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Local time for stable discontinuous superprocesses inone dimensionMarica Lewin aa Dept. of Mathematics , Technion—Israel Inst. of Technology , Haifa, 32000, IsraelPublished online: 03 Apr 2007.

To cite this article: Marica Lewin (1999) Local time for stable discontinuous superprocesses in one dimension, StochasticAnalysis and Applications, 17:1, 71-84, DOI: 10.1080/07362999908809588

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STOCHASTIC ANALYSIS AND APPLICATIONS, 17(1), 71-84 (1999)

Local Time for Stable Discontinuous Superprocesses in One Dimension

Marica Lewin Dept . of Mathematics

Technion-Israel Inst. of Technology Haifa 32000, Israel

Abstract

We investigate the existence of a local time process for a class of discontinuous measure branching processes in R. Formally, the local time process is defined as

L( t ) = (X,, 6) ds, where X, is the measure branching process and 6 stands for the I' Dirac function.

Actually,

~ ( t ) = ~2 ht(x . , c p , ~ . where cp, is a sequence of nice functions converging as generalized functions to 6. The limit is in the weak topology of Prohorov probability measures.

We prove that L( t ) exists and it has a cadlag version with Probability 1.

We attempt to investigate the existence of the local time process for a class of discon-

tinuous superprocesses, denoted in the literature also by MB (Measure Branching)

processes.

The local time and intersection local time for superprocesses were intensively studied

by Dynkin, Iscoe, Rosen, Adler and Lewin. It was proved that the local time for

continuous superprocesses exists up to dimension 3, and it is a continuous process

POI.

C o p ) r ~ g h t O 1999 by Marcel Dekker, Inc.

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First we introduce some notations and preliminaries.

Let R be the real line with the usual scalar product (x, y) = xy , x, y E R, and the d

usual norm 1x1. Rd is the Euclidean space of dimension d , with ( s , y) = szy, and 1 1 1

1

the norm / I s \ I = (x I:)' . If u : R - R is a Borel measurable function, and p a

Borel measure on R, then in the case u ,

p is integrable, we denote its integral over R by ( p , u ) , or u(x) p(dx). X will be J the Lebesgue measure on R, and 6, denotes the 6 function at x E R. If D c R+ x R

is an open set, and u : D - R, we write u ( t ) for the function whose value at x is

u ( t , r ) . Usually, by D we denote the domain of a function, operator, etc.

Two infinitesimal generators will be associated with the processes considered in this

note. The first is A, the Laplacian, which generates the semigroup associated with

the Brownian motion, St. The second operator is a fractional power of the Laplacian,

A, = 0 < CY < 2 [16], which generates the contraction semigroup Sr,

associated with a symmetric stable motion. St and Sf are convolution operators.

Generally, for covering both cases, we write A,, 0 < cr < 2.

We denote by C(R) the space of continuous bounded functions on R, equipped with

the usual sup norm topology. We denote by 1 1 p(x) I / = sup Ip(x)l. C r will be the xER

subset of functions on IR, whose members are infinitely differentiable functions with

a compact support. We have also,

We denote by D(R+, R ) the space of functions from R+ into R, which are cad-

lag (right continuous, possessing limits from the left) endowed with the Skorohod

topology [4].

L2(X) will be the space of square integrable functions with respect to the Lebesgue

measure A. L'(x)+ will be the subset of positive functions of L ~ ( x ) . L1(X), or simply

L1 will be the collection of all integrable functions and ) I [ I z will be the norm in L ~ .

We denote by M ( R ) the set of all positive Radon measures on the Borel o-algebra Dow

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DISCONTINUOUS SUPERPROCESSES 73

of R. M ( R ) is a complete, separable metric space when equipped with the vague

topology 1151. The set of positive Radon measures,

also carries the relative vague topology.

We write D1(T) for the set of all distributions (generalized functions) on T C R

In the following, ut f will be a nonlinear semigroup, called a cummulant semigroup,

a unique solution of the following nonlinear partial differential equation (P.D.E.)

for f E Cr(R), 0 < P 5 1.

ut determines uniquely a stochastic process (a, F, PA, Xt) taking values in a suitable

subset of M ( R ) , a process that is called a measure valued branching stable process

(or superstable process) driven by A,.

