

r < +∞ is obtained by repeating the same proof, and all estimates are independent of r.

By Corollary 3.1, Chapter XI of [43] (which can also be proved directly by the method

used below), if C > 0 is such that ‖y‖1 ≤ ‖y‖2/C for all y ∈ V2, and α = Cν/(2tr(Q1))

then

IE

∫ T

t

eαn‖X(s)‖21‖X(s)‖2

1ds ≤ enC1(‖x‖1,ν,T,R,tr(Q1)) (9.48)

for some constant C1(‖x‖1, ν, T, R, tr(Q1)). To prove (9.48) we take α = Cν/(4tr(Q1))

and apply Ito’s formula to get

eαn‖X(s)‖21 = eαn‖x‖

21 +

∫ s

t

2αneαn‖X(τ)‖21〈−νAX(τ) − B(X(τ),X(τ)) + f(τ),AX(τ)〉dτ

+

∫ s

t

2α√neαn‖X(τ)‖2

1〈AX(τ),Q12dW(τ)〉

+1

2n

∫ s

t

tr(Q[4α2n2AX(τ) ⊗ AX(τ) + 2αnA

]eαn‖X(τ)‖2

1

)dτ

≤ eαn‖x‖21 − αnν

∫ s

t

eαn‖X(τ)‖21‖AX(τ)‖2dτ +

αnR2

ν

∫ s

t

eαn‖X(τ)‖21dτ

+

∫ s

t

1

neαn‖X(τ)‖2

1(2α2n2tr(Q1)‖X(τ)‖21 + αntr(Q1))dτ

+

∫ s

t

2α√neαn‖X(τ)‖2

1〈AX(τ),Q12dW(τ)〉

39

≤ eαn‖x‖21 + (

αnR2

ν+ αtr(Q1))

∫ s

t

eαn‖X(τ)‖21dτ

+

∣∣∣∣∫ s

t

2α√neαn‖X(τ)‖2

1〈AX(τ),Q12dW(τ)〉

∣∣∣∣ . (9.49)

Therefore

IE supt≤τ≤s

eαn‖X(τ)‖21 ≤ eαn‖x‖

21 + (

αnR2

ν+ αtr(Q1))

∫ s

t

IE supt≤u≤τ

eαn‖X(u)‖21dτ

+2α√nIE sup

t≤s≤T

∣∣∣∣∫ s

t

eαn‖X(τ)‖21〈AX(τ),Q

12dW(τ)〉

∣∣∣∣ . (9.50)

Applying Burkholder-Davis-Gundy inequality, (9.48), and noticing that 2α = α we

obtain

2α√nIE sup

t≤s≤T

∣∣∣∣∫ s

t

eαn‖X(τ)‖21〈AX(τ),Q

12dW(τ)〉

∣∣∣∣

≤ 2α√n

(IE sup

t≤s≤T

∣∣∣∣∫ s

t

eαn‖X(τ)‖21〈AX(τ),Q

12dW(τ)〉

∣∣∣∣2) 1

2

≤ C2α√n

(tr(Q1)IE

∫ T

t

e2αn‖X(τ)‖21‖X(τ)‖2

1dτ

) 12

≤ C2α√n(tr(Q1))

12 enC1(‖x‖1,ν,T,R,tr(Q1)). (9.51)

Therefore plugging (9.51) into (9.50) we obtain

IE supt≤τ≤s

eαn‖X(τ)‖21 ≤ eαn‖x‖

21

(1 + C2α

√n(tr(Q1))

12 enC1(‖x‖1,ν,T,R,tr(Q1))

)

+(αnR2

ν+ αtr(Q1))

∫ s

t

IE supt≤u≤τ

eαn‖X(u)‖21dτ.

Estimate (9.48) now follows from Gronwall’s inequality.

For other results on 2-dimensional SNS equations we refer the readers to [3, 13, 34,

38, 43] and the references therein.

9.5 Existence of Laplace limit

Our first step is to establish the existence of the Laplace limit for the processes Xrn(·) at

single times, i.e. for the family Xn(T ), regarded as processes with values in H.

