数理解析研究所講究録
第 1904 巻 2014 年 35-45

35Super-Brownian motion in random environment
and its duality
Makoto Nakashima
nakamako@math.tsukuba.ac.jp,
Division of Mathematics, Graduate School of Pure and Applied Sciences
Mathematics,University of Tsukuba
Abstract
In [19], the author construct super-Brownian motion in random environment as the limit points of scaled branching random walks in random
environment which are solutions of an SPDE. To see its convergence, we
use the exponential dual process. In our case, the exponential dual processsatisfies a certain SPDE.

We denote bya probability space. Let, 1, 2,, and $\mathbb{Z}=\{0, \pm 1, \pm 2, \}$ . We denote by
the set of finite
Borel measures on with the topology by weak convergence. Let $C_{K}(S)$ be theset of continuous functions with support compact. If $F$ is a set of functions on, we writeorfor non-negative functions in $F.$$(\Omega, \mathcal{F}, P)$$\{1$, 2, 3,$\}$$\mathbb{N}=\{0$$\},$$\mathbb{N}^{*}=$$\mathcal{M}_{F}(S)$$S$$\mathbb{R}$1$F_{+}$$F^{+}$

Introduction

Dawson and Watanabe independently introduced super-Brownian motion [5, 23]
which was obtained as the limit of critical (or asymptotically critical) branching
Brownian motions (or branching random walks). Also, It is known that superBrownian motion appears as scaling limit of several models in physics or biology.
There are many books for introduction of super-Brownian motion [7, 10] anddealing with several aspects of it [8, 9, 12, 20].
There are several ways to characterize SBM, the unique solutions of martingale problem, non-linear PDE, etc. Here, we characterize it as the uniquesolution of the martingale problem:
Definition 1.1. We call a measure valued process $\{X_{t}(\cdot):t\in[0, \infty)\}$ superBrownian motion whenis the unique solution of the martingale problem$X_{t}$$\{\begin{array}{l}For all \phi\in \mathcal{D}(\Delta) ,Z_{t}(\phi):=X_{t}(\phi)-X_{0}(\phi)-\int_{0}^{t}\frac{1}{2}X_{s}(\triangle\phi)dsis an \mathcal{F}_{t}^{X}- continuous square- integrable martingale and\langle Z(\phi)\rangle_{t}=\int_{0}^{t}\gamma X_{s}(\phi^{2})ds,\end{array}$36
where $\gamma>0$ is a constant.
We are interested in the path property of super-Brownian motion on which
many researcher wrote papers. Here is one of them, absolute continuity andsingularity with respect to Lebesgue measure.
Theorem 1.2. [11, 20, 21] Assume $X$ is a Super-Brownian motion with $X_{0}=$, where.

$\mu$$\mu\in \mathcal{M}_{F}(\mathbb{R}^{d})$:-valued process(i) $(d=1)$ There exists an adapted continuous
the$t>0\}$ such that $X_{t}(dx)=u_{t}(x)dx$ for all $t>0$ P-a.s. andsatisfies$P$
SPDE (defined on the larger probability space$\{u_{t}$$C_{K}(\mathbb{R})$$u$$(\Omega’,$$\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+\sqrt{\gamma u}\dot{W},$$\mathcal{F}’,$$u_{0+}(dx)=\mu(dx)$ ,where $W$ is an white noise defined on the larger probability space(ii) $(d\geq 2)X_{t}$(SPDE)$(\Omega’,$$\mathcal{F}’,$$P$

is singular with respect to Lebesgue measure almost surely.

Remark: There are some results on the detailed path properties for $d\geq 2.$

We focus on (SPDE). (SPDE) is generally expressed as$\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+a(u)\dot{W}$where $a(u)$ is -valued continuous function on
There are some examples for (SPDE $(a)$ ):$\mathbb{R}$(SPDE $(a)$ ),$\mathbb{R}.$(a) If $a(u)=\lambda u$ , then the solution of (SPDE $(a)$ ) is the Cole-Hopf solution of
KPZ equation.

(b) If $a(u)=\sqrt{u-u^{2}}$ , then the solution of (SPDE $(a)$ ) appears as the densityof stepping-stone model.
Also, we constructed another example of (SPDE $(a)$ ) in [19].
Remark: The existence of solutions for (SPDE $(a)$ ) is studied in [14] withand the initial conditionsome assumptions on
In [19], we constructed some measure valued process as a limit points ofsome particle systems which satisfies an SPDE,$a$$\mu.$$\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+\sqrt{\gamma u+\beta^{2}u^{2}}\dot{W}.$In this article, we will give a review of the author’s paper [19].

