182,995 research outputs found
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
Harnack Inequalities for Stochastic Equations Driven by L\'evy Noise
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 L\'evy noise containing a
subordinate Brownian motion. The Harnack inequalities are new even for linear
equations driven by L\'evy noise, and the gradient estimate implied by our
log-Harnack inequality considerably generalizes some recent results on gradient
estimates and coupling properties derived for L\'evy processes or linear
equations driven by L\'evy noise. The main results are also extended to
semi-linear stochastic equations in Hilbert spaces.Comment: 15 page
Log-Sobolev inequalities: Different roles of Ric and Hess
Let be the diffusion semigroup generated by on a
complete connected Riemannian manifold with for some constants and the Riemannian
distance to a fixed point. It is shown that is hypercontractive, or the
log-Sobolev inequality holds for the associated Dirichlet form, provided
holds outside of a compact set for some
constant This indicates, at least in
finite dimensions, that and
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
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