By using coupling and Girsanov transformations, the dimensionfree Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the L p -norm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived.

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x) dx is a probability measure, and let α ∈ (0, 2). Explicit criteria are presented for the α-stable-like Dirichlet formto satisfy Poincaré-type (i.e., Poincaré, weak Poincaré and super Poincaré) inequalities. As applications, sharp functional inequalities are derived for the Dirichlet form with V having some typical growths. Finally, the main result of [15] on the Poincaré inequality is strengthened.

To characterize the Neumann problem for nonlinear Fokker-Planck equations, we investigate distribution dependent reflecting SDEs (DDRSDEs) in a domain. We first prove the well-posedness and establish functional inequalities for reflecting SDEs with singular drifts, then extend these results to DDRSDEs with singular or monotone coefficients, for which a general criterion deducing the well-posedness of DDRSDEs from that of reflecting SDEs is established. Moreover, three different types of exponential ergodicity are derived for DDRSDEs under dissipative, partially dissipative, and fully non-dissipative conditions respectively.

By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

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Let P 2 (R d ) be the space of probability measures on R d with finite second moment. The path independence of additive functionals of McKean-Vlasov SDEs is characterized by PDEs on the product space R d × P 2 (R d ) equipped with the usual derivative in space variable and Lions' derivative in distribution. These PDEs are solved by using probabilistic arguments developed from [2]. As consequence, the path independence of Girsanov transformations are identified with nonlinear PDEs on R d × P 2 (R d ) whose solutions are given by probabilistic arguments as well. In particular, the corresponding results on the Girsanov transformation killing the drift term derived earlier for the classical SDEs are recovered as special situations.