372Super-Brownian motion in random environment

Super-Brownian motion in random environment was originally introduced by
Mytnik [15]. He obtained super-Brownian motion in random environment as
the scaling limit of branching Brownian motion in random environment, where
random environment means that oﬀspring distributions depends on time-space
site.

2.1

Branching Brownian motion in random environment

Branching Brownian motion in random environment is defined by the following
rule.
(i) For each $N$ , particles locate in(ii) Each particle at time

time
bility$k+1$$\overline{n}$$\frac{k}{n}$$\{x_{1}, \cdots, x_{K_{n}}\}\subset \mathbb{R}^{d}$at time O.

independently performs Brownian motion up to

and then independently splits into two particles with proba-$\frac{1}{2}+\frac{\xi_{k}^{(n)}(x)}{2n^{1/2}}$or dies with probability$\frac{1}{2}-\frac{\xi_{k}^{(n)}(x)}{2n^{1/2}}$

)

where

$x$

is thesite which the particle reached at timeand$\xi_{k}^{(n)}(x)=(-\sqrt{n}\vee\xi_{k}(x))\wedge\sqrt{n}.$i.i. . random field which is defined by$\{\{\xi_{k}^{(n)}(x)\}_{x\in \mathbb{R}^{d}}, k\in \mathbb{N}\}$$\frac{k+1}{n}$

is

$d$

$\{\{\xi(k)\}_{x\in \mathbb{R}^{d}} :k\in \mathbb{N}\}$is i.i. . random field on$E[|\xi_{k}(x)|^{3}]<\infty$$d$for all$x\in \mathbb{R}^{d}$$P(\xi_{k}(x)>z)=P(\xi_{k}(x)<-z)$and$\mathbb{R}^{d}$such that$k\in \mathbb{N}.$for all$x\in \mathbb{R}^{d},$$z\in \mathbb{R}$Let $g_{n}(x, y)$ and $g(x, y)$ be the covariance functions ofspectively, that is, and$x_{k}^{(n)}(\cdot)$$k\in \mathbb{N}.$and$\xi_{k}$re-$g_{n}(x, y)=E[\xi_{k}^{(n)}(x)\xi_{k}^{(n)}(y)]$$g(x, y)=E[\xi_{k}(x)\xi_{k}(y)] x, y\in \mathbb{R}^{d}, k\in \mathbb{N}.$We assume that $g(x, y)$ is a continuous function with limit at infinity.
We identify the branching Brownian motion in random environment as themeasure valued process by$X_{t}^{(n)}(A)= \frac{1}{n}\#${particles locate infor any Borel set $A.$
Then, we have the following:$A$

at time

$t$

}

38
, where. Then,in
Theorem 2.1. Assume thatis the unique solution of the following martingale prob$X_{0}^{(n)}\Rightarrow X_{0}$$X^{(n)}\Rightarrow X$$\mathcal{M}_{F}(\mathbb{R}^{d})$$X\in C([O, \infty), \mathcal{M}_{F}(\mathbb{R}^{d}))$lem:$\{\begin{array}{l}For all \phi\in C_{b}^{2}(\mathbb{R}^{d}) ,Z_{t}(\phi)=X_{t}(\phi)-X_{0}(\phi)-\int_{0}^{t}X_{s}(\frac{1}{2}\triangle\phi)dsis an \mathcal{F}_{t}^{X} continuous square integrable martingale such that Z_{0}(\phi)=0 and\langle Z(\phi)\rangle_{t}=\int_{0}^{t}X_{s}(\phi^{2})ds+\int_{0}^{t}\int_{R^{d}xR^{d}}g(x, y)\phi(x)\phi(y)X_{s}(dx)X_{s}(dy)ds.\end{array}$

(2.1)

$u$

Also, Mytnik gave a remark that if $d=1$ and $g(x, y = \delta_{0}(x-y)$ and let
be a solution of SPDE$\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+\sqrt{u+u^{2}}\dot{W},$then $X_{t}(dx)=u(t, x)dx$ solves the martingale problem (2.1). Since $\delta_{0}(x-y)$
is not a continuous function any more, it is a “special case” of Mytnik’s result.
In [19], we obtain a measure valued process satisfying the special case as thescaling limit of some branching systems.

