1 Introduction
In this paper we consider the stochastic parabolic equation
where $(t,x)\in [0,T]\times \mathbb{R}$, μ is a general stochastic measure on the Borel σ-algebra on $\mathbb{R}$ (see Section 2), f, σ are measurable functions, $\mathcal{L}$ is the operator of the kind
Here a, b, c are defined on $[0,T]$. We prove that under certain conditions on a, b, c, f, σ the solution of (1), considered in the mild sense, tends to 0 a.s. uniformly on x. Note that regularity of the solution was proved in [2], the solution’s convergence in the case of integrator’s convergence was proved in [18] and the averaging principle for such an equation was established in [12].
(1)
\[ \left\{\begin{array}{l}\mathcal{L}u(t,x)dt+f(t,x,u(t,x))\hspace{0.1667em}dt+\sigma (t,x)\hspace{0.1667em}d\mu (x)=0\hspace{0.1667em},\\ {} u(0,x)={u_{0}}(x)\hspace{0.1667em},\end{array}\right.\](2)
\[ \mathcal{L}u(t,x)=a(t)\frac{{\partial ^{2}}u(t,x)}{{\partial ^{2}}x}+b(t)\frac{\partial u(t,x)}{\partial x}+c(t)u(t,x)-\frac{\partial u(t,x)}{\partial t}.\]The asymptotic behavior of the moments of solutions of a stochastic differential system driven by a Brownian motion was considered in [5]. The problem of the convergence of the solution of a nonautonomous logistic differential equation to zero as time coordinate goes to infinity, with disturbance of coefficients by white noise, centered and noncentered Poisson noises, was studied in [4]. Asymptotics of the solution of the stochastic heat equation with white noise, as time variable goes to infinity for the fixed spatial coordinate, was studied in [10] while asymptotic properties of the solution of the stochastic wave equation driven by a Lévy process were given in [7]. Behavior of solutions of different equations with a general stochastic measure when spatial coordinate goes to infinity was considered in [3] and [1]. In comparison to [15], where asymptotics of the heat equation driven by a general stochastic measure when time coordinate tends to infinity was considered, we study a more general parabolic equation.
The paper is organized as follows. Section 2 contains some general facts about stochastic measures and integrals with respect to them. In Section 3 we prove some technical facts and formulate the main result, which is proved in Section 4 jointly with the auxillary lemma.
2 Preliminaries
Let $(\Omega ,\mathcal{F},\mathsf{P})$ be a complete probability space and $\mathcal{B}$ be a Borel σ-algebra on $\mathbb{R}$. Denote by ${\mathsf{L}_{0}}={\mathsf{L}_{0}}(\Omega ,\mathcal{F},\mathsf{P})$ the set of all real-valued random variables defined on $(\Omega ,\mathcal{F},\mathsf{P})$. Convergence in ${\mathsf{L}_{0}}$ means the convergence in probability.
Definition 1.
A σ-additive mapping $\mu :\mathcal{B}\to {\mathsf{L}_{0}}$ is called stochastic measure (SM).
In other words, μ is a vector measure with values in ${\mathsf{L}_{0}}$. We do not assume any martingale properties or moment existence for SM.
Consider some examples of SMs. If ${M_{t}}$ is a square integrable martingal then $\mu (A)={\textstyle\int _{0}^{T}}{\mathbf{1}_{A}}(t)\hspace{0.1667em}d{M_{t}}$ is an SM. An α-stable random measure on $\mathcal{B}$ for $\alpha \in (0,1)\cup (1,2]$, as it is defined in [19, Sections 3.2–3.3], is an SM in sense of Definition 1. For a fractional Brownian motion ${W_{t}^{H}}$ with Hurst index $H>1/2$ and a bounded measurable function $f:[0,T]\to \mathbb{R}$ we can define an SM $\mu (A)={\textstyle\int _{0}^{T}}f(t){\mathbf{1}_{A}}(t)\hspace{0.1667em}d{W_{t}^{H}}$, see [13, Theorem 1.1]. Ref. [17] contains some other examples.
The integral ${\textstyle\int _{A}}g\hspace{0.1667em}d\mu $, where $g:\mathbb{R}\to \mathbb{R}$ is a deterministic measurable function, $A\in \mathcal{B}$ and μ is an SM, is defined and its basic properties are given in [11, Chapter 7]. In that paper the integral with respect to general stochastic measure was constructed and studied for μ defined on an arbitrary σ-algebra, but here we consider SM on Borel subsets of $\mathbb{R}$. Note that every bounded measurable g is integrable with respect to (w.r.t.) any μ.
In the sequel, μ denotes an SM, C and $C(\omega )$ denote positive constant and positive random constant, respectively, whose exact values are not important ($C<\infty $, $C(\omega )<\infty $ a.s.).
We use the following statement.
Lemma 1 (Lemma 3.1 in [14]).
Let ${\phi _{l}}:\hspace{2.5pt}\mathbb{R}\to \mathbb{R}$, $l\ge 1$, be measurable functions such that $\tilde{\phi }(x)={\textstyle\sum _{l=1}^{\infty }}|{\phi _{l}}(x)|$ is integrable w.r.t. μ on $\mathbb{R}$. Then
We consider the Besov spaces ${B_{22}^{\alpha }}([c,d])$, $0<\alpha <1$, with a standard norm
where
For any $j\in \mathbb{Z}$ and all $n\ge 0$, put
The following lemma is a key tool for estimates of the stochastic integral.
Lemma 2 (Lemma 3 in [16]).
Let Z be an arbitrary set, and the function $q(z,s):Z\times [j,j+1]\to \mathbb{R}$ be such that all paths $q(z,\cdot )$ are continuous on $[j,j+1]$. Denote
such that for all $\beta >0$, $\omega \in \Omega $, $z\in Z$
\[ {q_{n}}(z,s)=\sum \limits_{1\le k\le {2^{n}}}q\big(z,{d_{(k-1)n}^{(j)}}\big){\mathbf{1}_{{\Delta _{kn}^{(j)}}}}(s).\]
Then the random function
\[ \eta (z)={\int _{(j,j+1]}}\hspace{0.1667em}q(z,s)\hspace{0.1667em}d\mu (s),\hspace{2.5pt}z\in Z,\]
has a version
(4)
\[\begin{aligned}{}\widetilde{\eta }(z)& ={\int _{(j,j+1]}}\hspace{0.1667em}{q_{0}}(z,s)\hspace{0.1667em}d\mu (s)\\ {} & +\sum \limits_{n\ge 1}\Big({\int _{(j,j+1]}}\hspace{0.1667em}{q_{n}}(z,s)\hspace{0.1667em}d\mu (s)-{\int _{(j,j+1]}}\hspace{0.1667em}{q_{n-1}}(z,s)\hspace{0.1667em}d\mu (s)\Big)\end{aligned}\](5)
\[\begin{aligned}{}|\widetilde{\eta }(z)|\le |q(z,j)\mu ((j,j+1])|& +{\Big\{\sum \limits_{n\ge 1}{2^{n\beta }}\sum \limits_{1\le k\le {2^{n}}}|q(z,{d_{kn}^{(j)}})-q(z,{d_{(k-1)n}^{(j)}}){|^{2}}\Big\}^{1/2}}\\ {} & \times {\Big\{\sum \limits_{n\ge 1}{2^{-n\beta }}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big\}^{1/2}}.\end{aligned}\]From Theorem 1.1 [9] it follows that, for $\alpha =(\beta +1)/2$,
3 Formulation of the problem and the main result
We refer to the mild solution to (1), i.e. the measurable random function $u(t,x)=u(t,x,\omega ):[0,T]\times \mathbb{R}\times \Omega \to \mathbb{R}$ that satisfies
for each $(t,x)\in [0,+\infty )\times \mathbb{R}$ a.s. The properties of such solutions are considered in [2]. For example, solution of (7) exists, is unique and can be built as
where ${u^{(0)}}(t,x)=0$ and
The analogous iteration process for the stochastic heat equation is considered in more detail in [14].
