Can the Stochastic Wave Equation with Strong Drift Hit Zero?

Abstract

We study the stochastic wave equation with multiplicative noise and singular drift: \[ \partial_tu(t,x)=Δu(t,x)+u^{-α}(t,x)+g(u(t,x))\dot{W}(t,x) \] where $x$ lies in the circle $\mathbf{R}/J\mathbf{Z}$ and $u(0,x)>0$. We show that (i) If $0<\hat{I}\pm <1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$. (ii) If $\hat{I}\pm >3$ then with probability one, $u(t,x)\ne 0$ for all $(t,x)$.

35 pages