Nonuniqueness for nonnegative solutions of parabolic stochastic partial differential equations

Abstract

Pathwise nonuniqueness is established for nonnegative solutions of the parabolic stochastic pde \[ \frac{\partial X}{\partial t}=\frac{\Delta}{2}X+X^p\dot W+\psi,\quad X_0 \equiv0 \] where $\dot W$ is a white noise, $\psi\ge0$ is smooth, compactly supported and nontrivial, and $0<p<1/2$. We further show that any solution spends positive time at the $0$ function.