(7)
\[\begin{array}{l}\displaystyle u(t,x)={\int _{\mathbb{R}}}p(t,x;0,y){u_{0}}(y)\hspace{0.1667em}dy+{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x;s,y)f(s,y,u(s,y))\hspace{0.1667em}dy\\ {} \displaystyle +{\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x;s,y)\sigma (s,y)\hspace{0.1667em}ds,\end{array}\](9)
\[\begin{array}{l}\displaystyle {u^{(n)}}(t,x)\hspace{-0.1667em}=\hspace{-0.1667em}{\int _{\mathbb{R}}}p(t,x;0,y){u_{0}}(y)\hspace{0.1667em}dy+{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x;s,y)f(s,y,{u^{(n-1)}}(s,y))\hspace{0.1667em}dy\\ {} \displaystyle +{\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x;s,y)\sigma (s,y)\hspace{0.1667em}ds.\end{array}\]Let the coefficients of operator (2) satisfy the following assumptions.
Assumption 1 implies that $p(t,x;s,y)\hspace{-0.1667em}=\hspace{-0.1667em}p(t,x-y;s,0)$ for each $t,s\in [0,+\infty )$, $x,y\in \mathbb{R}$. We consider ${u_{0}}$, f, σ in (7) under the following conditions.
Assumption 3.
${u_{0}}:\mathbb{R}\times \Omega \to \mathbb{R}$ is measurable and for all $y,{y_{1}},{y_{2}}\in \mathbb{R}$
\[ |{u_{0}}(y,\omega )|\le C(\omega ),\hspace{1em}|{u_{0}}({y_{1}},\omega )-{u_{0}}({y_{2}},\omega )|\le {L_{{u_{0}}}}(\omega )|{y_{1}}-{y_{2}}{|^{\beta ({u_{0}})}},\]
where $C(\omega )$, ${L_{{u_{0}}}}(\omega )$ are random constants, $\beta ({u_{0}})\ge 1/2$.Assumption 4.
$f:{\mathbb{R}_{+}}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous, bounded, and
\[ |f(s,{y_{1}},{v_{1}})-f(s,{y_{2}},{v_{2}})|\le {L_{f}}\big(|{y_{1}}-{y_{2}}|+|{v_{1}}-{v_{2}}|\big)\]
for some constant ${L_{f}}$ and all $s\in {\mathbb{R}_{+}}$, ${y_{1}},{y_{2}},{v_{1}},{v_{2}}\in \mathbb{R}$.Assumption 5.
$\sigma :{\mathbb{R}_{+}}\times \mathbb{R}\to \mathbb{R}$ is measurable and
\[ |\sigma (s,y)|\le {C_{\sigma }}(s),\hspace{1em}|\sigma (s,{y_{1}})-\sigma (s,{y_{2}})|\le {L_{\sigma }}(s)|{y_{1}}-{y_{2}}{|^{\beta (\sigma )}},\]
for some constant $1/2<\beta (\sigma )<1$, all $s\in {\mathbb{R}_{+}}$, ${y_{1}},{y_{2}}\in \mathbb{R}$ and bounded functions ${C_{\sigma }},{L_{\sigma }}:{\mathbb{R}_{+}}\to \mathbb{R}$ such that
To proceed further, we need some statements about $\mathcal{L}$ and p. The following lemma [6, Theorem 10 §1] is formulated for our specific $\mathcal{L}$.
Lemma 4.
There exist positive constants ν, η, C such that for each $x,y\in \mathbb{R}$, $t>s>0$ the following estimates hold:
where ${E_{\alpha ,\beta }}(z)={\textstyle\sum _{k=0}^{\infty }}\frac{{z^{k}}}{\Gamma (\alpha k+\beta )}$ is the Mittag-Leffler function, $\phi <1$.
(10)
\[ |p(t,x;s,y)|\le C{(t-s)^{-1/2}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{\eta (t-s)}}{E_{\lambda ,\lambda +1}}(\tilde{C}{(t-s)^{\lambda }}),\](11)
\[ \Big|\frac{\partial p(t,x;s,y)}{\partial x}\Big|\le C{(t-s)^{-1}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{\eta (t-s)}}{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(t-s)^{\lambda }}),\](12)
\[\begin{array}{l}\displaystyle \Big|\frac{\partial p(t,x;s,y)}{\partial x}-\frac{\partial p(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\Big|\le C(\phi )|x-{x^{\prime }}{|^{\phi }}{(t-s)^{-3/2}}\times \\ {} \displaystyle \times \max \Big\{{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}},{e^{-\frac{\nu {({x^{\prime }}-y)^{2}}}{t-s}}}\Big\}{e^{\eta (t-s)}}{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }}),\end{array}\]Proof.
We represent p as
It is easy to calculate that
where $0<\nu <{(4{\sup _{t\in \mathbb{R}}}a(t))^{-1}}$. The solution of (13) can be rewritten as
where
where constants C, $\tilde{C}$ depend on λ. Taking the sum of (18) for each $k\ge 1$, we get the inequality
The following inequality plays an important role in further estimates
Now we use (19) and (20) to obtain (10).
