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 first extend the classical Ito stochastic integral to the case of measurable fields of Hilbert spaces. Then, a Kusuoka–Stroock formula on configuration space is proved. Using this formula, we study the fractional regularities of local times with jumps in the sense of the Malliavin calculus.

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Kusuoka-Stroock Formula on Configuration Space and Regularities of Local Times with Jumps

In this paper we show the existence and uniqueness for a class of density dependent SDEs with bounded measurable drift, where the existence part is based on Euler's approximation for density dependent SDEs and the uniqueness is based on the associated nonlinear Fokker-Planck equation. As an application, we obtain the well-posedness of a nonlinear Fokker-Planck equation.

Euler scheme for density dependent stochastic differential equations

We prove a Stroock-Varadhan type quasi-sure limit theorem for stochastic differential equations in the plane.

Quasi-sure analysis of two-parameter stochastic differential equations

In this paper, we prove the existence of smooth solutions in Sobolev spaces to fully nonlinear and nonlocal parabolic equations with critical index. Our argument is to transform the fully nonlinear equation into a quasi-linear nonlocal parabolic equation.

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

Under the bounded geometry assumption on Riemannian manifold M, a variational approximation for Fokker–Planck equation on M is constructed by the scheme of Jordan et al. in SIAM J Math Anal 29(1):1–17, 1998. Moreover, the uniqueness and global L
 p
 -estimate of the solution for 1 < p < dim(M)/(dim(M) − 1) are obtained for a broad class of potential.

Xicheng Zhang

Papers

Kusuoka-Stroock Formula on Configuration Space and Regularities of Local Times with Jumps

Euler scheme for density dependent stochastic differential equations

Quasi-sure analysis of two-parameter stochastic differential equations

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

Variational Approximation for Fokker–Planck Equation on Riemannian Manifold