One difficult problem for uniqueness in SPDEs, open for more than two decades, is to determine pathwise uniqueness in the SPDE of one-dimensional super-Brownian motion for nonnegative solutions. A recent work by Mueller, Mytnik and Perkins disproves, however, the stronger conjecture that the square-root diffusion coefficient of the SPDE is a robust mechanism to keep solutions of SPDEs unique in general, among other things. The counterexample studies an SPDE with the same coefficients as the SPDE of super-Brownian motion and signed solutions are allowed.
In this talk, I will discuss a result which continues the investigation of the SPDE of super-Brownian motion by Mueller et al. Only nonnegative solutions are considered, and now we introduce perturbation to the SPDE, which is realized by additional immigration mechanisms. I will first review the SPDE of super-Brownian motion and related aspects. I will then introduce SPDEs for super-Brownian motions with immigration and discuss a non-uniqueness result and its proof.