where the last equality is a consequence of
[8, chapter III, (1.15)]. From (15), (21) and the inequalities $t\ge s$ ${e^{\eta (t-s)}}\ge 1$, ${E_{\lambda ,\lambda +1}}(\tilde{C}{(t-s)^{\lambda }})\ge \frac{1}{\Gamma (\lambda +1)}$ we obtain (10). We prove (11) analogously, using
\[ p(t,x;s,y)=W(t,x;s,y)+{\int _{s}^{t}}d\theta {\int _{\mathbb{R}}}W(t,x;\theta ,\zeta )\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta \]
(see, for example, [6, (4.18)]). Here
the function $\Phi (t,x;s,y)$ is a solution of the integral equation
(13)
\[ \Phi (t,x;s,y)=\mathcal{L}W(t,x;s,y)+{\int _{s}^{t}}d\theta {\int _{\mathbb{R}}}\mathcal{L}W(t,x;\theta ,\zeta )\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta .\]
\[\begin{array}{l}\displaystyle \frac{\partial W(t,x;s,y)}{\partial x}=\frac{1}{\sqrt{4\pi (t-s)a(s)}}{e^{-\frac{{(x-y)^{2}}}{4(t-s)a(s)}}}\frac{y-x}{2(t-s)a(s)},\\ {} \displaystyle \frac{{\partial ^{2}}W(t,x;s,y)}{\partial {x^{2}}}=\frac{1}{\sqrt{4\pi (t-s)a(s)}}{e^{-\frac{{(x-y)^{2}}}{4(t-s)a(s)}}}\Big(\frac{{(x-y)^{2}}}{4{(t-s)^{2}}{a^{2}}(s)}-\frac{1}{2(t-s)a(s)}\Big),\\ {} \displaystyle \frac{\partial W(t,x;s,y)}{\partial t}=\frac{1}{\sqrt{4\pi (t-s)a(s)}}{e^{-\frac{{(x-y)^{2}}}{4(t-s)a(s)}}}\Big(\frac{{(x-y)^{2}}}{4{(t-s)^{2}}a(s)}-\frac{1}{2(t-s)}\Big).\end{array}\]
Using Assumption 1 and boundedness of the function ${x^{\alpha }}{e^{-x}}$ on $[0,+\infty )$ for arbitrary $\alpha >0$, we easily obtain that
\[\begin{array}{r}\displaystyle {\Phi _{1}}(t,x;s,y)=\mathcal{L}W(t,x;s,y),\hspace{1em}{\Phi _{k+1}}(t,x;s,y)\\ {} \displaystyle \hspace{1em}={\int _{s}^{t}}d\theta {\int _{\mathbb{R}}}\mathcal{L}W(t,x;\theta ,\zeta ){\Phi _{k}}(\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta .\end{array}\]
Using (14)–(17), we obtain that
\[\begin{array}{l}\displaystyle |{\Phi _{1}}(t,x;s,y)|\\ {} \displaystyle \le |a(t)-a(s)|\Big|\frac{{\partial ^{2}}W(t,x;s,y)}{\partial {x^{2}}}\Big|+|b(t)|\Big|\frac{\partial W(t,x;s,y)}{\partial x}\Big|+|c(t)||W(t,x;s,y)|\\ {} \displaystyle \le C{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}\big({(t-s)^{-3/2+\lambda }}+{(t-s)^{-1}}+{(t-s)^{-1/2}}\big)\\ {} \displaystyle \le C{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{(t-s)^{-3/2+\lambda }}{e^{{\eta _{1}}(t-s)}}.\end{array}\]
Analogously to [6, (4.58)] we show that
(18)
\[ |{\Phi _{k}}(t,x;s,y)|\le \frac{C}{\Gamma ((k-1)\lambda +\lambda )}{(\tilde{C}{(t-s)^{\lambda }})^{k-1}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{{\eta _{1}}(t-s)}}{(t-s)^{-3/2+\lambda }},\](19)
\[ |\Phi (t,x;s,y)|\le C{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{{\eta _{1}}(t-s)}}{(t-s)^{-3/2+\lambda }}{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }}).\](20)
\[ {\int _{\mathbb{R}}}{(t-\theta )^{-1/2}}{(\theta -s)^{-1/2}}{e^{-\nu \big(\frac{{(\zeta -y)^{2}}}{\theta -s}+\frac{{(x-\zeta )^{2}}}{t-\theta }\big)}}\hspace{0.1667em}d\zeta =\sqrt{\frac{\pi }{\nu }}{(t-s)^{-1/2}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}.\](21)
\[\begin{array}{l}\displaystyle \Big|{\int _{s}^{t}}d\theta {\int _{\mathbb{R}}}W(t,x;\theta ,\zeta )\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta \Big|\\ {} \displaystyle \le C{\int _{s}^{t}}d\theta {\int _{\mathbb{R}}}{(t-\theta )^{-\frac{1}{2}}}{e^{-\frac{\nu {(x-\zeta )^{2}}}{t-\theta }}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(\theta -s)^{\lambda }})}{{(\theta -s)^{3/2-\lambda }}}{e^{{\eta _{1}}(\theta -s)}}{e^{-\frac{\nu {(\zeta -y)^{2}}}{\theta -s}}}d\zeta \\ {} \displaystyle \le C{e^{{\eta _{1}}(t-s)}}{\int _{s}^{t}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(\theta -s)^{\lambda }})}{{(\theta -s)^{1-\lambda }}}d\theta {\int _{\mathbb{R}}}{(t-\theta )^{-\frac{1}{2}}}{(\theta -s)^{-\frac{1}{2}}}{e^{-\nu \big(\frac{{(\zeta -y)^{2}}}{\theta -s}+\frac{{(x-\zeta )^{2}}}{t-\theta }\big)}}d\zeta \\ {} \displaystyle =C{e^{{\eta _{1}}(t-s)}}{(t-s)^{-1/2}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{\int _{s}^{t}}{(\theta -s)^{-1+\lambda }}{E_{\lambda ,\lambda }}(\tilde{C}{(\theta -s)^{\lambda }})d\theta \\ {} \displaystyle =C{(t-s)^{-1/2+\lambda }}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{{\eta _{1}}(t-s)}}{E_{\lambda ,\lambda +1}}(\tilde{C}{(t-s)^{\lambda }}),\end{array}\](22)
\[ {\int _{0}^{z}}{E_{\rho ,\mu }}(\lambda {t^{\rho }}){t^{\mu -1}}dt={z^{\mu }}{E_{\rho ,\mu +1}}(\lambda {z^{\rho }})\]
\[ \frac{1}{\sqrt{\pi }}{\int _{0}^{z}}{E_{\rho ,\mu }}(\lambda {t^{\rho }}){(z-t)^{-1/2}}{t^{\mu -1}}dt={z^{\mu -1/2}}{E_{\rho ,\mu +1/2}}(\lambda {z^{\rho }})\]
[8, chapter III, (1.16) with $\alpha =1/2$] instead of (22).In the proof of (12) we use
where $A>0$. Firstly assume that ${({x^{\prime }}-x)^{2}}<A(t-s)$; we prove that for such t, x, s
Notice that
where ${\nu _{1}}>{\nu _{2}}>0$, C, ${\nu _{2}}$ do not depend on y and ${x^{\ast }}$, but C depends on A. We consider two cases.
Now we set $C=\max \{{C_{1}},{C_{2}}\}$, ${\nu _{2}}=\min \{{\nu _{21}},{\nu _{22}}\}$ and obtain (25). Therefore, the following estimate holds:
Consider the expression
Here we used the inequality $t-s-\frac{|{x^{\prime }}-x{|^{2}}}{2A}>\frac{t-s}{2}$ and (20). We estimate ${J_{2}}$ in analogous way using (25):
We apply (20) and (26) with $\tilde{A}=2A$ to prove the estimation for ${J_{3}}$:
for arbitrary $l\in (0,1/2)$. Now (24) follows from (26), (27), (28) and (29).
