1,983 research outputs found
Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups
We consider stochastic equations in Hilbert spaces with singular drift in the
framework of [Da Prato, R\"ockner, PTRF 2002]. We prove a Harnack inequality
(in the sense of [Wang, PTRF 1997]) for its transition semigroup and exploit
its consequences. In particular, we prove regularizing and ultraboundedness
properties of the transition semigroup as well as that the corresponding
Kolmogorov operator has at most one infinitesimally invariant measure
(satisfying some mild integrability conditions). Finally, we prove existence of
such a measure for non-continuous drifts
Log-Harnack Inequality for Stochastic Burgers Equations and Applications
By proving an -gradient estimate for the corresponding Galerkin
approximations, the log-Harnack inequality is established for the semigroup
associated to a class of stochastic Burgers equations. As applications, we
derive the strong Feller property of the semigroup, the irreducibility of the
solution, the entropy-cost inequality for the adjoint semigroup, and entropy
upper bounds of the transition density
Maximal regularity for Dirichlet problems in Hilbert spaces
We consider the Dirichlet problem in
\mathcal{O}, U=0 on . Here
where is a nondegenerate centered Gaussian measure in a Hilbert space
, is an Ornstein-Uhlenbeck operator, and is an
open set in with good boundary. We address the problem whether the weak
solution belongs to the Sobolev space . It is
well known that the question has positive answer if ; if
we give a sufficient condition in terms of geometric
properties of the boundary . The results are quite
different with respect to the finite dimensional case, for instance if
\mathcal{O} is the ball centered at the origin with radius we prove that
only for small
Existence of the Fomin derivative of the invariant measure of a stochastic reaction--diffusion equation
We consider a reaction--diffusion equation perturbed by noise (not
necessarily white). We prove existence of the Fomin derivative of the
corresponding transition semigroup . The main tool is a new estimate for
in terms of , where is the
invariant measure of
Asymptotic behavior of stochastic PDEs with random coefficients
We study the long time behavior of the solution of a stochastic PDEs with
random coefficients assuming that randomness arises in a different independent
scale. We apply the obtained results to 2D- Navier--Stokes equations
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