The Laplace functional of Xt is expressed by

As can be seen from the notation, we choose as an initial measure the Lebesgue

measure X(dx) = dx on R.

It is proved that if 0 < cy < 1, and P = 1, then for every fixed t > 0,

Px(Xt(dx) is singular with respect to dx) = 1 .

and if 1 < a 5 2, ,B = 1, then for every fixed t > 0, pA(xt (dx) ) is absolutely

continuous with respect to dx = 1 (see [17]).

Remark: If we treat the problem in R~ d 2 2, Xt is always singular with respect

to dx. We treat in this note the case 0 < p < 1, where the measures Xt(dx) are

singular with respect to the Lebesgue measure, but this aspect does not play any

role in the present note. Dow

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LEWIN

By {Ft}t>o we denote the collection of u-algebras defined by

which is the Bore1 u-algebra generated by (X,, p ) , s <_ t . Of course {3t) t>o is a

stochastic basis, included in 3. ( X t ) p o is a Markov process with respect to Ft, (a

strong Markov process).

By P t (x) we understand the normal density of probability

-2 1 Pt(x) = e r - m t > O

and Pe will be the density for the real stable process.

Remark: The problem can be treated more generally, if instead of A,, we work

with a strong generator of a Feller semigroup St.

The process Xt obtained via the Laplace functional (1) and (2) has always dis-

continuous trajectories for 0 < p < 1 (cadlag) w.p.1 and the second moment is

always infinite. X t has continuous trajectories and all the moments are finite if and

only if P = 1 (see [12]). Iscoe [9] defined the total weighted occupation time as a

superprocess Yt,

with the representation for the Laplace functional of Yt,

where u( t ) is the solution of the following evolution equation

p E D (A,)+ ( p is a positive function belonging to the domain of definition A,.

u ( t ) takes positive values.

Definition 2 (local t ime). W e consider a sequence cp,,,(y) of functions belonging

to Cp(R) converging to b ( x ) in the distributional sense, when E + 0+.

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DISCONTINUOUS SUPERPROCESSES 75

We define the process t

~ , ( t , X ) = J (xs, 0

The local time of X,, denoted by L ( t , x ) is the weak limit of the real processes.

L,(t, x), when e + O+, that is

lim L, ( t , x) = L( t , x) EF-iO+

By a weak limit we mean a limit in the weak topology of Prohorov probability

measures (see [4]). Of course, the limit (7) may or may not exist. We want to prove

the existence of the local time in a one dimensional case, according to the following

theorem.

T h e o r e m 1. The process L,(t, x ) converges weakly as E - O+ (for t > 0) to a cadlag

process L ( . , x ) , such that

where v is the unique solution of the singular P.D.E.

The distribution of L( . , x ) does not depend on x .

Proof: First, without loss of generality we assume that

~ , , ~ ( y ) 2 Ofor all x , y E IR .

Obviously by construction, LEn ( t , x ) , belongs to D([o, TI, R). To prove the weak

convergence of L,(t, x ) , we have to show for each sequence { E ~ ) ? = ~ , rn + 0 ,

i) for all f E R!, the sequence of Rk-valued random variables,

a,, = ( ~ , , ( t l , x ) , . . . , Lrn ( t k r I ) converge in distribution to a Rk-valued random

variable, not depending on the choice of

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ii) L,,, ( t , x ) is tight in D (10, TI, R) , for every x E R

Without loss of generality, we can assume x = 0. First, we prove i), and in order to

keep a simple notation, we verify i) for k = 2.

Let 8 E R: and f = { t l , t 2 ) , with 0 < t l 5 t2. Then,

Ex exp - (8, o,, ) = Ex exp (-01 Li; - 02Lt;)

Hence,

where uEn ( t ) is the solution of the equation

It results from a generalization of (4) and (5)[10].

is finite, for any t 2 E R.

For this purpose we show first the existence of a unique strong solution [3, page 601

for the nonlinear equation

- AU - d iP( t ) + ( ~ ~ l ~ , ~ , ~ ( t ~ - t ) + e2)6 , 0 5 t tl

u(0) = 0 (12)

in the space H-' (R) [7].

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DISCONTINUOUS SUPERPROCESSES 77

Remark : Without loss of generality we work with A. Usually the properties men-

tioned for A are inherited by A,.