(23)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \Big|\frac{\partial p(t,x;s,y)}{\partial x}-\frac{\partial p(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\Big|\\ {} & & \displaystyle \hspace{1em}=\Big|\frac{\partial p(t,x;s,y)}{\partial x}-\frac{\partial p(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\Big|{\mathbf{1}_{{({x^{\prime }}-x)^{2}}<A(t-s)}}\\ {} & & \displaystyle \hspace{2em}+\Big|\frac{\partial p(t,x;s,y)}{\partial x}-\frac{\partial p(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\Big|{\mathbf{1}_{{({x^{\prime }}-x)^{2}}\ge A(t-s)}},\end{array}\](24)
\[\begin{array}{r}\displaystyle \Big|\frac{\partial p(t,x;s,y)}{\partial x}-\frac{\partial p(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\Big|\le C(\phi ,A)|x-{x^{\prime }}{|^{\phi }}{(t-s)^{-3/2}}\times \\ {} \displaystyle \hspace{2em}\times {e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{\eta (t-s)}}{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }}).\end{array}\]
\[\begin{array}{l}\displaystyle \Big|\frac{\partial W(t,x;s,y)}{\partial x}-\frac{\partial W(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\Big|\le C\Big|{\int _{x}^{{x^{\prime }}}}\frac{1}{{(t-s)^{3/2}}}{e^{-\frac{\nu {(v-y)^{2}}}{t-s}}}\hspace{0.1667em}dv\Big|\\ {} \displaystyle \le C\frac{1}{{(t-s)^{3/2}}}{e^{-\frac{{\nu _{1}}{({x^{\ast }}-y)^{2}}}{t-s}}}{\int _{0}^{\frac{|x-{x^{\prime }}|}{2}}}{e^{-\frac{(\nu -{\nu _{1}}){w^{2}}}{t-s}}}\hspace{0.1667em}dw\\ {} \displaystyle \le C\frac{1}{{(t-s)^{3/2}}}{e^{-\frac{{\nu _{1}}{({x^{\ast }}-y)^{2}}}{t-s}}}{\int _{0}^{\frac{|x-{x^{\prime }}|}{2}}}{\Big(\frac{t-s}{{w^{2}}}\Big)^{l}}\hspace{0.1667em}dw\\ {} \displaystyle \le C\frac{1}{{(t-s)^{3/2-l}}}{e^{-\frac{{\nu _{1}}{({x^{\ast }}-y)^{2}}}{t-s}}}|x-{x^{\prime }}{|^{1-2l}},\end{array}\]
where $0\le l<1/2$, ${x^{\ast }}=\epsilon x+(1-\epsilon ){x^{\prime }}$, $0\le \epsilon \le 1$, $0<{\nu _{1}}<\nu $; here we used the fact that the function ${x^{\alpha }}{e^{-x}}$ is bounded on $[0,+\infty )$ for arbitrary $\alpha >0$. Now we show that ${({x^{\prime }}-x)^{2}}<A(t-s)$ implies
(25)
\[ {e^{-\frac{{\nu _{1}}{({x^{\ast }}-y)^{2}}}{t-s}}}\le C{e^{-\frac{{\nu _{2}}{(x-y)^{2}}}{t-s}}},\]-
2. $|x-y|>3|x-{x^{\prime }}|$. In this case we have the estimates:\[\begin{array}{l}\displaystyle {e^{-\frac{{\nu _{1}}{({x^{\ast }}-y)^{2}}}{t-s}}}={e^{-\frac{{\nu _{1}}{(x-y)^{2}}}{t-s}}}{e^{\frac{{\nu _{1}}(x-{x^{\ast }})(x+{x^{\ast }}-2y)}{t-s}}}\le {e^{-\frac{{\nu _{1}}{(x-y)^{2}}}{t-s}}}{e^{\frac{{\nu _{1}}|x-{x^{\prime }}|(2|x-y|+|x-{x^{\ast }}|)}{t-s}}}\\ {} \displaystyle <{e^{-\frac{{\nu _{1}}{(x-y)^{2}}}{t-s}}}{e^{\frac{7{\nu _{1}}{(x-y)^{2}}}{9(t-s)}}}={e^{-\frac{2{\nu _{1}}{(x-y)^{2}}}{9(t-s)}}}={C_{2}}{e^{-\frac{{\nu _{22}}{(x-y)^{2}}}{t-s}}}.\end{array}\]
(26)
\[ \Big|\frac{\partial W(t,x;s,y)}{\partial x}-\frac{\partial W(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\Big|\le C\frac{1}{{(t-s)^{3/2-l}}}{e^{-\frac{{\nu _{2}}{(x-y)^{2}}}{t-s}}}|x-{x^{\prime }}{|^{1-2l}}.\]
\[\begin{array}{l}\displaystyle \frac{\partial p(t,x;s,y)}{\partial x}-\frac{\partial p(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}=\bigg(\frac{\partial W(t,x;s,y)}{\partial x}-\frac{\partial W(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\bigg)\\ {} \displaystyle +{\int _{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}^{t}}d\theta {\int _{\mathbb{R}}}\frac{\partial W(t,x;\theta ,\zeta )}{\partial x}\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta \\ {} \displaystyle -{\int _{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}^{t}}d\theta {\int _{\mathbb{R}}}\frac{\partial W(t,{x^{\prime }};\theta ,\zeta )}{\partial {x^{\prime }}}\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta \\ {} \displaystyle +{\int _{s}^{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}}d\theta {\int _{\mathbb{R}}}\bigg(\frac{\partial W(t,x;\theta ,\zeta )}{\partial x}-\frac{\partial W(t,{x^{\prime }};\theta ,\zeta )}{\partial {x^{\prime }}}\bigg)\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta \\ {} \displaystyle ={J_{0}}+{J_{1}}+{J_{2}}+{J_{3}}.\end{array}\]
We estimate ${J_{1}}$ in the following way:
(27)
\[\begin{array}{l}\displaystyle |{J_{1}}|=\bigg|{\int _{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}^{t}}d\theta {\int _{\mathbb{R}}}\frac{\partial W(t,x;\theta ,\zeta )}{\partial x}\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta \bigg|\\ {} \displaystyle \le C{\int _{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}^{t}}d\theta {\int _{\mathbb{R}}}{e^{-\nu \big(\frac{{(\zeta -y)^{2}}}{\theta -s}+\frac{{(x-\zeta )^{2}}}{t-\theta }\big)}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(\theta -s)^{\lambda }})}{(t-\theta ){(\theta -s)^{3/2-\lambda }}}{e^{{\eta _{1}}(\theta -s)}}d\zeta \\ {} \displaystyle \le C{e^{{\eta _{1}}(t-s)}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }})}{{(t-s)^{1/2}}}{\int _{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}^{t}}{(t-\theta )^{-1/2}}{(\theta -s)^{\lambda -1}}\hspace{0.1667em}d\theta \\ {} \displaystyle \le C{e^{{\eta _{1}}(t-s)}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }})}{{(t-s)^{1/2}}}{\Big(t-s-\frac{|{x^{\prime }}-x{|^{2}}}{2A}\Big)^{\lambda -1}}{\Big(\frac{|{x^{\prime }}-x{|^{2}}}{2A}\Big)^{1/2}}\\ {} \displaystyle \le C{e^{{\eta _{1}}(t-s)}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{(t-s)^{-3/2+\lambda }}{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }})|x-{x^{\prime }}|.\end{array}\](28)
