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 $(t,x)$.
Citation
Kevin Lin. Carl Mueller. "Can the stochastic wave equation with strong drift hit zero?." Electron. J. Probab. 24 1 - 26, 2019. https://doi.org/10.1214/19-EJP279
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