We consider the Sobolev spaces HA(R), HP1(R), L2(R). HA is a Hilbert space with

the norm

where 11 u / I 2 = (k 1 ~ ( x ) 1 2 d x ) ' (u t L ~ ( A ) ) .

H-' is the set of all distributions (generalized functions) which can be represented

as

D' means the derivative of order 1, 1 E N [7] and it is the dual space of Hi (R) [7]

H-'(R) will also be considered a Hilbert space endowed with the negative norm of

Lax [Yoshida] or some other equivalent norm.

The distribution S E H-'(R), and for this reason we look for a strong solution of

(12) in H-'(R). The Laplace operator (-A) is the canonical isomorphism from

Hi(R) onto H-'(R) . H-'(R) is a Hilbert space with the inner product

((u, v)) = (-A-'u, v) for all u, v E H-' (R) (13)

It is proved [3] that A is a subdifferential mapping of a proper, convex function, so

(-A) is maximal monotone in H-' x H-'. We check that b(v) = u!,?', 0 < P < 1

is also maximal monotone in HP'(R) x HV'(R). This happens because the real

function cp(x) = XI+', 0 < P < 1 is monotone increasing from R+ onto R+.

H-' is a Hilbert space, so (-A) and b(u) are maximal accretive in H-' x H-' and

the intersection of their domain is non empty, D(-A) n D(b) # 4. So, we conclude

that the sum of the two operators, the nonlinear operator -Au + b(u) is maximal

accretive in H-', so A(u) - b(u) is maximal dissipative in H-'.

The function f (1) = 81110,tll(t~-t)+826, for t E [O, t 2 ] which belongs to L' [(0, t2 ) , H-'1

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78 LEWIN

According to a theorem of existence [3, page 131, Th. 2.21 for quasi-autonomous

differential equations, the equation(l2) has a unique solution u(t) in H-'(R). u(t)

is the uniform limit of un(t), solutions of (10) in H-'(R).

From the definition of the solution, we know that u(t) E Hi (R), a.e. in t . For (11)

to be valid for every t E R+, we have to show that u(t, x) is a function of x for all

t R + , and even more, that ~ ( t , x) E L'(R) for all t E R+. First we prove that u(t)

is a function, more precisely that u E c[[o, I], L'(R)], for every t E R+. Denoting

the free term of (10) by f,(s) and using the mild form of ( lo) , we obtain

The positivity of un(t) implies that

and also the positivity of u( t )

Taking into account the expression of fn(s) we can write

where M is a majorant for 6'l18E[0,tll + 6'2110,t2~. TO continue the proof, we first show

that

I s s ~ n ds 1 1 2 5 A (16)

for every t E R+, with A being a positive constant.

Because the limit of p,, (2) to 6(0) does not depend on the choice of cp,, , we choose

as 9%" (x), PE"(2).

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DISCONTINUOUS SUPERPROCESSES 79

The last equality is obtained from [16], using the Fourier transform of the convolution

of P,(x) and P,, (2).

The real function e+ is increasing, as a function o f t , for t > 0, so, we obtain

So, we proved (16).

For every function $ E C r ( R ) and 7 a generalized function, we denote by (7, $),

their natural pairing. If un( t ) is considered as a generalized function for every

$ E C r ( R ) , we get

for all n E N and for every t E R+, where un( t ) is the solution of ( l o ) , with PEn(x) ,

as ' ~ n ( z ) .

We already proved that lim un( t ) = u( t ) in H- ' (R ) , so u( t ) is the limit of u,(t), n - w

for every t E [O,tz] in the distributional sense. Taking the limit in (19) we obtain

In particular, we get

The relation (21) and the following Lemma 1 from [7] help us to conclude that u( t ) ,

the solution of (12), is a function belonging to L'(R) for every t E [0, tz].