Theorem 9.6. Let r ∈ (0,+∞). Let f : [0, T ] → V be continuous, g ∈ Lipb(H), and

tr(Q1) < +∞. Then the functions

urn(t,x) = − 1

nlog IE[e−ng(X

rn(T ))]

40

are bounded uniformly in n, there exist a constant D1 and, for every R > 0, a constant

D2 = D2(R) such that

|urn(t,x)−urn(s,y)| ≤ D1‖x−y‖+D2(max‖x‖1, ‖y‖1)|t−s|12 for x,y ∈ V, t, s ∈ [0, T ],

(9.52)

and the urn are the unique bounded viscosity solutions of

(urn)t +12n

tr (QD2urn) − 12‖Q 1

2Durn‖2 − 〈νAx + Br(x,x), Durn〉 + 〈f(t), Durn〉 = 0,

urn(T,x) = g(x) in (0, T ) × V

(9.53)

satisfying (9.33).

Proof. First of all we notice that (9.52) follows from (9.43) and (9.44). This in particular

implies that urn satisfies (9.33). The uniform boundedness is obvious and uniqueness will

follow from Theorem 9.3 once we show that the urn solve (9.53).

We will only show that the urn are subsolutions of (9.53) indicating how to prove the

supersolution property as its proof is similar. To do this we suppose that urn−h(‖ ·‖1)−ϕhas a local maximum at (t,x). We can assume that the maximum is strict and global.

Moreover, in light of the boundedness of urn, we can always assume that ϕ, ϕt, Dϕ,D2ϕ

are bounded and uniformly continuous and that h(r) = r2 for big r. Then

urn(t+ ε,Xrn(t+ ε))− h(‖Xr

n(t+ ε)‖1)−ϕ(t+ ε,Xrn(t+ ε)) ≤ urn(t,x)− h(‖x‖1)−ϕ(t,x).

Therefore

vn(t+ ε,Xrn(t+ ε))

vn(t,x)≥ e−n(h(‖Xr

n(t+ε)‖1)+ϕ(t+ε,Xrn(t+ε)))en(h(‖x‖1)+ϕ(t,x)),

where vn = e−nurn. Taking the expectation of both sides and using the Markov property

of Xrn(s) we now have

e−n(h(‖x‖1)+ϕ(t,x)) ≥ IEe−n(h(‖Xrn(t+ε)‖1)+ϕ(t+ε,Xr

n(t+ε))).

Therefore, using the above and Ito’s formula, we obtain

41

0 ≥ IE1

ε

e−n(h(‖Xr

n(t+ε)‖1)+ϕ(t+ε,Xrn(t+ε))) − e−n(h(‖x‖1)+ϕ(t,x))

= IE1

ε

∫ t+ε

t

−e−n(h(‖Xrn(τ)‖1)+ϕ(τ,Xr

n(τ)))nϕt(τ,Xrn(τ))dτ

−IE1

ε

∫ t+ε

t

e−n(h(‖Xrn(τ)‖1)+ϕ(τ,Xr

n(τ)))〈nh′(‖Xr

n(τ)‖1)

‖X(τ)‖1

AXrn(τ) + nDϕ(τ,Xr

n(τ)),

−νAXrn(τ) − Br(Xr

n(τ),Xrn(τ)) + f(τ)〉dτ

+IE1

2nε

∫ t+ε

t

e−n(h(‖Xrn(τ)‖1)+ϕ(τ,Xr

n(τ)))tr

[Q

((nh′(‖Xr

n(τ)‖1)

‖Xrn(τ)‖1

AXrn(τ)

+nDϕ(τ,Xrn(τ))

)⊗(nh′(‖Xr

n(τ)‖1)

‖Xrn(τ)‖1

AXrn(τ) + nDϕ(τ,Xr

n(τ))

)

−nh′′(‖Xr

n(τ)‖1)

‖Xrn(τ)‖2

1

AXrn(τ) ⊗ AXr

n(τ) + nh′(‖Xr

n(τ)‖1)

‖Xrn(τ)‖3

1

AXrn(τ) ⊗ AXr

n(τ)

−nh′(‖Xr

n(τ)‖1)

‖Xrn(τ)‖1

A− nD2ϕ(τ,Xrn(τ))

)]dτ. (9.54)

Now, using the boundedness of ϕ and its derivatives, the special form of h and the

fact that h′(r)/r > γ > 0, (9.6), and the assumptions on f and Q1, after some tedious

calculations we obtain that

1

εIE

∫ t+ε

t

e−n(h(‖Xrn(τ)‖1)+ϕ(τ,Xr

n(τ)))‖AXrn(τ)‖2dτ ≤ C (9.55)

for some constant C independent of ε. Let now M > ‖x‖1 be such that if

Ω =

ω : sup

t≤τ≤T‖Xr

n(τ)‖1 ≤M

then P(Ω) > 0. It then follows from (9.55) that

1

εIE

∫ t+ε

t

‖AXrn(τ)‖2χΩdτ ≤ C1 (9.56)

Therefore there exists a sequence tn → t such that Xrn(tn) are bounded in L2(Ω,V2).

