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 $\frac{\partial X}{\partial t}=\fracΔ{2}X+X^p\dot W+ψ,\ X_0\equiv 0$ where $\dot W$ is a white noise, $ψ\ge 0$ is smooth, compactly supported and non-trivial, and $0<p<1/2$. We further show that any solution spends positive time at the 0 function.

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