\[\begin{array}{l}\displaystyle |{J_{2}}|=\bigg|{\int _{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}^{t}}d\theta {\int _{\mathbb{R}}}\frac{\partial W(t,{x^{\prime }};\theta ,\zeta )}{\partial {x^{\prime }}}\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta \bigg|\\ {} \displaystyle \le C{e^{{\eta _{1}}(t-s)}}{e^{-\frac{{\nu _{2}}{(x-y)^{2}}}{t-s}}}{(t-s)^{-3/2+\lambda }}{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }})|x-{x^{\prime }}|.\end{array}\](29)
\[\begin{array}{l}\displaystyle |{J_{3}}|\le {\int _{s}^{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}}d\theta {\int _{\mathbb{R}}}\bigg|\frac{\partial W(t,x;\theta ,\zeta )}{\partial x}-\frac{\partial W(t,{x^{\prime }};\theta ,\zeta )}{\partial {x^{\prime }}}\bigg||\Phi (\theta ,\zeta ;s,y)|\hspace{0.1667em}d\zeta \\ {} \displaystyle \le C|x-{x^{\prime }}{|^{1-2l}}{\int _{s}^{t-\frac{|{x^{\prime }}-x{|^{2}}}{2A}}}\hspace{-5.0pt}d\theta {\int _{\mathbb{R}}}\frac{{e^{-{\nu _{2}}\big(\frac{{(\zeta -y)^{2}}}{\theta -s}+\frac{{(x-\zeta )^{2}}}{t-\theta }\big)}}{E_{\lambda ,\lambda }}(\tilde{C}{(\theta -s)^{\lambda }})}{{(t-\theta )^{3/2-l}}{(\theta -s)^{3/2-\lambda }}}{e^{{\eta _{1}}(\theta -s)}}d\zeta \\ {} \displaystyle \le C|x-{x^{\prime }}{|^{1-2l}}{e^{{\eta _{1}}(t-s)}}{e^{-\frac{{\nu _{2}}{(x-y)^{2}}}{t-s}}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }})}{{(t-s)^{1/2}}}{\int _{s}^{t}}{(t-\theta )^{l-1}}{(\theta -s)^{\lambda -1}}\hspace{0.1667em}d\theta \\ {} \displaystyle \le C|x-{x^{\prime }}{|^{1-2l}}{e^{{\eta _{1}}(t-s)}}{e^{-\frac{{\nu _{2}}{(x-y)^{2}}}{t-s}}}{(t-s)^{-3/2+l+\lambda }}{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }}),\end{array}\]Let ${({x^{\prime }}-x)^{2}}\ge A(t-s)$. This implies
where $l\in (0,1)$. On the other hand,
(30)
\[\begin{array}{l}\displaystyle \bigg|\frac{\partial W(t,x;s,y)}{\partial x}-\frac{\partial W(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\bigg|\le \bigg|\frac{\partial W(t,x;s,y)}{\partial x}\bigg|+\bigg|\frac{\partial W(t,{x^{\prime }};s,y)}{\partial {x^{\prime }}}\bigg|\\ {} \displaystyle \le C{(t-s)^{-1}}\max \Big\{{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}},{e^{-\frac{\nu {({x^{\prime }}-y)^{2}}}{t-s}}}\Big\}\\ {} \displaystyle \le C{(t-s)^{-3/2+l}}{\Big(\frac{|{x^{\prime }}-x|}{\sqrt{A}}\Big)^{1-2l}}\max \Big\{{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}},{e^{-\frac{\nu {({x^{\prime }}-y)^{2}}}{t-s}}}\Big\},\end{array}\]
\[\begin{array}{l}\displaystyle \Big|{\int _{s}^{t}}d\theta {\int _{\mathbb{R}}}\frac{\partial W(t,x;\theta ,\zeta )}{\partial x}\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta \Big|\\ {} \displaystyle \le C{\int _{s}^{t}}d\theta {\int _{\mathbb{R}}}{e^{-\nu \big(\frac{{(\zeta -y)^{2}}}{\theta -s}+\frac{{(x-\zeta )^{2}}}{t-\theta }\big)}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(\theta -s)^{\lambda }})}{(t-\theta ){(\theta -s)^{3/2-\lambda }}}{e^{{\eta _{1}}(\theta -s)}}d\zeta \\ {} \displaystyle \le C{e^{{\eta _{1}}(t-s)}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }})}{{(t-s)^{1/2}}}{\Big(\frac{|{x^{\prime }}-x|}{\sqrt{A}}\Big)^{1-2l}}\hspace{-5.0pt}{\int _{s}^{t}}\hspace{-2.0pt}{(\theta -s)^{-1+\lambda }}{(t-\theta )^{-1+l}}d\theta \\ {} \displaystyle =C{e^{{\eta _{1}}(t-s)}}{(t-s)^{-3/2+l+\lambda }}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{\Big(\frac{|{x^{\prime }}-x|}{\sqrt{A}}\Big)^{1-2l}}{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }}),\end{array}\]
and the same estimates hold for ${\textstyle\int _{s}^{t}}d\theta {\textstyle\int _{\mathbb{R}}}\frac{\partial W(t,{x^{\prime }};\theta ,\zeta )}{\partial {x^{\prime }}}\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta $. Using (30), (24) and (23), we obtain (12); in (23) we can set, for example, $A=1$. □The main result of the paper is this theorem.
Theorem 1.
Let Assumptions 1–5 hold. Then there exists the version of the solution of (7) such that for each $\omega \in \Omega $:
4 Proof of the auxillary lemma and the main result
To prove Theorem 1, we consider the integral
Lemma 5.
Assume Assumptions 1, 2, 5 hold. Then there exists a version ${\nu _{1}}(t,x)$ of integral (32) such that for each $\omega \in \Omega $
Proof.
It follows from Lemma 2 and (6) that a version ${\nu _{1}}(t,x)$ of integral (32) exists such that for each $x\in \mathbb{R}$, $t\ge 0$, $\omega \in \Omega $
where
Next we prove that ${\nu _{1}}(t,x)$ satisfies (33). In order to estimate the Besov norm on $[j,j+1]$, we consider
For the derivatives of ${\eta _{\varepsilon }}(v)$ we have the following well-known inequalities:
where the constant C does not depend on ε. Let us prove, for example, the first inequality in (36):
for each $(t,x)\in {\Theta _{MN}^{\gamma }}$. Let $\gamma \in (0,1/3)$ be fixed; consider $t>3\gamma $. Thus, ${I_{2}}$ can be rewritten in following way:
Now we estimate ${I_{22}}$. We get
For further estimates we use the function
Let $M>2\gamma $, $N>1+2\gamma $. Then function $\hat{p}$ has properties, which are analogouos to the properties of $\tilde{p}$. For example, $\hat{p}(t,x;s,y,h)=p(t,x;s,y+h)-p(t,x;s,y)$ when $(t,x)\in {\Omega _{MN}}$, $\hat{p}$ is bounded on $([0,T]\times \mathbb{R})$, where $T>0$, $\hat{p}=0$ when $(t,x)\in \big([0,+\infty )\times \mathbb{R}\big)\setminus {\Omega _{MN}^{2\gamma }}\cup \{s\ge t\}$. Now we estimate $\mathcal{L}\hat{p}$. Notice that $\mathcal{L}\tilde{p}=0$ outside the set ${\Theta _{MN}^{\gamma }}$, and for each $(t,x)\in {\Theta _{MN}^{\gamma }}$ the following estimates hold:
We rewrite ${I_{1}}$ as
${I_{12}}$ is estimated in the following way:
where $l\in (0,1/2)$. We can choose l, ϕ such that $1-2l=\beta (\sigma )$, $\phi =\beta (\sigma )$.
