Abstract
In this paper we solve the Kolmogorov equation and, as a consequence, the martingale problem corresponding to a stochastic differential equation of type dX t =AX t dt+b(X t )dt+dY t , on a Hilbert space E, where (Y t ) t≥0 is a Levy process on E,A generates a C 0-semigroup on E and b:E→E. Our main point is to allow unbounded A and also singular (in particular, non-continuous) b. Our approach is based on perturbation theory of C 0-semigroups, which we apply to generalized Mehler semigroups considered on L 2(μ), where μ is their respective invariant measure. We apply our results, in particular, to stochastic heat equations with Levy noise and singular drift.
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Lescot, P., Röckner, M. Perturbations of Generalized Mehler Semigroups and Applications to Stochastic Heat Equations with Levy Noise and Singular Drift. Potential Analysis 20, 317–344 (2004). https://doi.org/10.1023/B:POTA.0000009814.54278.34
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DOI: https://doi.org/10.1023/B:POTA.0000009814.54278.34