Xicheng Zhang

Bio: Xicheng Zhang is an academic researcher from Wuhan University. The author has contributed to research in topics: Stochastic differential equation & Uniqueness. The author has an hindex of 32, co-authored 143 publications receiving 2932 citations. Previous affiliations of Xicheng Zhang include Huazhong University of Science and Technology & University of Lisbon.

Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients

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.

Heat kernels and analyticity of non-symmetric jump diffusion semigroups

Abstract: Let \(d\geqslant 1\) and \(\alpha \in (0, 2)\). Consider the following non-local and non-symmetric Levy-type operator on \({\mathbb R}^d\): $$\begin{aligned} {\fancyscript{L}}^\kappa _{\alpha }f(x):=\hbox {p.v.}\int _{{\mathbb R}^d}(f(x+z)-f(x)) \frac{\kappa (x,z)}{ |z|^{d+\alpha }} {\mathord {\mathrm{d}}}z, \end{aligned}$$ where \(0<\kappa _0\leqslant \kappa (x,z)\leqslant \kappa _1, \kappa (x,z)=\kappa (x,-z)\), and \(|\kappa (x,z)-\kappa (y,z)|\leqslant \kappa _2|x-y|^\beta \) for some \(\beta \in (0,1)\). Using Levi’s method, we construct the fundamental solution (also called heat kernel) \(p^\kappa _\alpha (t, x, y)\) of \({\fancyscript{L}}^\kappa _\alpha \), and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates. We also show that \(p^\kappa _\alpha (t, x, y)\) is jointly Holder continuous in \((t, x)\). The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of \({\fancyscript{L}}^\kappa _{\alpha }\) gives rise a Feller process \(\{X, {\mathbb P}_x, x\in {\mathbb R}^d\}\) on \({\mathbb R}^d\). We determine the Levy system of \(X\) and show that \({\mathbb P}_x\) solves the martingale problem for \(({\fancyscript{L}}^\kappa _{\alpha }, C^2_b({\mathbb R}^d))\). Furthermore, we show that the \(C_0\)-semigroup associated with \({\fancyscript{L}}^\kappa _\alpha \) is analytic in \(L^p ({\mathbb R}^d)\) for every \(p\in [1,\infty )\). A maximum principle for solutions of the parabolic equation \(\partial _t u ={\fancyscript{L}}^\kappa _\alpha u\) is also established. As an application of the main result of this paper, sharp two-sided estimates for the transition density of the solution of \({\mathord {\mathrm{d}}}X_t = A(X_{t-}) {\mathord {\mathrm{d}}}Y_t\) is derived, where \(Y\) is a (rotationally) symmetric stable process on \({\mathbb R}^d\) and \(A(x)\) is a Holder continuous \(d\times d\) matrix-valued function on \({\mathbb R}^d\) that is uniformly elliptic and bounded.

Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations

Abstract: In this paper, using weak convergence method, we prove a large deviation principle of Freidlin-Wentzell type for the stochastic tamed 3D Navier-Stokes equations driven by multiplicative noise, which was investigated in (Rockner and Zhang in Probab. Theory Relat. Fields 145(1–2), 211–267, 2009).

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.