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Pathwise nonuniqueness for the SPDEs of some super-Brownian motions with immigration. (English) Zbl 1337.60135
Summary: We prove pathwise nonuniqueness for stochastic partial differential equations (SPDEs) for some one-dimensional super-Brownian motions with immigration. In contrast to a closely related case investigated by C. Mueller et al. [Ann. Probab. 42, No. 5, 2032–2112 (2014; Zbl 1301.60080)], 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 are 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.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60J68 Superprocesses 60J65 Brownian motion 35R60 PDEs with randomness, stochastic partial differential equations 35K05 Heat equation
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