2.2

branching random walks in random environment

Although there are a lot of definition of branching random walks in random
environment, ours is the one introduced in [2]. Let
be large enough. We
consider the system where particles move on and the process evolves according
to the following rule:$N\in \mathbb{N}$$\mathbb{Z}$(i) There are particles at$\{x_{1},$$)x_{M_{N}}\}$

at time O.

(ii) If a particle locates at site
at time , then it moves to a uniformly
chosen hearest neighbor site and split into two particles with probability, where jump and branchor dies out with probability
particles,
ing system are independent of eachare$\{1, -1\}$ -valued i.i. . random variables with $P(\xi(n, x)=1)=P(\xi(n, x)=$, and $\beta>0$ is constant.
$x\in \mathbb{Z}$$n$$\frac{1}{2}-\frac{\beta\xi(n,x)}{2N^{1/4}}$$\frac{1}{2}+\frac{\beta\xi(n,x)}{2N^{1/4}}$$\{\xi(n, x):(n, x)\in \mathbb{N}\cross \mathbb{Z}\}$$d$$-1)= \frac{1}{2}$Remark: In our model, random environment is given by branching mechanics which are updated for each site and each time.
Remark: $N$ is the scaling parameter which tends to infinity later. Also, weemphasize that the fluctuations of oﬀspring distributions are diﬀerent from theones in [15].
We don’t give the mathematically rigorous definition in this paper.

39
2.3

Super-Brownian motion in random environment

In this subsection, we introduce super-Brownian motion in random environment.
Super-Brownian motion is obtained as the limit of scaled critical branching
Brownian motions (branching random walks). When we look at our model, themean number of oﬀsprings from one particle is 1, so that we can regard ourmodel as “critical” branching random walks in random environment in somesense. We will try to obtain the scaled limit process.
We denote by
the number of particles at site at time . We define$X_{t}^{(N)}(dx)$by$B_{n,x}^{(N)}$

$n$

$x$

$X_{0}^{(N)}(dx)= \frac{1}{N}\sum_{i=1}^{M_{N}}\delta_{x_{i}}(dx)$,$X_{t}^{(N)}(dx)= \frac{1}{N}\sum_{y\in \mathbb{Z}}B_{\lfloor tn\rfloor,y}^{(N)}\delta_{y}(N^{1/2}dx)$

More simply, we can express the definition of
be a Borel set in . Then,$X_{t}^{(N)}(\cdot)$.

as follows: Let$A\in \mathcal{B}(\mathbb{R})$$\mathbb{R}$$X_{t}^{(N)}(A)= \frac{\#\{particleslocatesinN^{1/2}Aattime\lfloor Nt\rfloor\}}{N}.$In [19], we have the following result.

Theorem 2.2. If, thenin
is $C$ -relatively
compact. Moreover, if we denote bya limit point, thenis absolutely
continuous with respect to Lebesgue measure for all $t>0$ P-a.s. and its density$u(t, x)$ satisfies SPDE
$X_{0}^{(N)}\Rightarrow X_{0}$$\mathcal{M}_{F}(\mathbb{R})$$\{X_{t}(\cdot)\}$$\{X^{(N)} :N\in \mathbb{N}^{*}\}$$X_{t}$$\frac{\partial u}{\partial t}=\frac{1}{2}\triangle u+\sqrt{u+2\beta^{2}u^{2}}\dot{W}, u_{0+}dx=\delta_{0}(dx)$Formally,$\{X_{t}(\cdot) :t\geq 0\}$

.

(2.2)

is a solution of the following martingale problem:$\{\begin{array}{l}For all \phi\in \mathcal{D}(\triangle) ,Z_{t}(\phi):=X_{t}(\phi)-X_{0}(\phi)-\int_{0}^{t}\frac{1}{2}X_{S}(\triangle\phi)dsis an \mathcal{F}_{t}^{X}- continuous square- integrable martingale and\langle Z(\phi)\rangle_{t}=\int_{0}^{t}X_{s}(\phi^{2})ds+2\beta^{2}\int_{0}^{t}\int_{\mathbb{R}\cross \mathbb{R}}\delta_{x-y}\phi(x)\phi(y)X_{s}(dx)X_{s}(dy)ds.\end{array}$

To be rigorous,$\{X_{t}(\cdot) :t\geq 0\}$is a solution of the following martingale problem:$\{\begin{array}{l}For all \phi\in \mathcal{D}(\Delta) ,Z_{t}(\phi):=X_{t}(\phi)-X_{0}(\phi)-\int_{0}^{t}\frac{1}{2}X_{s}(\Delta\phi)dsis an \mathcal{F}_{t}^{X}- continuous square- integrable martingale and\langle Z(\phi)\rangle_{t}=\int_{0}^{t}X_{s}(\phi^{2})ds+2\beta^{2}\int_{0}^{t}\int_{\mathbb{R}}\phi^{2}(x)X_{s}^{2}(x)dxds.\end{array}$

