Showing papers by "Xicheng Zhang published in 2011"
TL;DR: In this article, the stochastic homeomorphism flow property and the strong Feller property for deterministic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients were proved.
Abstract: In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Rockner's results in [10] to the case of non-constant diffusion coefficients.
134 citations
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TL;DR: In this article, the authors prove the well-posedness of stochastic 2D Navier-Stokes equation driven by general L\'evy processes (in particular, $\alpha$-stable processes), and obtain the existence of invariant measures.
Abstract: In this note we prove the well-posedness for stochastic 2D Navier-Stokes equation driven by general L\'evy processes (in particular, $\alpha$-stable processes), and obtain the existence of invariant measures.
32 citations
TL;DR: In this paper, a stochastic Lagrangian particle path approach for the Navier-Stokes equations is presented, based on which a self-contained proof for the existence of local unique solution for the fractal Navier Stokes equation with initial data in Ω(W 1,p) is provided, and in the case of two dimensions or large viscosity, the existence is also obtained.
Abstract: In this article we study the fractal Navier-Stokes equations by using stochastic Lagrangian particle path approach in Constantin and Iyer \cite{Co-Iy}. More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Levy processes. Basing on this representation, a self-contained proof for the existence of local unique solution for the fractal Navier-Stokes equation with initial data in $\mW^{1,p}$ is provided, and in the case of two dimensions or large viscosity, the existence of global solution is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Levy processes with time dependent and discontinuous drifts is proved.
32 citations
TL;DR: In this paper, it was shown that stochastic differential equations with Sobolev drift on a Riemannian manifold admit a unique invertible flow which is quasi-invariant with respect to ν.
13 citations
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TL;DR: In this paper, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts using the Malliavin calculus.
Abstract: By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack inequalities are derived.
8 citations
TL;DR: In this article, the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE with unbounded and non-smooth coefficients were proved, and the L 1 -integrability and a general maximal principle for generalized solutions of generalized SPDEs were established.
7 citations
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TL;DR: In this article, the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index was proved based on the apriori H\"older estimate for advection fractional-diffusion equation established by Silvestre \cite{Si2}.
Abstract: In this paper we prove the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index. The proof relies on the apriori H\"older estimate for advection fractional-diffusion equation established by Silvestre \cite{Si2}.
2 citations
01 Jan 2011
TL;DR: For a nonlinear stochastic partial differential equation driven by linear multiplicative space-time white noises, the authors proved that there exists a bicontinuous version of the solution with respect to the initial value and the time variable.
Abstract: For a nonlinear stochastic partial differential equation driven by linear multiplicative space-time white noises, we prove that there exists a bicontinuous version of the solution with respect to the initial value and thetime variable
1 citations
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TL;DR: In this article, a large deviation principle of Freidlin-Wentzell's type for multivalued stochastic differential equations with monotone drifts was proved for a class of SDEs with reflection in a convex domain.
Abstract: We prove a large deviation principle of Freidlin-Wentzell’s type for the multivalued stochastic differential equations with monotone drifts, which in particular contains a class of SDEs with reflection in a convex domain.
1 citations
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TL;DR: In this article, the authors prove the exponential ergodicity of the transition probabilities of solutions to elliptic multivalued stochastic differential equations and prove that the transition probability of solutions is exponential.
Abstract: We prove the exponential ergodicity of the transition probabilities of solutions to elliptic multivalued stochastic differential equations.