Stochastic Partial Differential Equations with Two Reflecting Walls

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 2, Page 129--144.

Otobe, Yoshiki
Stochastic Partial Differential Equations with Two Reflecting Walls
We study stochastic partial differential equations (SPDEs) driven by space-time white noise with two reflecting smooth walls $h_1$ and $h_2$. If the solution stays in the open interval $(h_1(x,t), h_2(x,t))$, the dynamics obeys a usual type of SPDEs, and at a point where the value of the solution is $h_1$ or $h_2$, we add forces in order to prevent it from exiting the interval $[h_1, h_2]$. We will first show the existence and uniqueness of the solutions, and secondly study the stationary distribution of the dynamics and corresponding Dirichlet forms.