Can the stochastic wave equation with strong drift hit zero?

Abstract

We study the stochastic wave equation with multiplicative noise and singular drift:

\[\partial _{t}u(t,x)=\Delta u(t,x)+u^{-\alpha }(t,x)+g(u(t,x))\dot{W} (t,x)\]

where $x$ lies in the circle $\mathbf{R} /J\mathbf{Z} $ and $u(0,x)>0$. We show that

(i) If $0<\alpha <1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$.

(ii) If $\alpha >3$ then with probability one, $u(t,x)\ne 0$ for all $(t,x)$.