(34)
\[\begin{array}{l}\displaystyle |{\nu _{1}}(t,x)|\le \sum \limits_{j\in \mathbb{Z}}|q(t,x,j)\mu ((j,j+1])|+C\sum \limits_{j\in \mathbb{Z}}\| q(t,x,\cdot ){\| _{{B_{22}^{\alpha }}([j,j+1])}}\\ {} \displaystyle \times {\left({\sum \limits_{n=1}^{\infty }}{2^{-n(2\alpha -1)}}{\sum \limits_{k=1}^{{2^{n}}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\right)^{1/2}}\\ {} \displaystyle \le {\Big(\sum \limits_{j\in \mathbb{Z}}|q(t,x,j){|^{2}}\Big)^{1/2}}\Big(\sum \limits_{j\in \mathbb{Z}}|\mu {((j,j+1]){|^{2}}\Big)^{1/2}}\\ {} \displaystyle +C{\Big(\sum \limits_{j\in \mathbb{Z}}\| q(t,x,\cdot ){\| _{{B_{22}^{\alpha }}([j,j+1])}^{2}}\Big)^{1/2}}{\Big(\sum \limits_{j\in \mathbb{Z}}{\sum \limits_{n=1}^{\infty }}{2^{-n(2\alpha -1)}}{\sum \limits_{k=1}^{{2^{n}}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big)^{1/2}},\end{array}\]
\[\begin{array}{l}\displaystyle |q(t,x,y+h)-q(t,x,y)|\le {\int _{0}^{t}}|p(t,x;s,y+h)-p(t,x;s,y)|\hspace{0.1667em}|\sigma (s,y)|\hspace{0.1667em}ds\\ {} \displaystyle +{\int _{0}^{t}}|p(t,x;s,y+h)|\hspace{0.1667em}|\sigma (s,y+h)-\sigma (s,y)|\hspace{0.1667em}ds={I_{1}}+{I_{2}},\end{array}\]
where $y,\hspace{2.5pt}y+h\in [j,j+1]$. Denote
\[\begin{aligned}{}{\Omega _{MN}}& =\big([s,+\infty )\times \mathbb{R}\big)\setminus \big([s,s+M)\times (y-N,y+N)\big),\\ {} {\Omega _{MN}^{\gamma }}& =\{(t,x):d({\Omega _{MN}},(t,x))\le \gamma \},\\ {} {\eta _{1}}(v)& =C{e^{\frac{1}{|v{|^{2}}-1}}}{\mathbf{1}_{\{|v|<1\}}};\hspace{1em}v\in {\mathbb{R}^{2}},\hspace{2.5pt}{\int _{{\mathbb{R}^{2}}}}{\eta _{1}}(v)\hspace{0.1667em}dv=1,\\ {} {\eta _{\varepsilon }}(v)& ={\varepsilon ^{-2}}{\eta _{1}}(v{\varepsilon ^{-1}}),\\ {} {\Theta _{MN}^{\gamma }}& =({\Omega _{MN}^{2\gamma }}\setminus {\Omega _{MN}})\cap \{t>s\},\end{aligned}\]
where M, N, $\gamma >0$. To estimate ${I_{2}}$ we introduce the function
where
\[ \Psi (t,x;s,y)={\int _{{\Omega _{MN}^{\gamma }}}}{\eta _{\gamma }}(t-{v_{1}},x-{v_{2}})\hspace{0.1667em}dv,\hspace{1em}v=({v_{1}},{v_{2}}).\]
It is obvious that, for each $0<\gamma <\min \{M/2,N/2\}$ and arbitrary fixed pair $(s,y)$, $\tilde{p}$ belongs to a class ${C^{1,2}}([0,+\infty )\times \mathbb{R})$ as a function of $(t,x)$. It is easy to obtain that $\tilde{p}(t,x;s,y)=p(t,x;s,y)$ if $(t,x)\in {\Omega _{MN}}$ and $\tilde{p}(t,x;s,y)=0$ if $(t,x)\in \big([0,+\infty )\times \mathbb{R}\big)\setminus {\Omega _{MN}^{2\gamma }}\cup \{s\ge t\}$. Moreover, boundedness of $p(t,x;s,y)$ on ${\Omega _{MN}^{2\gamma }}\cap \{t\le T\}$ for each $T>0$ and the fact that $|\Psi |\le 1$ imply boundedness of $\tilde{p}(t,x;s,y)$ on $([0,T]\times \mathbb{R})$. Now we estimate $\mathcal{L}\tilde{p}$. Taking into consideration properties of $\tilde{p}$, it is easy to see that $\mathcal{L}\tilde{p}=0$ outside the set ${\Theta _{MN}^{\gamma }}$. And inside ${\Theta _{MN}^{\gamma }}$,
(35)
\[ \mathcal{L}\tilde{p}=\mathcal{L}(p\Psi )=\Psi \mathcal{L}p+p\mathcal{L}\Psi +2a\frac{\partial p}{\partial x}\frac{\partial \Psi }{\partial x}\stackrel{\mathcal{L}p=0}{=}p\mathcal{L}\Psi +2a\frac{\partial p}{\partial x}\frac{\partial \Psi }{\partial x}.\](36)
\[ \Big|\frac{\partial {\eta _{\varepsilon }}(v)}{\partial {v_{i}}}\Big|\le C{\varepsilon ^{-3}},\hspace{1em}\Big|\frac{{\partial ^{2}}{\eta _{\varepsilon }}(v)}{\partial {v_{i}^{2}}}\Big|\le C{\varepsilon ^{-4}},\hspace{1em}i=1,2,\]
\[ \Big|\frac{\partial {\eta _{\varepsilon }}(v)}{\partial {v_{i}}}\Big|\le {\varepsilon ^{-3}}\underset{|v|\le 1}{\max }\Big|\frac{\partial {\eta _{1}}(v)}{\partial {v_{i}}}\Big|=C{\varepsilon ^{-3}},\]
where we used that ${\eta _{1}}\in {C^{\infty }}({\mathbb{R}^{2}})$. From (36) and the fact that ${\eta _{\varepsilon }}=0$ outside the ball with radius ε it follows that
\[ \Big|\frac{\partial \Psi }{\partial x}\Big|\le C{\gamma ^{-1}},\hspace{1em}\Big|\frac{\partial \Psi }{\partial t}\Big|\le C{\gamma ^{-1}},\hspace{1em}\Big|\frac{{\partial ^{2}}\Psi }{\partial {x^{2}}}\Big|\le C{\gamma ^{-2}}.\]
Using estimates (35), (10) and (11) we obtain the inequality
(37)
\[ |\mathcal{L}\tilde{p}|\le C{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{\eta (t-s)}}\bigg({\gamma ^{-2}}\frac{{E_{\lambda ,\lambda +1}}(\tilde{C}{(t-s)^{\lambda }})}{\sqrt{t-s}}+{\gamma ^{-1}}\frac{{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(t-s)^{\lambda }})}{t-s}\bigg)\]
\[\begin{array}{r}\displaystyle {I_{2}}={\int _{0}^{t-3\gamma }}|p(t,x;s,y+h)|\hspace{0.1667em}|\sigma (s,y+h)-\sigma (s,y)|\hspace{0.1667em}ds\\ {} \displaystyle \hspace{2em}+{\int _{t-3\gamma }^{t}}|p(t,x;s,y+h)|\hspace{0.1667em}|\sigma (s,y+h)-\sigma (s,y)|\hspace{0.1667em}ds={I_{21}}+{I_{22}}.\end{array}\]
