Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift

Consider the stochastic evolution equation in a separable Hilbert space with a nice multiplicative noise and a locally Dini continuous drift. We prove that for any initial data the equation has a unique (possibly explosive) mild solution. Under a reasonable condition ensuring the non-explosion of the solution, the strong Feller property of the associated Markov semigroup is proved. Gradient estimates and log-Harnack inequalities are derived for the associated semigroup under certain global conditions, which are new even in finite-dimensions.Comment: 36 page

Log-Sobolev inequalities: Different roles of Ric and Hess

Let $P_t$ be the diffusion semigroup generated by $L:=\Delta +\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(\sigma ^2\rho_o^2+c)$ for some constants $\sigma, c>0$ and $\rho_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided $-\operatorname {Hess}_V\ge\delta$ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}.$ This indicates, at least in finite dimensions, that $\operatorname {Ric}$ and $-\operatorname {Hess}_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.Comment: Published in at http://dx.doi.org/10.1214/08-AOP444 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds

By constructing a coupling with unbounded time-dependent drift, dimension-free Harnack inequalities are established for a large class of stochastic differential equations with multiplicative noise. These inequalities are applied to the study of heat kernel upper bound and contractivity properties of the semigroup. The main results are also extended to reflecting diffusion processes on Riemannian manifolds with nonconvex boundary.Comment: Published in at http://dx.doi.org/10.1214/10-AOP600 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Coupling and Applications

This paper presents a self-contained account for coupling arguments and applications in the context of Markov processes. We first use coupling to describe the transport problem, which leads to the concepts of optimal coupling and probability distance (or transportation-cost), then introduce applications of coupling to the study of ergodicity, Liouville theorem, convergence rate, gradient estimate, and Harnack inequality for Markov processes.Comment: 16 page