Non-uniqueness for non-negative solutions of parabolic stochastic partial differential equations

Pathwise non-uniqueness is established for non-negative solutions of the parabolic stochastic pde $$\frac{\mathrm{\partial }X}{\mathrm{\partial }t}=\frac{\mathrm{\Delta }}{2}X+X^p\dot{W}+\psi ,X_0\equiv 0$$ where $\dot{W}$ is a white noise, $\psi \ge 0$ is smooth, compactly supported and non-trivial, and $0<p<1\mathrm{/}2$. We further show that any solution spends positive time at the 0 function.