Uniqueness in stochastic PDE’s (SPDE’s) concerns their completeness and can induce the fundamental Markov property of solutions. Up to now, to determine uniqueness in SPDE’s with non-Lipschitz diffusion coefficients, only very few cases can be handled. The arguments are a kind of exact calculations, and they are known for being non-robust whenever perturbation of coefficients is imposed. On the other hand, it has been believed that the classical Yamada- Watanabe theorem, which gives a sharp condition on Hölder continuity of coefficients for uniqueness in stochastic differential equations, should have analogues in the context of SPDE’s.

 

One important problem in this direction, open for more than two decades, is whether pathwise uniqueness in the SPDE of one-dimensional super-Brownian motion holds. A recent work by Mueller, Mytnik and Perkins sheds light on this difficult problem, proving, however, that pathwise uniqueness in some closely related SPDE’s fails. In contrast to these particular SPDE’s, the SPDE’s of one-dimensional super-Brownian motions with immigration share more properties with the SPDE of super-Brownian motion but pose different difficulties in settling the question whether pathwise uniqueness still fails.

 

I will first review the SPDE of super-Brownian motion, some notions of uniqueness and related results. I will then introduce a class of super-Brownian motions with immigration and discuss a non-uniqueness result for the corresponding SPDE’s. In the rest of this talk, I will explain the key arguments of its proof.