We estimate the first summand using the function $\tilde{p}(t,x;s,y)$ for $M=N=3\gamma $. Note that $t-s<3\gamma $ on the set ${\Theta _{MN}^{\gamma }}$; moreover, $t-s>\gamma $ or $|x-y|>\gamma $. In the first case we have the following consequence of (37):
\[ |\mathcal{L}\tilde{p}|\le C{e^{3\eta \gamma }}{\gamma ^{-5/2}}\big({E_{\lambda ,\lambda +1}}(\tilde{C}{(3\gamma )^{\lambda }})+{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(3\gamma )^{\lambda }})\big).\]
In the second case,
\[ |\mathcal{L}\tilde{p}|\le C{e^{3\eta \gamma }}{\gamma ^{-7/2}}\big({E_{\lambda ,\lambda +1}}(\tilde{C}{(3\gamma )^{\lambda }})+{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(3\gamma )^{\lambda }})\big).\]
Anyway,
\[ |\mathcal{L}\tilde{p}|\le C(\gamma )\hspace{2.5pt}\forall \hspace{0.1667em}(t,x)\in [0,+\infty )\times \mathbb{R}.\]
Using Lemma 3 for arbitrary $T>t$, we obtain
where the constant C does not depend on T. On the other hand, taking into account that $\tilde{p}(t,x;s,y)=p(t,x;s,y)$ if $t-s>3\gamma $ we obtain
(38)
\[ |{I_{21}}|\le C(\gamma ){h^{\beta (\sigma )}}{e^{-{c_{0}}t}}{t^{2}}\to 0,\hspace{2.5pt}t\to \infty .\](39)
\[\begin{array}{r}\displaystyle |{I_{22}}|\le C{h^{\beta (\sigma )}}{\int _{t-3\gamma }^{t}}\frac{1}{\sqrt{t-s}}{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{\eta (t-s)}}{E_{\lambda ,\lambda +1}}(\tilde{C}{(t-s)^{\lambda }}){L_{\sigma }}(s)\hspace{0.1667em}ds\\ {} \displaystyle \hspace{1em}\le C{h^{\beta (\sigma )}}\sqrt{\gamma }{e^{3\gamma \eta }}{E_{\lambda ,\lambda +1}}(\tilde{C}{(3\gamma )^{\lambda }})\underset{s\in [t-3\gamma ,t]}{\sup }|{L_{\sigma }}(s)|\to 0,\hspace{0.1667em}t\to \infty .\end{array}\]
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& \displaystyle \mathcal{L}\hat{p}& \displaystyle =\mathcal{L}(({p_{y+h}}-{p_{y}})\Psi )\\ {} & & \displaystyle =\Psi \mathcal{L}({p_{y+h}}-{p_{y}})+({p_{y+h}}-{p_{y}})\mathcal{L}\Psi +2a\frac{\partial ({p_{y+h}}-{p_{y}})}{\partial x}\frac{\partial \Psi }{\partial x}\\ {} & & \displaystyle \stackrel{\mathcal{L}p=0}{=}({p_{y+h}}-{p_{y}})\mathcal{L}\Psi +2a\frac{\partial ({p_{y+h}}-{p_{y}})}{\partial x}\frac{\partial \Psi }{\partial x},\end{array}\]
where for convenience we denote ${p_{y}}=p(t,x;s,y)$. (11) and (12) imply that
\[\begin{array}{l}\displaystyle \Big|\frac{\partial p(t,x;s,y+h)}{\partial x}-\frac{\partial p(t,x;s,y)}{\partial x}\Big|\\ {} \displaystyle \le C{h^{\phi }}{(t-s)^{-3/2}}\underset{y\in [j,j+1]}{\sup }{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{\eta (t-s)}}{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }});\\ {} \displaystyle \big|p(t,x;s,y+h)-p(t,x;s,y)\big|=\Big|{\int _{x-y-h}^{x-y}}\frac{\partial p(t,v;s,0)}{\partial v}\hspace{0.1667em}dv\Big|\\ {} \displaystyle \le {\int _{x-y-h}^{x-y}}\frac{1}{t-s}{e^{-\frac{\nu {v^{2}}}{t-s}}}{e^{\eta (t-s)}}{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(t-s)^{\lambda }})\hspace{0.1667em}dv\\ {} \displaystyle \le \frac{h}{t-s}\underset{y\in [j,j+1]}{\sup }{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{\eta (t-s)}}{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(t-s)^{\lambda }}).\end{array}\]
Therefore, for each $(t,x)\in {\Theta _{MN}^{\gamma }}$,
(40)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& \displaystyle |\mathcal{L}\hat{p}|& \displaystyle \le \\ {} & & \displaystyle \le C{h^{\phi }}\underset{y\in [j,j+1]}{\sup }{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}{e^{\eta (t-s)}}\\ {} & & \displaystyle \hspace{1em}\times \bigg({\gamma ^{-2}}\frac{{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(t-s)^{\lambda }})}{t-s}+{\gamma ^{-1}}\frac{{E_{\lambda ,\lambda }}(\tilde{C}{(t-s)^{\lambda }})}{{(t-s)^{3/2}}}\bigg).\end{array}\]
\[\begin{array}{r}\displaystyle {I_{1}}={\int _{0}^{t-3\gamma }}|p(t,x;s,y+h)-p(t,x;s,y)|\hspace{0.1667em}|\sigma (s,y)|\hspace{0.1667em}ds\\ {} \displaystyle \hspace{2em}+{\int _{t-3\gamma }^{t}}|p(t,x;s,y+h)-p(t,x;s,y)|\hspace{0.1667em}|\sigma (s,y)|\hspace{0.1667em}ds={I_{11}}+{I_{12}}.\end{array}\]
Consider the function $\hat{p}(t,x;s,y,h)$ for $M=3\gamma $, $N=3\gamma +1$. We estimate ${I_{11}}$ analogouosly to ${I_{21}}$. It follows from (40) and Lemma 3 that
\[ |\hat{p}(t,x;s,y,h)|\le C(\gamma ){h^{\phi }}{e^{-{c_{0}}t}}t\hspace{2.5pt}\forall (t,x)\in [0,+\infty )\times \mathbb{R}.\]
On the other hand, the equality $\hat{p}(t,x;s,y,h)=p(t,x;s,y+h)-p(t,x;s,y)$ for $t-s>3\gamma $ implies that
(41)
\[ |{I_{11}}|\le C(\gamma ){h^{\phi }}{e^{-{c_{0}}t}}{t^{2}}\to 0,\hspace{2.5pt}t\to \infty .\](42)