This implies that there exists a subsequence, still denoted by tn, such that Xrn(tn) Y

in L2(Ω,V2) for some Y ∈ L2(Ω,V2). But Xrn(tn) → x in L2(Ω,H) and so x = Y ∈ V2.

It now remains to pass to the liminf in (9.54) as ε → 0. This can be done exactly as

in the proof of Theorem 6.3 of [25]. To simplify the arguments, once we have established

that x ∈ V2 we can modify the function h for big r again so that we still have the strict

and global maximum at (t,x) and hence (9.54) still holds with the new h, however now

h, h′, h′(r)/r, h′′ are bounded and uniformly continuous. There are more terms to take

42

care here than in the proof of Theorem 6.3 of [25], however all functions involved are

bounded and uniformly continuous so the passage to the liminf is in fact even easier. We

then obtain, denoting ψ(s,y) = h(‖y‖1) + ϕ(s,y),

0 ≥ −ne−nψ(t,x)

(ψt(t,x) + 〈−νAx − Br(x,x) + f(t), Dψ(t,x)〉

+1

2ntr[Q(−nDψ(t,x) ⊗Dψ(t,x) +D2ψ(t,x)

)]). (9.57)

This proves the claim.

There is one technical difference in the proof of the supersolution property. If urn +

h(‖ · ‖1) − ϕ has a minimum at (t,x) we should modify h to be h(r) = c1r2/4 for big r,

where c1 is the constant from (9.47) to guarantee the integrability of various terms in the

supersolution version of (9.54). The rest of the proof is easier as there is no need to take

the set Ω since in place of (9.55) we will simply get

1

εIE

∫ t+ε

t

‖AXrn(τ)‖2dτ ≤ C.

If we pass to the limit as n→ +∞ we formally obtain the HJB equation

(vr)t − 12‖Q 1

2Dvr‖2 − 〈νAx + Br(x,x), Dvr〉 + 〈f(t), Dvr〉 = 0,

vr(T,x) = g(x) in (0, T ) × V.

(9.58)

As in Section 7 it corresponds to the optimal control problem where for 0 ≤ t ≤ T,x ∈ H

and z(·) ∈ L2(t, T ;H), the state equation is

dds

Xr(s) = −νAXr(s) − Br (Xr(s)) + f(s) + Q12 z(s) for s > t,

X(t) = x,

(9.59)

and we minimize the cost functional

J(t,x; z(·)) =

∫ T

t

1

2‖z(s)‖2ds+ g(Xr(T )) (9.60)

over all z(·) ∈ L2(t, T ;H) or equivalently over all

z(·) ∈Mt := z(·) ∈ L2(t, T ;H) :

∫ T

t

‖z(s)‖2ds ≤ K = 2‖g‖∞. (9.61)

If we denote by vr the value function for the problem then

vr(t,x) = infz(·)∈L2(t,T ;H)

J(t,x; z(·)) = infz(·)∈Mt

J(t,x; z(·)). (9.62)

43

The next proposition collects the continuity estimates for solutions of (9.59). Its proof

follows easily from standard results on Navier-Stokes equations (see for instance [41]) and

arguments similar to these used to obtain Proposition 9.5 (see also [24]) and thus is

omitted.

Proposition 9.7. Let r ∈ (0,+∞) and 0 ≤ t ≤ T . Let tr(Q1) < +∞, z(·) ∈Mt, and let

f : [0, T ] → V be continuous and such that

‖f(t)‖1 ≤ R for all t ∈ [0, T ].

Then:

(i) For every x ∈ V there exists a unique strong solution Xr(·) = Xr(·; t,x) of (9.59).

The solution satisfies

‖Xr(s)‖21 + ν

∫ s

t

‖Xr(τ)‖22dτ

≤ ‖x‖21 + C(ν, T, ‖x‖1, R, ‖A

12 Q

12 ‖, ‖z(·)‖L2(t,T ;H))(s− t)

12 .

(9.63)

(ii) For every initial conditions x,y ∈ V

‖Xr(s) − Yr(s)‖ ≤ C(ν, r, T )‖x − y‖, (9.64)

where Xr(·) = Xr(·; t,x),Yr(·) = Yr(·; t,y) are solutions of (9.59) with initial

conditions x and y respectively.

