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Structural properties of semilinear SPDEs driven by cylindrical stable processes. (English) Zbl 1231.60061
This paper is concerned with structural properties of solutions to semilinear stochastic equations \(dX_t=AX_t dt + F(X_t)\,dt + d Z_t\), \(X_0=x\in H\), in a real separable Hilbert space \(H\) driven by a cylindrical \(\alpha\)-stable (\(\alpha\in(0,2)\)) process \(Z=(Z_t)_{t\geq 0}\) taking values in a Hilbert space \(U\) typically greater than \(H\). The operator \(A\), possibly unbounded, generates a \(C^0\)-semigroup on \(H\) and \(F:H\to H\) is bounded and Lipschitz continuous.
The authors start studying the linear case when \(F=0\) and establish the existence of a solution taking values in \(H\), its measurability and Markovianity. Then, in the semilinear case, they study Markovianity, irreducibility, stochastic continuity, Feller and strong Feller properties for the solutions, and investigate the integrability of trajectories. The main results are gradient estimates for the associated transition semigroup, from which they deduce the strong Feller property and the time regularity of trajectories.
Applications to a stochastic heat equation with Dirichlet boundary conditions, bounded Lipschitz nonlinearities and cylindrical \(\alpha\)-stable noise process are discussed.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J75 Jump processes (MSC2010)
47D07 Markov semigroups and applications to diffusion processes
35R60 PDEs with randomness, stochastic partial differential equations
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