Reflected Stochastic Burgers Equation with Jumps

Abstract

This paper is concerned with the reflected stochastic Burgers equation driven both by the Brownian motion and by the Poisson random measure. The existence and uniqueness of solutions are established. The penalization method plays an important role.

Appendix A Stochastic Burgers Equation with Jumps

Lemma A.1

Assume that $g,\sigma$ and $\gamma$ satisfy (H.2) and $f{:}\mathbb{R}\rightarrow \mathbb{R}$ is a Lipschitz function. Then the following equation has a unique solution $u\in S^2_{{\mathcal {F}}}([0,T];\,H)\cap L^2_{{\mathcal {F}}}([0,T];\,V){:}$

$$\begin{aligned} \left\{ \begin{array}{l} \hbox{d}u(t,x)=\frac{\partial ^2 u(t,x)}{\partial x^2}\hbox{d}t+\frac{\partial g(u(t,x))}{\partial x}\hbox{d}t+f(u(t,x))\hbox{d}t\\ \qquad+\sum\limits^d_{k=1}\sigma _k(u(t,x))\hbox{d}B_k(t)+\int _{U}\gamma (u(t-,x),z){\tilde{N}}(\hbox{d}t,\hbox{d}z);\\ u(0,\cdot )=u_0(\cdot );\\ u(t,0)=u(t,1)=0,~ t\in \mathbb{R}_+. \end{array} \right. \end{aligned}$$

(A.1)

Proof

From Theorem 3.1 in [22], we know that (A.1) has a unique local solution u which satisfies $E[u(t\wedge \tau _{\infty })|^2_H]<\infty ,$ where $\tau _{\infty }=\sup _{k\geqslant 1}\tau _k$ with $\tau _k=\inf \{t\geqslant 0{:}|u(t)|_H\geqslant k\}\wedge T$. Next, we will prove that u is the global solution. It suffices to show that ${\mathbb {P}}(\tau _{\infty }=T)=1$. To this end, we apply Itô’s formula to obtain

$$\begin{aligned}& | u(t) |^2_{H}+2\int ^t_0\Vert u(s)\Vert ^2\hbox{d}s\nonumber \\ =&| u(0) |^2_{H} +2\int ^t_0\int ^1_0 u(s,x) \frac{\partial g(u(s,x))}{\partial x}\hbox{d}x\hbox{d}s+2\sum ^d_{k=1}\int ^t_0\langle u(s),\sigma _k(u(s))\rangle _{H} \hbox{d}B_k(s)\nonumber \nonumber \\ &+2\int ^t_0\langle u(s),f(u(s))\rangle _{H} \hbox{d}s+\sum ^d_{k=1}\int ^t_0| \sigma _k(u(s)) |^2_{H} \hbox{d}s +\int ^t_0\int _{U} | \gamma (u(s-),z) |^2_{H}\nu (\hbox{d}z)\hbox{d}s\nonumber \\& +\int ^t_0\int _{U}( |u(s-)+ \gamma (u(s-),z)|^2_{H}-| u(s-)|^2_{H}){\tilde{N}}(\hbox{d}s,\hbox{d}z). \end{aligned}$$

From the arguments of Lemma 2, we can prove

$$\begin{aligned} E\left[\sup _{0\leqslant t\leqslant T}| u(t) |^2_{H}\right]+E\Big [\int ^{{T}}_0\Vert u^n(t)\Vert ^2 \hbox{d}t\Big ]<\infty . \end{aligned}$$

Note that

$${\mathbb{P}}(\tau _{k} \leqslant {\mkern 1mu} T) = {\mathbb{P}}(\left| {u(\tau _{k} )} \right|_{H} \geqslant k) \leqslant \frac{{E\left[ {\mathop {\sup }\limits_{{0 \leqslant {\kern 1pt} t \leqslant {\kern 1pt} T}} \left| {u(t)} \right|_{H}^{2} } \right]}}{{k^{2} }}.$$

Letting $k\rightarrow \infty$, we have ${\mathbb {P}}(\tau _{\infty }=T)=1$.