Can the Stochastic Wave Equation with Strong Drift Hit Zero?

TL;DR

This paper investigates the conditions under which solutions to a stochastic wave equation with singular drift hit zero, revealing that the likelihood depends on the parameter alpha, with certain regimes almost surely avoiding zero.

Contribution

The study provides a rigorous analysis of the stochastic wave equation with multiplicative noise and singular drift, establishing conditions for hitting zero based on the parameter alpha.

Findings

For 0<alpha<1, solutions hit zero with positive probability.

For alpha>3, solutions almost surely do not hit zero.

The behavior of solutions is critically dependent on the value of alpha.

Abstract

We study the stochastic wave equation with multiplicative noise and singular drift: \[ \partial_tu(t,x)=\Delta u(t,x)+u^{-\alpha}(t,x)+g(u(t,x))\dot{W}(t,x) \] 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, $u(t,x)\ne0$ for all $(t,x)$.