SPDEs with two reflecting walls and two singular drifts

Abstract

We study SPDEs with two reflecting walls $\Lambda^1$, $\Lambda^2$ and two singular drifts $\frac{c_1}{(X-\Lambda^1)^{\vartheta}}$, $\frac{c_2}{(\Lambda^2-X)^{\vartheta}}$, driven by space-time white noise. First, we establish the existence and uniqueness of the solutions $X$ for $\vartheta\geq 0$. Second, we obtain the following pathwise properties of the solutions $X$. If $\vartheta>3$, then a.s. $\Lambda^1<X<\Lambda^2$ for all $t\geq0$; If $0<\vartheta<3$, then $X$ hits $\Lambda^1$ or $\Lambda^2$ with positive probability in finite time. Thus $\vartheta=3$ is the critical parameter for $X$ to hit reflecting walls.