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

The existence and uniqueness of mild solutions are proved for a class of degenerate stochastic differential equations on Hilbert spaces where the drift is Dini continuous in the component with noise and Holder continuous of order larger than 2 3 in the other component. In the finite-dimensional case the Dini continuity is further weakened. The main results are applied to solve second order stochastic systems driven by spacetime white noises.

Degenerate SDEs in Hilbert spaces with rough drifts

We investigate stochastic differential equations with jumps and irregular coefficients, and obtain the existence and uniqueness ofgeneralized stochastic flows. Moreover, we also prove the existence and uniqueness of $L^p$-solutions or measure-valued solutionsfor second order integro-differential equation of Fokker-Planck type.

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Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type

In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\bf R}^d$: $$ d X_{t}= \sigma_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^d, $$ where $Z$ is a $d$-dimensioanl nondegenerate $\alpha$-stable-like process with $\alpha \in(0,2)$ (including cylindrical case), and uniform in $t\geq 0$, $x\mapsto \sigma_t(x): {\bf R}^d\to {\bf R}^d\otimes {\bf R}^d$ is Lipchitz and uniformly elliptic and $x\mapsto b_t (x)$ is $\beta$-order Holder continuous with $\beta\in(1-\alpha/2,1)$. Under these assumptions, we show the above SDE has a unique strong solution for every starting point $x \in {\bf R}^d$. When $\sigma_t (x)={\bf I}_{d\times d}$, the $d\times d$ identity matrix, our result in particular gives an affirmative answer to the open problem of Priola (2015).

Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts

In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray’s solution of 3D Navier–Stokes equations. More precisely, for any Leray’s solution $$\mathbf{u }$$
 of 3D-NSE and each $$(s,x)\in \mathbb {R}_+\times \mathbb {R}^3$$
 , we show the existence of weak solutions to the following SDE, which have densities $$\rho _{s,x}(t,y)$$
 belonging to $$\mathbb {H}^{1,p}_q$$
 with $$p,q\in [1,2)$$
 and $$\frac{3}{p}+\frac{2}{q}>4$$
 : $$\begin{aligned} \text {d} X_{s,t}=\mathbf{u } (s,X_{s,t})\text {d} t+\sqrt{2\nu }\text {d} W_t,\ \ X_{s,s}=x,\ \ t\geqslant s, \end{aligned}$$
 where W is a three dimensional standard Brownian motion, $$\nu >0$$
 is the viscosity constant. Moreover, we also show that for Lebesgue almost all (s, x), the solution $$X^n_{s,\cdot }(x)$$
 of the above SDE associated with the mollifying velocity field $$\mathbf{u }_n$$
 weakly converges to $$X_{s,\cdot }(x)$$
 so that X is a Markov process in almost sure sense.

Stochastic Lagrangian Path for Leray’s Solutions of 3D Navier–Stokes Equations

https://hal.archives-ouvertes.fr/hal-02864994/document

Xicheng Zhang

Papers

Degenerate SDEs in Hilbert spaces with rough drifts

Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type

Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts

Stochastic Lagrangian Path for Leray’s Solutions of 3D Navier–Stokes Equations

Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift