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We refine and generalize several interpolation inequalities bounding the L p norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich distance to μ on a smooth weighted Riemannian manifold satisfying CD (0 , ∞) condition. [ABSTRACT FROM AUTHOR]

By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker–Planck equations for probability measures (μt)t≥0 on the path space C := C([−r0,0];Rd), is analyzed: ∂tμ(t) = L∗t,μtμt, t ≥ 0, where μ(t) is the image of μt under the projection C ∋ ξ ↦ ξ(0) ∈ Rd, and Lt,μ(ξ) := 1/2 d∑i,j=1aij(t,ξ,μ)∂2/∂ξ(0)i∂ξ(0)j + d∑i=1bi(t,ξ,μ) ∂/∂ξ(0)i, t ≥ 0,ξ ∈ C ,μ ∈ PC. Under reasonable conditions on the coefficients aij and bi, the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions. [ABSTRACT FROM AUTHOR]

The distribution dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a nonlinear PDE. Due to the distribution dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a nonlinear semigroup P t ∗ on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the nonlinear semigroup P t ∗ using coupling by change of measures. The main results are illustrated by homogeneous Landau equations. [ABSTRACT FROM AUTHOR]

The hypercontractivity is proved for the Markov semigroup associated with a class of stochastic Hamiltonian systems on Hilbert spaces. Consequently, the Markov semigroup converges exponentially to the invariant probability measure in entropy and is compact for large time. These strengthen the hypocoercivity results derived in the literature. Since the log-Sobolev inequality is invalid, we introduce a new argument to prove the hypercontractivity using coupling and dimension-free Harnack inequality. The main results are illustrated by concrete examples of the kinetic Fokker–Planck equation and highly degenerate diffusion processes. [ABSTRACT FROM AUTHOR]

An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated with a class of functional stochastic differential equations. Consequently, the semigroup P t converges exponentially to its unique invariant probability measure μ in both L 2 ( μ ) and the totally variational norm ‖ ⋅ ‖ var , and it is compact in L 2 ( μ ) for sufficiently large t > 0 . This provides a natural class of non-symmetric Markov semigroups which are compact for large time but non-compact for small time. A semi-linear model which may not satisfy this sufficient condition is also investigated. As the associated Dirichlet form does not satisfy the log-Sobolev inequality, the standard argument using functional inequalities does not work. [ABSTRACT FROM AUTHOR]

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semilinear stochastic equations in Hilbert spaces. [ABSTRACT FROM AUTHOR]

By using lower bound conditions of the Lévy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Lévy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Lévy processes is derived. [ABSTRACT FROM PUBLISHER]

Abstract: By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack inequalities are derived. [Copyright &y& Elsevier]

Abstract: By using Malliavin calculus, explicit derivative formulae are established for a class of semi-linear functional stochastic partial differential equations with additive or multiplicative noise. As applications, gradient estimates and Harnack inequalities are derived for the semigroup of the associated segment process. [Copyright &y& Elsevier]

By constructing successful couplings, the derivative formula, gradient estimates and Harnack inequalities are established for the semigroup associated with a class of degenerate functional stochastic differential equations. [ABSTRACT FROM AUTHOR]

Abstract: By constructing successful couplings for degenerate diffusion processes, explicit derivative formula and Harnack type inequalities are presented for solutions to a class of degenerate Fokker–Planck equations on . The main results are also applied to the study of gradient estimate, entropy/transportation-cost inequality and heat kernel inequalities. [Copyright &y& Elsevier]

Abstract: By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solution. The dimension-free Harnack inequality in the sense of Wang (1997) is also investigated. [Copyright &y& Elsevier]

Abstract: Some equivalent gradient and Harnack inequalities of a diffusion semigroup are presented for the curvature-dimension condition of the associated generator. As applications, the first eigenvalue, the log-Harnack inequality, the heat kernel estimates, and the HWI inequality are derived by using the curvature-dimension condition. The transportation inequality for diffusion semigroups is also investigated. [Copyright &y& Elsevier]

Abstract: On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup are proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for the Neumann heat kernel w.r.t. a volume type measure μ and for K a constant, the curvature condition together with the convexity of the boundary is equivalent to the heat kernel entropy inequality: where ρ is the Riemannian distance. The main result is partly extended to manifolds with non-convex boundary and applied to derive the HWI inequality. [ABSTRACT FROM AUTHOR]

Abstract: Counterexamples are constructed to show that when the second fundamental form of the boundary is bounded below by a negative constant, any curvature lower bound is not enough to imply the log-Sobolev inequality. This indicates that in the study of functional inequalities on non-convex manifolds, the concavity of the boundary cannot be compensated by the positivity of the curvature. Next, when the boundary is merely concave on a bounded domain, a criterion on the log-Sobolev inequality known for convex manifolds is proved. Finally, when the concave part of the boundary is unbounded, a Sobolev inequality for a weighted volume measure is established, which implies an explicit sufficient condition for the log-Sobolev inequality to hold on non-convex manifolds. [Copyright &y& Elsevier]

Abstract: As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333–1350], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, this paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev–Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results. [Copyright &y& Elsevier]

Harnack inequalities are established for a class of generalized Mehler semigroups, which in particular imply upper bound estimates for the transition density. Moreover, Poincare´ and log-Sobolev inequalities are proved in terms of estimates for the square field operators. Furthermore, under a condition, well-known in the Gaussian case, we prove that generalized Mehler semigroups are strong Feller. The results are illustrated by concrete examples. In particular, we show that a generalized Mehler semigroup with an α-stable part is not hyperbounded but exponentially ergodic, and that the log-Sobolev constant obtained by our method in the special Gaussian case can be sharper than the one following from the usual curvature condition. Moreover, a Harnack inequality is established for the generalized Mehler semigroup associated with the Dirichlet heat semigroup on (0,1). We also prove that this semigroup is not hyperbounded. [Copyright &y& Elsevier]

Abstract: We consider stochastic equations in Hilbert spaces with singular drift in the framework of [G. Da Prato, M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2) (2002) 261–303]. We prove a Harnack inequality (in the sense of [F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417–424]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure μ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure μ for noncontinuous drifts. [Copyright &y& Elsevier]

Abstract: Using the coupling by parallel translation, along with Girsanov''s theorem, a new version of a dimension-free Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curvature bounded below by , where is a constant and is the Riemannian distance function to a fixed point o on the manifold. As an application, in the symmetric case, a Li–Yau type heat kernel bound is presented for such semigroups. [Copyright &y& Elsevier]