(2.3)

40
We shall call solutions of the above martingale problem super-Brownian
motion in random environment.
Also, we are interested in uniqueness of solutions of (2.3). Before giving ananswer, we introduce a notation. Let
be the set of rapidly decreasing$C_{rap}^{+}(\mathbb{R})$functions, that is$C_{rap}^{+}( \mathbb{R})=\{g\in C^{+}(\mathbb{R}):|g|_{p}\equiv\sup_{x\in \mathbb{R}}e^{p|x|}|g(x)|<\infty, \forall p>0\}.$The following theorem gives us an answer.
Theorem 2.3. Solutions of martingale problem (2.3) is unique if $X_{0}(dx)=$$u_{0}(x)dx$, then $X^{(N)}\Rightarrow X$ in. Moreover, if
for, where $X$ is a solution of the martingale problem of (2.3).
$X_{0}\in \mathcal{M}_{F}(\mathbb{R})$$u_{0}\in C_{rap}^{+}(\mathbb{R})$$C([O, \infty), \mathcal{M}_{F}(\mathbb{R}))$

3

Uniqueness

Although there are several definition of the uniqueness for SPDE, we consider
the uniqueness in law for our model. The readers can refer some papers on theuniqueness (in law or pathwise) of the solutions of (SPDE $(a)$ ) $[13$ , 16, 17, 18
In most cases, H\"older continuity of
influences on the uniqueness. Actually,, 1 . In our case, the H\"older
the uniqueness in law holds when $a(u)=u^{\gamma},$contiuity ofis so that we can conjecture the uniqueness in law does hold.
We suppose thatis a solution of (2.3).
The main idea to prove the uniqueness of solutions of the martingale problem(2.3) is to prove the existence of the “dual” process $\{Y_{t} : t\geq 0\}$ , which is-valued process and satisfies the equation
$].$

$a$

$\gamma\in$$a$$[ \frac{1}{2}$$]$$\frac{1}{2}$$X_{t}$$C_{rap}^{+}(\mathbb{R})$$E[\exp(-\langle Y_{t}, X_{0} =E[\exp(-\langle\phi, X_{t} (3.1)$$\mu\rangle=\int_{R}\phi(x)\mu(dx)$ for, wherefor each.
In particular, a solution of the SPDE$\phi\in C_{rap}^{+}(\mathbb{R})$$\langle\phi,$$\phi\in C_{b}(\mathbb{R})$and$\mu\in$$\mathcal{M}_{F}(\mathbb{R})$$\frac{\partial Y}{\partial t}=\frac{1}{2}\Delta Y_{t}-\frac{1}{2}Y_{t}^{2}-\sqrt{2}|\beta|Y_{t}\tilde{W},$is a “dual” process of $\{X_{t} : t\geq 0\}$ . Indeed, if
follows from Ito’s lemma that

$Y_{0}(x)=\phi(x)$$Y_{t}\in C_{+}^{1}(\mathbb{R})$(3.2)

for all $t\geq 0$ , then it$\exp(-\langle Y_{t-s}, X_{s}\rangle)=\exp(-\langle Y_{t}, X_{0}\rangle)-\int_{0}^{s}\langle\frac{1}{2}Y_{t-u}^{2},$ $X_{u}\rangle\exp(-\langle Y_{t-u}, X_{u}\rangle)du$$- \int_{0}^{s}\beta^{2}\langle Y_{t-u}^{2},$$X_{u}^{2} \rangle\exp(-\langle Y_{t-u}, X_{u}\rangle)du+\int_{0}^{s}\langle\frac{1}{2}\Delta Y_{t-u},$ $X_{u}\rangle\exp(-\langle Y_{t-u}, X_{u}\rangle)$$- \int_{0}^{s}\langle\frac{1}{2}\Delta Y_{t-u}, X_{u}\rangle\exp(-\langle Y_{t-u}, X_{u}\rangle)$41$+ \int_{0}^{s}\langle Y_{t-u}^{2},$$\frac{1}{2}X_{u}+\beta^{2}X_{u}^{2}\rangle\exp(-\langle Y_{t-u}, X_{u}\rangle)du+($Then, taking expectation and letting $s=t,$ $E[\exp(-\langle Y_{0},$However, we find thatis not diﬀerentiable for any
so we need to approximate
bywill omit a proof of the statement that

