Harnack inequality and applications for stochastic generalized porous media equations

Abstract

By using coupling and Girsanov transformations, the dimension-free 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.

arXiv commentPublished at http://dx.doi.org/10.1214/009117906000001204 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (47)

• SPDE in Hilbert Space with Locally Monotone Coefficients
• Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise
• Harnack Inequalities for Functional SDEs with Multiplicative Noise and Applications
• Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds
• Harnack Inequality and Applications for Stochastic Evolution Equations with Monotone Drifts
• Harnack Inequality and Strong Feller Property for Stochastic Fast-Diffusion Equations
• Existence and Uniqueness of Solutions to Nonlinear Evolution Equations with Locally Monotone Operators
• Harnack Type Inequalities and Applications for SDE Driven by Fractional Brownian Motion
• Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts
• Derivative Formula and Harnack Inequality for Linear SDEs Driven by Lévy Processes
• Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations
• Nonlinear Fokker-Planck equations for Probability Measures on Path Space and Path-Distribution Dependent SDEs
• Ergodicity for Stochastic Porous Media Equations
• Harnack Inequalities for Stochastic Equations Driven by Lévy Noise
• Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups
• Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises
• Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups
• Asymptotic Log-Harnack Inequality for Monotone SPDE with Multiplicative Noise
• Hypercontractivity for Functional Stochastic Differential Equations
• Harnack Inequality for Semilinear spdes with multiplicative noise
• Exponential Convergence of Non-Linear Monotone SPDEs
• Harnack Inequalities and Applications for Ornstein-Uhlenbeck Semigroups with Jump
• Log-Harnack inequality for reflected SPDEs driven by multiplicative noises and its applications
• Strong Feller Property and Irreducibility for Non-Linear Monotone SPDEs
• Degenerate Fokker-Planck Equations : Bismut Formula, Gradient Estimate and Harnack Inequality
• Harnack Inequalities for Ornstein-Uhlenbeck Processes Driven by Lévy Processes
• Harnack Inequalities and Applications for Stochastic Differential Equations Driven by Fractional Brownian Motion
• Asymptotic Couplings by Reflection and Applications for Non-Linear Monotone SPDES
• Mild Solutions and Harnack Inequality for Functional SPDEs with Dini Drift
• Log-Harnack Inequality and Exponential Ergodicity for Distribution Dependent CKLS and Vasicek Model
• Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in $L^p$
• Ergodicity for singular-degenerate porous media equations
• Harnack and Super Poincaré Inequalities for Generalized Cox-Ingersoll-Ross Model
• Asymptotic Log-Harnack Inequality and Applications for Stochastic Systems of Infinite Memory
• Derivative Formula and Harnack Inequality for Degenerate Functional SDEs
• Log-Harnack Inequality for Stochastic Differential Equations in Hilbert Spaces and its Consequences
• Estimates for Invariant Probability Measures of Degenerate SPDEs with Singular and Path-Dependent Drifts
• Well-posedness and Regularity for Distribution Dependent SPDEs with Singular Drifts
• Stochastic generalized porous media equations driven by Lévy noise with increasing Lipschitz nonlinearities
• Gradient estimates for semigroups associated with stochastic differential equations driven by cylindrical Lévy processes
• The global random attractor for a class of stochastic porous media equations
• Harnack Inequality for Functional SDEs with Bounded Memory
• Dimension-independent Harnack inequalities for subordinated semigroups
• A Modified log-Harnack inequality and asymptotically strong Feller property
• Poincare Inequality on the Path Space of Poisson Point Processes
• Heat kernel upper bounds under the generalized curvature(-dimension) inequality
• Harnack Inequalities and Applications for Multivalued Stochastic Evolution Equations