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

K. Burdzy, C. Mueller, E. A. Perkins

Published 2010-08-12, updated 2011-03-21Version 4

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