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.

Stochastic Equations in Infinite Dimensions

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Stochastic equations in infinite dimensions

Stochastic differential equations and diffusion processes

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.

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

Strong solutions of SDES with singular drift and Sobolev diffusion coefficients

Euler schemes and large deviations for stochastic Volterra equations with singular kernels

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.

Heat kernels and analyticity of non-symmetric jump diffusion semigroups

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).

Xicheng Zhang

Papers

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

Strong solutions of SDES with singular drift and Sobolev diffusion coefficients

Euler schemes and large deviations for stochastic Volterra equations with singular kernels

Heat kernels and analyticity of non-symmetric jump diffusion semigroups

Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations