Summary
A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupR t are obtained. We show thatR t is a compactC 0-semigroup in all Sobolev spacesW n,p which are built on its invariant measure μ. Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inL p(μ) spaces and spacesW 1,p. As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to μ. It is shown also that the density of this measure with respect to μ is inL p(μ) for allp≧1.
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This work was done during the first author's stay at UNSW supported by ARC Grant 150.346 and the second author's stay at Łódź University supported by KBN Grant 2.1020.91.01
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Chojnowska-Michalik, A., Goldys, B. Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces. Probab. Th. Rel. Fields 102, 331–356 (1995). https://doi.org/10.1007/BF01192465
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DOI: https://doi.org/10.1007/BF01192465
Mathematics Subject Classification (1979)
- 60H15