Smooth densities of stochastic differential equations forced by degenerate stable type noises

Using the Bismut's approach to Malliavin calculus, we introduce a simplified Malliavin matrix ([11]) for stochastic differential equations (SDEs) force by degenerate stable like noises. For the degenerate SDEs driven by Wiener noises, one can derive a Norris type lemma and use it \emph{iteratively} to prove the smoothness of density functions. Unfortunately, Norris type lemma is very hard to be iteratively applied to SDEs with stable like noises. In this paper, we derive a simple inequality as a replacement and use it to show that two families of degenerate SDEs with stable like noises admit smooth density functions. One family is the linear SDEs studied by Priola and Zabczyk ([13]), under some additional assumption we can iteratively use the inequality to get the smoothness of the density. The other family is the general SDEs with stable like noises, we can apply this inequality only \emph{one} time and thus derive that the SDEs admit smooth density if the \emph{first} order Lie brackets span $\R^d$. The crucial step in this paper is estimating the smallest eigenvalue of the simplified Malliavin matrix, which only uses some elementary facts of Poisson processes and undergraduate level ordinary differential equations.Comment: Added some references by Bismut which include integration by parts of SDEs forced by stable like noises, Rewrited the abstract, Corrected some very small error

Derivative Formula and Applications for Hyperdissipative Stochastic Navier-Stokes/Burgers Equations

By using coupling method, a Bismut type derivative formula is established for the Markov semigroup associated to a class of hyperdissipative stochastic Navier-Stokes/Burgers equations. As applications, gradient estimates, dimension-free Harnack inequality, strong Feller property, heat kernel estimates and some properties of the invariant probability measure are derived

Ergodicity of infinite white α\alpha-stable Systems with linear and bounded interactions

We proved the existence of an infinite dimensional stochastic system driven by white $\alpha$-stable noises ($1<\alpha \leq 2$), and prove this system is strongly mixing. Our method is by perturbing Ornstein-Uhlenbeck $\alpha$-stable processes.Comment: 20 page

Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises

We prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes are forced) via Kolmogorov equation approach.Comment: 31 pp. Corrected several errors in the original versio