Stochastic equations in infinite dimensions

We consider the system of stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $\alpha$-stable process $Z_t$ in $\mathbb{R}^d$. We assume that $A(x) = (a_{ij}(x))$ is diagonal and $a_{ii}(x)$ are bounded away from zero, from infinity and H\"older continuous. We construct transition density $p^A(t,x,y)$ of the process $X_t$ and show sharp two-sided estimates of this density. We also prove H\"older and gradient estimates of $x \to p^A(t,x,y)$. Our approach is based on the method developed by Chen and Zhang.

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Transition density estimates for diagonal systems of SDEs driven by cylindrical $\alpha$-stable processes

Ergodicity of stochastic Magneto-Hydrodynamic equations driven by α-stable noise

By studying the infinite dimensional Kolmogorov equation with non-local operator, we show the pathwise uniqueness for stochastic partial differential equation driven by cylindrical α-stable process with Holder continuous drift.

Pathwise Uniqueness for a Class of SPDEs Driven by Cylindrical α-Stable Processes

It is shown that the transition semigroup (P t ) t≥0 corresponding to stochastic dissipative equations driven by α-stable is strong Feller and irreducible for α ∈ (1, 2). This result ensures the ergodicity for the equation.

Xiaobin Sun

Papers

Ergodicity of Stochastic Dissipative Equations Driven by α-Stable Process