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 $\vartheta \ge 0$. Second, we obtain the following pathwise properties of the solutions $X$. If $\vartheta >3$, then a.s. ${\Lambda }^1<X<{\Lambda }^2$ for all $t\ge 0$; 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.