Harnack Inequality and Applications for SDEs Driven by G-Brownian Motion

In this paper,Wang's Harnack and shift Harnack inequality for a class of stochastic differential equations driven by G-Brownian motion are established.The results generalize the ones in the linear expectation setting.Moreover,some applications are also given.

Harnack inequality and derivative formula for SDE driven by fractional Brownian motion

In the paper, Harnack inequality and derivative formula are established for stochastic differential equation driven by fractional Brownian motion with Hurst parameter H < 1/2. As applications, strong Feller property, log-Harnack inequality and entropy-cost inequality are given.

On Heat Kernel Estimates and Parabolic Harnack Inequality for Jump Processes on Metric Measure Spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.

Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds

Let M be a noncompact complete Riemannian manifold.In this paper,we consider the following nonlinear parabolic equation on M ut(x,t)=△u(x,t)+au(x,t)ln u(x,t)+bu^α(x,t).We prove a Li–Yau type gradient estimate for positive solutions to the above equation;as an application,we also derive the corresponding Harnack inequality.These results generalize the corresponding ones proved by Li(J Funct Anal 100:233–256,1991).

Harnack inequality and gradient estimate for G-SDEs with degenerate noise

In this paper,the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of the measure.Moreover,for any bounded and continuous function f,the gradient estimate∣∇P_(t)f∣≤c(p,t)(Pt|f|p)^(1/p),p>1,t>0 is obtained for the associated nonlinear semigroup P¯t.As an application of the Harnack inequality,we prove the existence of the weak solution to degenerate G-SDEs under some integrable condition.Finally,an example is presented.All of the above results extend the existing ones in the linear expectation setting.

Shift Harnack inequality and integration by parts formula for semilinear stochastic partial differential equations

Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F. -Y. Wang [Ann. Probab., 2012, 42（3）： 994-1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non- Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.

Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise

In this paper,the Harnack and shift Harnack inequalities for functional G-SDEs with degenerate noise are derived by the method of coupling by change of measure.Moreover,the gradient estimate for the associated nonlinear semigroup P_(t) is obtained.All of the above results extend the existed results in linear expectation setting.

HARNACK'S INEQUALITY FOR GENERALIZED SUBELLIPTIC SCHRDINGER OPERATORS

We prove a uniform Harnack μu = 0, where △G is a sublaplacian, μ is scale-invariant Kato condition. inequality for nonnegative solutions of △u - a non-negative Radon measure and satisfying

HARNACK TYPE INEQUALITIES FOR SDES DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH MARKOVIAN SWITCHING

In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H is supposed to be in(1/2, 1). As a direct application, the strong Feller property is presented.

HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS

We prove a Harnack inequality for positive harmonic functions on graphs whichis similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean valueinequality of nonnegative subharmonic functions on graphs.

Remarks on the Harnak Inequality for Local-Minima of Scalar Integral Functionals with General Growth Conditions

In this paper we proof a Harnack inequality and a regularity theorem for local-minima of scalar intagral functionals with general growth conditions.

AN INTEGRATION BY PARTS FORMULA FOR STOCHASTIC HEAT EQUATIONS WITH FRACTIONAL NOISE

In this paper,we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling.As an application,we also obtain the shift Harnack inequalities.

Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.

A sharp gradient estimate for the weighted p-Laplacian

Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation △μ,pu=-λμ,p｜u｜^p-2ufor p ∈ （1, 2）. We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..

LOWER BOUNDS FOR SUP+INF AND SUP*INF AND AN EXTENSION OF CHEN-LIN RESULT IN DIMENSION 3

We give two results about Harnack type inequalities. First, on Riemannian surfaces, we have an estimate of type sup ＋ inf. The second result concern the solutions of prescribed scalar curvature equation on the unit ball of Rn with Dirichlet condition. Next, we give an inequality of type （supK ^u）^2s-1 × infπu ≤ c for positive solutions of △u = V u^5 on Ω belong toR^3, where K is a compact set of Ω and V is s-Holderian, s ∈] - 1/2, 1]. For the case s = 1/2 and Ω = S3, we prove that, if minΩ u 〉 m 〉 0 （for some particular constant m 〉 0）, and the H¨olderian constant A of V tends to 0 （in certain meaning）, we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of Ω.

An Elliptic Gradient Estimate for A Non-homogeneous Heat Equation on Complete Noncompact Manifolds

Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we will give the elliptic gradient estimate for positive smooth solutions to the non-homogeneous heat equation(?_t-△)u(x, t) = A(x, t)when the metric evolves under the Ricci flow. As applications, we get Harnack inequalities to compare solutions at the same time.

A Note on Harnack Type Inequality for the Gaussian Curvature Flow

In this short note we present a new Harnack expression for the Gaussian curvature flow, which is modeled from the shrinking self similiar solutions. As applications we give alternate proofs of Chow’s Harnack inequality and entropy estimate.

Stability of elliptic Harnack inequalities Dedicated to Professor Jia-an Yan on the Occasion of His 80th Birthday

We survey some recent progress in the study of stability of elliptic Harnack inequalities under form-bounded perturbations for strongly local Dirichlet forms on complete locally compact separable metric spaces.

THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS

This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains.After establishing the weak maximum principles,the global boundary Holder estimates and the boundary Harnack inequalities of the equations,we show that all solutions bounded from below are linear combinations of two special solutions(exponential growth at one end and exponential decay at the other)with a bounded solution to the degenerate equations.

Harnack and HWI Inequalities on Infinite-Dimensional Spaces

In this paper, the dimensional-free Harnack inequalities are established on infinite-dimen- sional spaces. More precisely, we establish Harnack inequalities for heat semigroup on based loop group and for Ornstein-Uhlenbeck semigroup on the abstract Wiener space. As an application, we establish the HWI inequality on the abstract Wiener space, which contains three important quantities in one inequality, the relative entropy ＂H＂, Wasserstein distance ＂W＂, and Fisher information ＂T＂.