Lemma 1. Let 0 be an open subset of R , and F be in D' (0) . Let 1 < p 5 cm, and

+ $ = 1. Then F is a function from LP(0) 27 and only if there is a finite constant

K , such that

I(F, *) I 5 K 1 1 !& 11 / , for all !& E C r ( O ) . (22)

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80 LEWIN

We show now that u(t) E L1(R), for every t E [0, t z ] The convergence of un(t) to

u(t) in H-'(R) for every t E [0, t2] and the fact that u(t) is a function, implies the

existence of subsequence, denoted also by un(t, x) converging a.e. in x to u(t , x) for

all t E [0,t2]. Also,

Then, by a Fat6u lemma, we obtain

Hence, ~ ( t ) E L'(R), for all t E [O,tz], u(t)dX is finite. So, ( 22 ) L

and ( 2 2 ) implies that the relation (11) takes place. Part i) of the criterion of weak

convergence for the sequence L,, ( t , x) = ( Y ( t ) , P,, ( 2 ) ) is satisfied. It remains to

prove the tightness of the sequence LEn(t, x) in D ( [ o , T I , R ) We shall do it for

We shall apply the "Aldous principle for tightness i n D(R+, R) that is, a sequence

&(t ) is tight in D(Rt, R) , if

I) for each fixed t , y,(t) is tight i n R

II) given a sequence of stopping t imes T,, bounded b y T , and pn a sequence of pos-

itive numbers converging to 0, then, y,(~,, + p,) - y , ( ~ ~ ) converges i n probability

as n -t co.

We now prove a stronger form of u, which is

lim E ( I Y " ( T ~ + pn) - Y " ( T ~ ) I ) = 0 n-+m

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DISCONTINUOUS SUPERPROCESSES 8 1

From the construction of L,,(t, x), it is obvious that I) takes place. So we have to

prove for every Ft stopping time rn 5 T, we have

which will imply condition (23).

Without loss of generality we choose cp,,(x) as being PEn(x). PEn(x) is positive,

( X , , PEn(x)) remains positive for V s E R+, representing the integral of a positive

function with respect to a positive measure. So, we have to prove

By the strong Markov property of the process X t , and the fact that the conditional

mean of X, has the multiplicative property [14] we obtain

the last equality is obtained by changing the order of integration.

Taking into account that r, is a Ft stopping time, the function e 5 is increasing

as a function of s, for s 2 0, and the expression of S,, we obtain by

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82 LEWIN

We mentioned that the moments of order 2 and higher numbers do not exist for

(X t , c p ) , for all t E R+, cp E CoM,, but we have

sup E(Xt,cp) :$ for O < B < p . O<t<T

The relation ( 2 8 ) was proved in the Dawson book [ 6 ] . Hence,

1+p ib sup E(x,,,; e-" ) j sup [E(x , , epX2 ) ] n n

< { E sup (xt, epZ2 ) )A - O<t<T

The first inequality of ( 2 9 ) was obtained by applying the Holder inequality for the

expectation

We assign, (X,,,, e-"'), to f, and we choose, the constant function g = 1, p = I+/?.

To get the second inequality we used the fact that the stopping times have to be

bounded, r n ( w ) 5 T with Probability 1.

From the general theory of superprocesses, it is established that

is a martingale, for every yo E C r ( R ) .

Hence,

< K [ ( E sup ~ ( ~ , t , e - ~ ~ ) - ~ ~ ( ~ ~ , ~ ~ e - ~ ~ ) d s ~ ~ + ~ OSt<T 0

K is a positive constant.

Using Doob's inequality [6] on the first term and Jensen's inequality on the second,

we continue.

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The last conclusion results from some simple calculations and (29 ) .

Combining (30)-(32) we get

sup E(x,, e-z2 ) < co (33 ) n

for any sequence of stopping times T, 5 T

Now we take the limit in (26 ) for h + 0 and n -+ m, use (27 ) and (33 ) , then we

obtain (25) .

Hence we proved both conditions i) and ii) which ensure the weak convergence

of L,( t , x ) , we have established the existence of the local time L ( t , x ) . L ( t , x) is a

process with the state space D ( R + , R ) . By proving the existence of a unique solution

for equation (12 ) , which for every t E R+ is an L' n L' function, we established the

existence and uniqueness of a solution for ( 9 ) .

where u E ( t r x , 6') solves ( 5 ) with p = BP,(x).

Taking the limit in (34 ) for E -+ 0 , for every t E R+, 6' E R+, we obtain (9), which

concludes the proof of our theorem.

Remark: This result may be extended to dimensions d = 2 , 3 but we need different

techniques for proving the tightness.

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[2] Aldous, D.: Stopping T imes and Tightness, Ann. Prob. 6 (1978)

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[14] Watanabe, S.: A Limit Theorem for Branching Processes and Continuous State Branching Processes, J . Math., Kyoto Univ., 8 (1968), 141-167.

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