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Harnack inequality and applications for stochastic evolution equations with monotone drifts
 J. Evol. Equat
"... As a Generalization to [37] where the dimensionfree Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity proper ..."
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Cited by 22 (6 self)
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As a Generalization to [37] where the dimensionfree Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reactiondiffusion equations, stochastic porous media equations and the stochastic pLaplace equation in Hilbert space.
Harnack Inequalities and Applications for Multivalued Stochastic Evolution Equations
, 2009
"... By the method of coupling and Girsanov transformation, Harnack inequalities [F.Y. Wang, 1997] and strong Feller property are proved for the transition semigroup associated with the multivalued stochastic evolution equation on a Gelfand triple. The concentration property of the invariant measure for ..."
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Cited by 6 (0 self)
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By the method of coupling and Girsanov transformation, Harnack inequalities [F.Y. Wang, 1997] and strong Feller property are proved for the transition semigroup associated with the multivalued stochastic evolution equation on a Gelfand triple. The concentration property of the invariant measure for the semigroup is investigated. As applications of Harnack inequalities, explicit upper bounds of the L pnorm of the density, contractivity, compactness and entropycost inequality for the semigroup are also presented.
Dimensionfree Harnack inequality and its applications, Front
 Math. China
, 2006
"... This paper presents a selfcontained account concerning a dimensionfree Harnack inequality and its applications. This new type of inequality not only implies heat kernel bounds as the classical LiYau’s Harnack inequality did, but also provides a direct way to describe various dimensionfree proper ..."
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Cited by 5 (1 self)
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This paper presents a selfcontained account concerning a dimensionfree Harnack inequality and its applications. This new type of inequality not only implies heat kernel bounds as the classical LiYau’s Harnack inequality did, but also provides a direct way to describe various dimensionfree properties of finite and infinitedimensional diffusion semigroups. We will start from a standard weighted Laplace operator on a Riemannian manifold with curvature bounded from below, then move further to the unbounded below curvature case and infinitedimensional settings.
Derivative formula, integration by parts formula and applications for SDEs driven by fractional Brownian motion
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A criterion for Talagrand's quadratic transportation cost inequality
, 2008
"... We show that the quadratic transportation cost inequality T2 is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. In particular if a logarithmic Sobolev inequality implies T2, we are able to give examples for which T2 holds but the logarithmic Sobol ..."
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Cited by 1 (1 self)
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We show that the quadratic transportation cost inequality T2 is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. In particular if a logarithmic Sobolev inequality implies T2, we are able to give examples for which T2 holds but the logarithmic Sobolev inequality does not hold. This answers to a question left open by Otto and Villani [21] and Bobkov, Gentil and Ledoux [4], and furnishes (in a Riemannian setting) the analogue of the well known criterion by Bobkov and Götze for the linear transportation cost inequality T1 [5] (also see [12]). The main ingredient in the proof is a new family of inequalities, called modified quadratic transportation cost inequalities in the spirit of the modified logarithmicSobolev inequalities by Bobkov and Ledoux [6], that are shown to hold as soon as a Poincaré inequality is satisfied.
Harnack Inequality and Applications for Stochastic Differential Equations with Jumps ∗
, 2008
"... Gradient estimates and a Harnack inequality are established for the semigroup associated to stochastic differential equations driven by Poisson processes. As applications, estimates of the transition probability density, the compactness and ultraboundedness of the semigroup are studied in terms of t ..."
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Gradient estimates and a Harnack inequality are established for the semigroup associated to stochastic differential equations driven by Poisson processes. As applications, estimates of the transition probability density, the compactness and ultraboundedness of the semigroup are studied in terms of the corresponding invariant measure.
mn header will be provided by the publisher Estimates of the First Neumann Eigenvalue and the LogSobolev Constant on NonConvex Manifolds
, 2006
"... this paper a number of explicit lower bounds are presented for the first Neumann eigenvalue on nonconvex manifolds. The main idea to derive these estimates is to make a conformal change of the metric such that the manifold is convex under the new metric, which enables one to apply known results obt ..."
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this paper a number of explicit lower bounds are presented for the first Neumann eigenvalue on nonconvex manifolds. The main idea to derive these estimates is to make a conformal change of the metric such that the manifold is convex under the new metric, which enables one to apply known results obtained in the convex case. This method also works for more general functional inequalities. In particular, some explicit lower bounds are presented for the logSobolev constant on nonconvex manifolds. 1
EntropyCost Inequalities for Diffusion Semigroups with Curvature Unbounded Below ∗
, 2007
"... The weighted logSobolev inequality and the entropycost inequality are established for a class of diffusion semigroups with curvature unbounded below. Concrete examples are presented to illustrate the main results. AMS subject Classification: 58G32, 60J60 ..."
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The weighted logSobolev inequality and the entropycost inequality are established for a class of diffusion semigroups with curvature unbounded below. Concrete examples are presented to illustrate the main results. AMS subject Classification: 58G32, 60J60
Submitted to the Annals of Probability HARNACK INEQUALITY AND APPLICATIONS FOR STOCHASTIC GENERALIZED POROUS MEDIA EQUATIONS ∗
"... 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 Lpnorm of the density as ..."
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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 Lpnorm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived. 1. Introduction. The dimensionfree Harnack inequality
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, 2005
"... www.elsevier.com/locate/bulsci Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below ✩ ..."
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www.elsevier.com/locate/bulsci Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below ✩