$X_{t}$$x\in \mathbb{R}$$Y_{t}$

martingale part .
$)$$=E[\exp(-Y_{t},$ $X_{0}$such that $Y_{t}(x)\neq 0$. We$Y_{t}^{\epsilon}(x)= \int_{\mathbb{R}}\frac{1}{\sqrt{2\pi\epsilon}}Y_{t}(x+y)\exp(-L^{2})dy$$Y_{t}$$\lim_{\epsilonarrow 0}E[\exp(-\langle Y_{t}^{\epsilon}, X_{0} =E[\exp(-\langleY_{t}, X_{0} =E[\exp(-\langle Y_{0}, X_{t}$(3.3)

for any $t\in[0, \infty$ ) and $Y_{0}(x)=\phi(x)\in C_{rap}^{+}(\mathbb{R})$ . We remark that when we prove(3.3), we have used estimates coming from branching random walks in random
environment. It implies that we don’t still prove the uniqueness of solutions of
the martingale problem (2.3) for. However, if $X_{(}dx$ ) $=\psi(x)dx$
for, then we can prove (3.3) directly by using the properties of
The existence of nonnegative solutions to (3.2) for the case where
follows from [22] by using Dawson’s Girsanov theorem[6]. Indeed, theexistence and the uniqueness of nonnegative solutions to$X_{0}\in \mathcal{M}_{F}(\mathbb{R})$$\psi\in C_{rap}^{+}(\mathbb{R})$$X_{t}.$$Y_{0}\in$$C_{rap}^{+}(\mathbb{R})$$\tilde{Y}_{0}(x)=\phi(x) , \frac{\partial}{\partial t}\tilde{Y}_{t}(x)=\frac{1}{2}\Delta\tilde{Y}_{t}(x)+\sqrt{2}|\beta|\tilde{Y}_{t}(x)\tilde{W}(t, x)$.

has been already known, whereis a time-space white noise independent of$X[1$ , 22 . We denote by
the law of . Let
be the probability measurewith Radon-Nikodym derivatives$\tilde{W}$$\tilde{Y}$$]$$P_{Y}$$P_{Y}-$$\frac{dP_{Y}}{dP_{\tilde{Y}}}|_{\mathcal{F}_{r^{Y}}^{-}}=\exp(\frac{\gamma}{2\sqrt{2}|\beta|}\int_{0}^{t}\int_{\mathbb{R}}\tilde{Y}_{s}(y)\tilde{W}(ds, dy)-\frac{\gamma^{2}}{16\beta^{2}}\int_{0}^{t}\int_{\pi}\tilde{Y}_{s}^{2}(y)dyds)$

Then, under.

satisfies (3.2) and is also a-valued process. Thus,
we constructed a solution to (3.2). Especially, we remark that the solutions to(3.2) satisfy for $t\geq 0$$\tilde{Y}$$\tilde{Y}$$P_{Y},$$C_{rap}^{+}(\mathbb{R})$$Y_{t}(x)= \int_{\mathbb{R}}p_{t}(x+y)\phi(y)dy-\frac{\gamma}{2}\int_{0}^{t}\int_{\pi}p_{t-s}(x+y)Y_{s}^{2}(y)dyds$$+ \sqrt{2}|\beta|\int_{0}^{t}\int_{\mathbb{R}}p_{t-s}(x+y)Y_{s}(y)\tilde{W}(ds,dy)$where$p_{t}(x)= \frac{1}{\sqrt{2\pi t}}\exp(-\frac{x^{2}}{2t})$

for $t>0$ and

The following lemma tells us thatmartingale problem:

,

(3.4)

$x\in \mathbb{R}.$$(\{Y_{t}\}_{t\geq 0}, \mathcal{F}_{t}^{Y}, P_{Y})$is a solution to the$\{\begin{array}{l}For all \psi\in C_{b}^{2}(\mathbb{R}) ,\tilde{Z}_{t}(\psi)=\langle Y_{t}, \psi\rangle-\langle Y_{0}, \psi\rangle+\frac{\gamma}{2}\int_{0}^{t}\langle Y_{s}^{2}, \psi\rangle ds-\int_{0}^{t}\langle Y_{s}, \frac{1}{2}\triangle\psi\rangle dsis an \vee\prime--Yt- continuous square integrable martingale and\langle\tilde{Z}(\psi)\rangle_{t}=2\beta^{2}\int_{0}^{t}\langle Y_{s}^{2}, \psi^{2}\rangle ds\end{array}$