\[\begin{array}{l}\displaystyle |{I_{12}}|\le C{\int _{t-3\gamma }^{t}}\hspace{0.1667em}ds{\int _{x-y-h}^{x-y}}\frac{{C_{\sigma }}(s)}{t-s}{e^{-\frac{\nu {v^{2}}}{t-s}}}{e^{\eta (t-s)}}{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(t-s)^{\lambda }})\hspace{0.1667em}dv\\ {} \displaystyle \le C{e^{3\eta \gamma }}{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(3\gamma )^{\lambda }})\underset{s\in [t-3\gamma ,t]}{\sup }|{C_{\sigma }}(s)|{\int _{0}^{h/2}}{v^{-2l}}\hspace{0.1667em}dv{\int _{t-3\gamma }^{t}}{(t-s)^{1-l}}\hspace{0.1667em}ds\\ {} \displaystyle =C{e^{3\eta \gamma }}{E_{\lambda ,\lambda +1/2}}(\tilde{C}{(3\gamma )^{\lambda }})\underset{s\in [t-3\gamma ,t]}{\sup }|{C_{\sigma }}(s)|{\gamma ^{1-l}}{h^{1-2l}}\to 0,\hspace{2.5pt}t\to \infty ,\end{array}\]Now assume that for each $y\in [j,j+1]$ and some $m\in \mathbb{N}$ the following inequality holds:
Then we consider the functions $\tilde{p}(t,x;s,y)$ and $\hat{p}(t,x;s,y,h)$ with $M=3\gamma $ and $N=3\gamma +m$. For such M and N, provided that (43) holds, $(t,x)\in {\Omega _{MN}}$. Moreover, using (37), (40), the fact that (43) implies
Note that we estimate $|q(t,x,y)|$ analogously to ${I_{2}}$. From (38), (39), (41), (42), (44), (45) it follows that there exists ${G_{\gamma }}(t):[0,+\infty )\to [0,+\infty )$ such that ${G_{\gamma }}(t)\to 0$, $t\to \infty $, and
\[ \underset{y\in [j,j+1]}{\sup }{e^{-\frac{\nu {(x-y)^{2}}}{t-s}}}\le {e^{-\frac{{\nu _{1}}{m^{2}}}{3\gamma }}}\underset{y\in [j,j+1]}{\sup }{e^{-\frac{(1-{\nu _{1}}){(x-y)^{2}}}{t-s}}}\hspace{1em}\forall (t,x)\in {\Theta _{MN}^{\gamma }},\hspace{2.5pt}0<{\nu _{1}}<\nu ,\]
and Lemma 3, we obtain:
\[\begin{array}{l}\displaystyle |p(t,x;s,y)|\le C(\gamma ){e^{-{c_{0}}t}}t{e^{-\frac{{\nu _{1}}{m^{2}}}{3\gamma }}},\\ {} \displaystyle |p(t,x;s,y+h)-p(t,x;s,y)|\le C(\gamma ){h^{\beta (\sigma )}}{e^{-{c_{0}}t}}t{e^{-\frac{{\nu _{1}}{m^{2}}}{3\gamma }}}.\end{array}\]
Now it is easy to estimate ${I_{1}}$, ${I_{2}}$:
\[\begin{array}{l}\displaystyle {w_{2}}(q,r)\le {G_{\gamma }}(t){r^{\beta (\sigma )}}\hspace{2.5pt}\forall t,\hspace{0.1667em}r\ge 0,j\in \mathbb{Z},x\in \mathbb{R};\\ {} \displaystyle |q(t,x,j)|\le {G_{\gamma }}(t)\hspace{2.5pt}\forall t\ge 0,j\in \mathbb{Z},x\in \mathbb{R};\\ {} \displaystyle {w_{2}}(q,r)\le {G_{\gamma }}(t){r^{\beta (\sigma )}}{m^{-1}}\hspace{2.5pt}\forall t,\hspace{0.1667em}r\ge 0,j\in \mathbb{Z},x\in \mathbb{R}:\underset{y\in [j,j+1]}{\max }|x-y|\ge m+1;\\ {} \displaystyle |q(t,x,j)|\le {G_{\gamma }}(t){m^{-1}}\hspace{2.5pt}\forall t\ge 0,j\in \mathbb{Z},x\in \mathbb{R}:\underset{y\in [j,j+1]}{\max }|x-y|\ge m+1.\end{array}\]
From this it follows that for each $\alpha \in (1/2,\beta (\sigma ))$ the following inequalities hold:
\[\begin{array}{l}\displaystyle \| q(t,x,\cdot ){\| _{{B_{22}^{\alpha }}([j,j+1])}}\le C{G_{\gamma }}(t)\hspace{2.5pt}\forall t\ge 0,j\in \mathbb{Z},x\in \mathbb{R};\\ {} \displaystyle \| q(t,x,\cdot ){\| _{{B_{22}^{\alpha }}([j,j+1])}}\le C{G_{\gamma }}(t){m^{-1}}\\ {} \displaystyle \forall t\ge 0,j\in \mathbb{Z},x\in \mathbb{R}:\underset{y\in [j,j+1]}{\max }|x-y|\ge m+1.\end{array}\]
These estimates imply
\[\begin{array}{l}\displaystyle \sum \limits_{j\in \mathbb{Z}}|q(t,x,j){|^{2}}\le C{G_{\gamma }^{2}}(t)+C{G_{\gamma }^{2}}(t)\sum \limits_{m\in \mathbb{N}}\frac{1}{{m^{2}}}=C{G_{\gamma }^{2}}(t),\\ {} \displaystyle \sum \limits_{j\in \mathbb{Z}}\| q(t,x,\cdot ){\| _{{B_{22}^{\alpha }}([j,j+1])}^{2}}\le C{G_{\gamma }^{2}}(t)+C{G_{\gamma }^{2}}(t)\sum \limits_{m\in \mathbb{N}}\frac{1}{{m^{2}}}=C{G_{\gamma }^{2}}(t).\end{array}\]
for each $x\in \mathbb{R}$. On the other hand,
\[\begin{array}{l}\displaystyle \sum \limits_{j\in \mathbb{Z}}|\mu ((j,j+1]){|^{2}}<\infty \hspace{2.5pt}\text{a.s.,}\\ {} \displaystyle \sum \limits_{j\in \mathbb{Z}}{\sum \limits_{n=1}^{\infty }}{\sum \limits_{k=1}^{{2^{n}}}}{2^{-n(2\alpha -1)}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}<\infty \hspace{2.5pt}\text{a.s.}\end{array}\]
Therefore, for each version that satisfies (34) we have
Taking a supremum on x and sending t to infinity, we obtain the statement of the lemma. □Now we return to the proof of Theorem 1.
Proof.
We use the iteration process (9). For each $n\in \mathbb{N}$ we consider the function
\[ {\nu _{2}^{(n)}}(t,x)={\int _{\mathbb{R}}}p(t,x;0,y){u_{0}}(y)\hspace{0.1667em}dy+{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x;s,y)f(s,y,{u^{(n-1)}}(s,y))\hspace{0.1667em}dy.\]
From [6, Theorem 2 §4] it follows that the function ${\nu _{2}^{(n)}}$ is a solution, bounded on $[0,T]\times \mathbb{R}$, of the Cauchy problem
for each $\omega \in \Omega $, $T>0$. Using Lemma 3, we obtain
Now from (8) and (46) it follows that
Taking a supremum on x, sending t to infinity and using Lemma 5, we obtain the statement of the theorem. □
