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

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.

Published at http://dx.doi.org/10.1214/14-AOP962 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\epsilon$-Hölder diffusion coefficients
• Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case

Cited by in corpus (4)

• Long-time behavior for subcritical measure-valued branching processes with immigration
• Pathwise Uniqueness of the Solutions of Stochastic Heat Equation with Square-root Coefficient
• Exponential ergodicity of stochastic heat equations with Hölder coefficients
• A distribution-function-valued SPDE and its applications