42
Lemma 3.1. Let. Let
tive solution to (3.2). Then, we have that$\phi\in C_{rap}^{+}(\mathbb{R})$$(\{Y_{t}\}_{t\geq 0}, \mathcal{F}^{Y}, \{\mathcal{F}_{t}^{Y}\}_{t\geq 0}, P_{Y})$$E_{Y}[ \int_{R}Y_{t}(x)dx]\leq\int_{\mathbb{R}}\phi(x)dx$

be a nonnega-

,

(3.5)

and
$E_{Y}[ \int_{0}^{t}\int_{\mathbb{R}}Y_{s}^{p}(x)dxds]<\infty$

(3.6)

,

for all $0\leq t<\infty$ andProof. (3.5) is clear from (3.4). Let $0\leq t\leq T$ . Also, we have that$p\geq 1.$$Y_{t}^{p}(x) \leq C(p, \beta)\{(\int_{R}p_{t}(x+y)\phi(y)dy)^{p}+(\int_{0}^{t}\int_{R}p_{t-s}(x+y)Y_{s}(y)\tilde{W}(ds, dy))^{p}\}.$We define$T( \ell)=\inf\{t\geq 0 :\sup_{x}e^{|x|}|Y_{t}(x)|>\ell\}.$We remark that $T(\ell)arrow\infty P_{Y}-a.s$ . assincefor all $t\geq 0$$P_{Y}-a.s$ .
H\"older’sThen, we have by
inequality and Burkholder-Daivs-Gundy
inequality that$\ellarrow\infty$$Y_{t}\in C_{rap}^{+}(\mathbb{R})$$E_{Y}[Y_{t}^{p}(x):t\leq T(\ell)]$$\leq C(p, \beta)E_{Y}[(\int_{N}p_{t}(x+y)\phi(y)dy)^{p}+(\int_{0}^{t}\int_{R}1\{t\leq T(\ell)\}p_{t-s}^{2}(x+y)Y_{s}^{2}(y)dyds)^{2}R]$$\leq C(p, \beta)(\int_{\mathbb{R}}p_{t}(x+y)\phi(y)dy)^{p}$$+C(p, \beta)E_{Y}[(\int_{0}^{t}\int_{\mathbb{R}}1\{t\leq T(\ell)\}p_{t-s}^{2}(x+y)Y_{s}^{p}(y)dyds)^{2}R(\int_{0}^{t}\int_{R}p_{t-s}^{2}(x+y)dyds)^{S-1}]$$\leq C(p, \beta)(\int_{R}p_{t}(x+y)\phi(y)dy)^{p}$$+C(p, \beta)t^{R_{\frac{-2}{4}}}\int_{0}^{t}\int_{\mathbb{R}}(t-s)^{-\frac{1}{2}}p_{t-s}(x+y)E_{Y}[Y_{s}^{p}(y):t\leq T(\ell)]dyds,$. Intewhere we have used thatandgrating on over and letting $\nu(s, t, \ell, p)=\int_{R}E_{Y}[Y_{s}^{p}(x) : t\leq T(\ell)]dx$ , then$\nu(s, t, \ell,p)<\infty$ by definition and we have that$p_{s}^{2}(x)\leq Cs^{-\frac{1}{2}}p_{s}(x)$$x$$\int_{0}^{t}\int_{R}p_{s}^{2}(x)dxds\leq Ct^{\frac{1}{2}}$$\mathbb{R}$$\nu(t, t, \ell,p)\leq C(p, \beta, T)(1+\int_{0}^{t}(t-s)^{-\frac{1}{2}}\nu(s, t, \ell,p)ds)$,

43
where we have used $\sup_{t\leq T}\int_{\mathbb{R}}(\int_{\pi}p_{t}(x+y)\phi(y)dy)^{2}dx<\infty$ . It follows from
Lemma 4.1 in [14] that$\nu(t, t,\ell,p)\leq C(p, \beta, T, Y_{0})\exp(C(p, \beta, T, Y_{0})t^{\frac{1}{2}})$

for $t\leq T.$

Since the right hand side does not depend on , it follows from the monotone
convergence theorem that$\ell$$\int_{N}E_{Y}[Y_{t}^{p}(x)]dx\leq C(p, \beta, T, Y_{0})$and$\int_{0}^{T}\int_{\mathbb{R}}E_{Y}[Y_{t}^{p}(x)]dxdt\leq C(p, \beta, T, Y_{0})T.$$\square$

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