Pathwise nonuniqueness for the SPDEs of some super-Brownian motions with immigration

TL;DR

This paper demonstrates pathwise nonuniqueness in certain one-dimensional super-Brownian motion SPDEs with immigration, introducing a novel coupling method called continuous decomposition that could benefit broader SPDE research.

Contribution

It establishes pathwise nonuniqueness for specific super-Brownian motion SPDEs with immigration and introduces the continuous decomposition coupling method for analyzing such equations.

Findings

Proves pathwise nonuniqueness in the considered SPDEs.

Develops a novel continuous decomposition coupling method.

Highlights differences in solution properties compared to related models.

Abstract

We prove pathwise nonuniqueness in the stochastic partial differential equations (SPDEs) for some one-dimensional super-Brownian motions with immigration. In contrast to a closely related case investigated by Mueller, Mytnik and Perkins [Ann. Probab. 42 (2014) 2032-2112], the solutions of the present SPDEs are assumed to be nonnegative and have very different properties including uniqueness in law. In proving possible separation of solutions, we derive delicate properties of certain correlated approximating solutions, which is based on a novel coupling method called continuous decomposition. In general, this method may be of independent interest in furnishing solutions of SPDEs with intrinsic adapted structure.oximating solutions, which may be of independent interest in the study of superprocesses with immigration.