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Harnack inequality and strong Feller property for stochastic fast-diffusion equations. (English) Zbl 1151.60032
As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equation, Ann. Probab. 35, No. 4, 1333–1350 (2007; Zbl 1129.60060)], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, in this very nice paper the authors obtain analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation has weaker dissipativity than the porous medium equation, such results are harder to prove. As a compensation for the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some convincing concrete examples are presented to illustrate the main results.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35Q35 PDEs in connection with fluid mechanics
47D07 Markov semigroups and applications to diffusion processes
60J35 Transition functions, generators and resolvents
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