(iii)

‖Xr(s) − x‖ ≤ C(ν, T, R, ‖x‖1, ‖z(·)‖L2(t,T ;H), tr (Q))(s− t)12 . (9.65)

Moreover for every M > 0 there exists a modulus ωx,M such that if ‖z(·)‖L2(t,T ;H) ≤M then

‖Xr(s) − x‖1 ≤ ωx,M(s− t). (9.66)

Theorem 9.8. Let r ∈ (0,+∞). Let f : [0, T ] → V be continuous, let g ∈ Lipb(H), and

let tr(Q1) < +∞. Then the value function vr is bounded, there exist a constant D1 and,

for every R > 0, a constant D2 = D2(R) such that

|vr(t,x)−vr(s,y)| ≤ D1‖x−y‖+D2(max‖x‖1, ‖y‖1)|t−s|12 for x,y ∈ V, t, s ∈ [0, T ],

(9.67)

and vr is the unique bounded viscosity solutions of (9.58) satisfying (9.33).

44

Proof. The proof is very similar to the proof of Theorem 7.3. Estimate (9.67) is a conse-

quence of (9.64) and (9.65) as in the proof of Theorem 7.3.

As regards the rest, again we will only sketch the proof of the supersolution property.

Let now vr + h(‖x‖1) − ϕ have a local minimum at (t,x). By dynamic programming

principle for every 0 < ε < T − t there exists a control zε(·) such that

vr(t,x) ≥∫ t+ε

t

‖zε(s)‖2ds+ vr(t+ ε,Xr(t+ ε)) − ε2

Denote ψ(s,y) = ϕ(s,y) − h(‖y‖1) and h(y) := h(‖y‖1). Then we have

ε ≥ 1

ε(vr(t+ ε,Xr

ε(t+ ε)) − vr(t,x)) +1

2ε

∫ t+ε

t

‖zε(s)‖2ds

≥ 1

ε(ϕ(t+ ε,Xr

ε(t+ ε)) − ϕ(t,x) − h(‖Xrε(t+ ε)‖1) + h(‖x‖1)) +

1

2ε

∫ t+ε

t

‖zε(s)‖2ds

≥ 1

ε

∫ t+ε

t

[ϕt(s,X

rε(s)) + 〈−νAXr

ε(s) − Br(Xrε(s)) + f(s), Dψ(s,Xr

ε(s))〉

+〈Q 12 zε(s), Dψ(s,Xr

ε(s))〉 +1

2‖zε(s)‖2

]ds

≥ 1

ε

∫ t+ε

t

[ϕt(s,X

rε(s)) + 〈−νAXr

ε(s) − Br(Xrε(s)) + f(s), Dψ(s,Xr

ε(s))〉

−1

2‖Q 1

2Dψ(s,Xrε(s))‖2

]ds

. (9.68)

We remind that 〈Br(Xrε(s)), Dh(X

rε(s))〉 = 0. Using the fact that ‖zε(·)‖L2(t,t+ε;H) are

bounded, (9.63), the properties of test functions and by now familiar computations, the

above inequality implies that

1

ε

∫ t+ε

t

‖AXrε(s)‖2ds ≤ C

for some constant C. Therefore there exists a sequence sεn → t as εn → 0 such that

‖AXrεn(sεn)‖ ≤ C. This implies that up to a subsequence, still denoted by sεn, Xr

εn(sεn)

y in V2 for some y ∈ V2. However, by (9.65), Xrεn(sεn) → x in H and so x = y ∈ V2.

We can now pass to the liminf as ε→ 0 in (9.68) using the fact that ‖zε(·)‖L2(t,t+ε;H)

are bounded, (9.63), (9.66) and arguments similar to these needed to obtain the same fact

in the proof of Theorem 9.6 (see the proof of Theorem 6.3 of [25]). We then obtain

0 ≥ ϕt(t,x) + 〈−νAx − Br(x) + f(t), Dψ(t,x)〉 − 1

2‖Q 1

2Dψ(t,x)‖2

which shows that vr is a supersolution. The uniqueness follows from Theorem 9.3 since

vr is Lipschitz continuous in x.

In light of Theorems 9.4, 9.6 and 9.8 we can therefore conclude the following.

