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A stochastic heat equation with the distributions of Lévy processes as its invariant measures. (English) Zbl 1157.60063

Summary: We consider a linear heat equation on a half line with an additive noise chosen properly in such a manner that its invariant measures are a class of distributions of Lévy processes. Our assumption on the corresponding Lévy measure is, in general, mild except that we need its integrability to show that the distributions of Lévy processes are the only invariant measures of the stochastic heat equation.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
35R60 PDEs with randomness, stochastic partial differential equations
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