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

Probabilistic approach for semi-linear stochastic fractal equations

We establish H\"older regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: $$ {\rm d} X_t=\sigma (t, X_{t-}){\rm d} Z_t+b (t, X_t){\rm d} t,\ \ X_0=x\in{\mathbb R}^d, $$ where $( Z_t)_{t\geq 0}$ is a $d$-dimensional cylindrical $\alpha$-stable process with $\alpha \in (0, 2)$, $\sigma (t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ is bounded measurable, uniformly nondegenerate and Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is bounded $\beta$-H\"older continuous in $x$ uniformly in $t$ with $\beta\in[0,1]$ satisfying $\alpha+\beta>1$. Moreover, we also show the existence and regularity of the distributional density of $X (t, x)$. Our proof is based on Littlewood-Paley's theory.

H\"older regularity and gradient estimates H\"older regularity and gradient estimates for SDEs driven by cylindrical $\alpha$-stable processes

In this paper we prove the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index. The proof relies on the apriori H\"older estimate for advection fractional-diffusion equation established by Silvestre \cite{Si2}.

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

Under full Hormander's conditions, we prove the strong Feller property of the semigroup determined by an SDE driven by additive subordinate Brownian motion, where the drift is allowed to be arbitrarily growth For this, we extend a criterion due to Malicet-Poly \cite{Ma-Po} and Bally-Caramellino \cite{Ba-Ca} about the convergence of the laws of Wiener functionals in total variations Moreover, the example of a chain of coupled oscillators is verified

Strong Feller properties for degenerate SDEs with jumps

In this paper we establish the local and global well-posedness of weak and strong solutions to second order fractional mean-field SDEs with singular/distribution interaction kernels and measure initial value, where the kernel can be Newton or Coulomb potential, Riesz potential, Biot-Savart law, etc. Moreover, we also show the stability, smoothness and the short time singularity and large time decay estimates of the distribution density. Our results reveal a phenomenon that for {\it nonlinear} mean-field equations, the regularity of the initial distribution could balance the singularity of the kernel. The precise relationship between the singularity of kernels and the regularity of initial values are calculated, which belongs to the subcritical regime in the scaling sense. In particular, our results provide a microscopic probabilistic explanation and establish a unified treatment for many physical models such as the fractional Vlasov-Poisson-Fokker-Planck system, the vorticity formulation of 2D-fractal Navier-Stokes equations, surface quasi-geostrophic models, fractional porous medium equation with viscosity, etc.

Xicheng Zhang

Papers

Probabilistic approach for semi-linear stochastic fractal equations

H\"older regularity and gradient estimates H\"older regularity and gradient estimates for SDEs driven by cylindrical $\alpha$-stable processes

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

Strong Feller properties for degenerate SDEs with jumps

Second order fractional mean-field SDEs with singular kernels and measure initial data