Perturbations of generalized Mehler semigroups and applications to stochastic heat equations with Levy noise and singular drift
The authors study the stochastic differential equation \(dX_t=AX_t dt+b(X_t) dt+dY_t\) on a Hilbert space \(E\), where \((Y_t)_{t\geq 0}\) is a Levy process on \(E\), \(A\) generates a \(C_0\)-semigroup on \(E\) and \(b:E\to E\) is a singular drift. Using the perturbation theory of \(C_0\)-semigroups (applied to generalized Mehler semigroups), the authors construct a strong Markov process solving the above equation in the sense of martingal problem. Under some constraint, the uniqueness in the sense of Markov selections is proved as well. An application to stochastic heat equations with Levy noise and singular drift is included.
scientific article; zbMATH DE number 2057484