45

Corollary 9.9. Let r ∈ (0,+∞), f : [0, T ] → V be continuous, g ∈ Lipb(H), and

tr(Q1) < +∞. Then

limn→+∞

‖urn − vr‖∞ = 0. (9.69)

As in Section 7 we now denote

Vn(t)g(x) =1

nlog IE

(e−ng(Xn(t)) : Xn(0) = x

),

where Xn is the solution of (9.1) on [0,+∞). Repeating almost exactly the proof of Lemma

7.5 we obtain.

Lemma 9.10. Let t ≥ 0, f : [0,+∞) → V be continuous, tr(Q1) < +∞, and g ∈Lipb(H). Let Xn(t) = Xn(t; 0,x). Then for every M > 0 there exists r > 0 such that

V (t)g(x) := limn→∞

Vn(t)g(x) = −vr(0,x) = −v+∞(0,x) (9.70)

uniformly on ‖x‖1 ≤ M, where vr and v+∞ are defined with terminal time T = t. In

particular V (t)g is Lipschitz continuous in the ‖ · ‖ norm on bounded subsets of V.

We can now argue as in the proofs of Lemma 7.6 and Proposition 7.7 to obtain the

following equivalent version of the latter. We notice that V now corresponds to H in

Section 7 and H corresponds to H−1 and they are related by the operator B = A−1

(which should not be confused with the Euler operator B) so the setup is the same. The

proof of Proposition 9.11 is therefore omitted.

Proposition 9.11. Let t > 0, f : [0,+∞) → V be continuous, let tr(Q1) < +∞ and let

g be bounded and weakly sequentially continuous on V. Then:

(i) For every n the function Vn(t)g is weakly sequentially continuous on V.

(ii) For every x ∈ V

V (t)g(x) = limn→+∞

Vn(t)g(x)

exists and is uniform on bounded subsets of V. In particular V (t)g(x) is weakly sequen-

tially continuous on V.

(iii) If gn are weakly sequentially continuous on V, such that ‖gn‖∞ ≤ M for n ≥ 1 and

gn → g uniformly on bounded subsets of V then

limn→+∞

Vn(t)gn(x) = V (t)g(x) (9.71)

uniformly on bounded subsets of V.

46

9.6 Large deviation principle

Having the existence of Laplace limit at single times in the form of Proposition 9.11, the

rest of the proof of the large deviation principle literally follows the steps of Section 8.

Since the proofs are the same or very similar we will only sketch their main points.

Again, because of continuity of the paths, it is enough to show that Xn(·) satisfies

large deviation principle in DH[0,+∞).

The existence of the multiple Laplace limit for 0 ≤ t1 ≤ ... ≤ tm and g1, ..., gm ∈Cb(H) follows from Proposition 9.11 and the Markov property of the Xn(·) as in Section

8, and we also obtain

limn→+∞

1

nlog IE[e−n(g1(Xn(t1))+...+gm(Xn(tm)))]

= V (t1)(g1 − V (t2 − t1)(g2 − ...− V (tm − tm−1)gm)...)(x).

The proof of exponential tightness of Xn(·) in DH[0,+∞) uses Theorem 8.1 applied

to Z = H. To show the exponential compact containment condition (8.3) it is enough to

choose KM,T = x : ‖x‖1 ≤ r for a sufficiently big r > 0 and use (9.46). As regards

condition (ii) there we have the following lemma.

Lemma 9.12. Let

A = m∑

i=1

gi(‖x−xi‖) : m ∈ N,xi ∈ V, gi ∈ C2([0,+∞)), g′i(0) = 0, gi, g′i, g

′′i are bounded

.

Then A is closed under addition and isolates points in H. Moreover if g ∈ A then for

every r > 0 there exists C(g, r) ≥ 0 such that

sup‖x‖1≤r

‖Dg(x)‖1 ≤ C(g, r). (9.72)

To conclude that Xn(·) is exponentially tight in DH[0,+∞), following the proof of

Theorem 8.3 it is now enough to show that if g ∈ A then for T, s > 0, λ ∈ R there exist

random variables γn(s, λ, T ), nondecreasing in s and satisfying (8.5) and (8.6). The proof

of this is very similar to the proof of Theorem 8.3 if we use (9.72. We leave the details to

the readers.

Summarizing we have shown that if f : [0,+∞) → V is continuous, tr(Q1) < +∞and x ∈ V, then the sequence Xn(·; 0,x) of solutions of SNS equation (9.1) satisfies

the large deviation principle in C([0,+∞);H).

47

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