## SEARCH

#### Institution

##### ( see all 14)

- Huazhong University of Science and Technology 7 (%)
- Wuhan University 6 (%)
- The University of New South Wales 4 (%)
- Purdue University 3 (%)
- Universität Bielefeld 3 (%)

#### Author

##### ( see all 9)

- Zhang, Xicheng [x] 14 (%)
- Röckner, Michael 3 (%)
- Chen, Zhen-Qing 2 (%)
- Cruzeiro, Ana Bela 2 (%)
- Ren, Jiagang 2 (%)

## CURRENTLY DISPLAYING:

Most articles

Fewest articles

Showing 1 to 10 of 14 matching Articles
Results per page:

## Large Deviations for Multivalued Stochastic Differential Equations

### Journal of Theoretical Probability (2010-12-01) 23: 1142-1156 , December 01, 2010

We prove a large deviation principle of Freidlin–Wentzell type for multivalued stochastic differential equations with monotone drifts that in particular contain a class of SDEs with reflection in a convex domain.

## Hölder Estimates for Nonlocal-Diffusion Equations with Drifts

### Communications in Mathematics and Statistics (2014-12-01) 2: 331-348 , December 01, 2014

We study a class of nonlocal-diffusion equations with drifts, and derive a priori $$\Phi $$ -Hölder estimate for the solutions by using a purely probabilistic argument, where $$\Phi $$ is an intrinsic scaling function for the equation.

## Fundamental Solutions of Nonlocal Hörmander’s Operators

### Communications in Mathematics and Statistics (2016-09-01) 4: 359-402 , September 01, 2016

Consider the following nonlocal integro-differential operator: for $$\alpha \in (0,2)$$ , $$\begin{aligned} {\mathcal {L}}^{(\alpha )}_{\sigma ,b} f(x):=\text{ p.v. } \int _{{\mathbb {R}}^d-\{0\}}\frac{f(x+\sigma (x)z)-f(x)}{|z|^{d+\alpha }}{\mathord {\mathrm{d}}}z+b(x)\cdot \nabla f(x), \end{aligned}$$ where $$\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d$$ and $$b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$$ are smooth and have bounded first-order derivatives, and p.v. stands for the Cauchy principal value. Let $$B_1(x):=\sigma (x)$$ and $$B_{j+1}(x):=b(x)\cdot \nabla B_j(x)-\nabla b(x)\cdot B_j(x)$$ for $$j\in {\mathbb {N}}$$ . Under the following Hörmander’s type condition: for any $$x\in {\mathbb {R}}^d$$ and some $$n=n(x)\in {\mathbb {N}}$$ , $$\begin{aligned} {\mathrm {Rank}}[B_1(x), B_2(x),\ldots , B_n(x)]=d, \end{aligned}$$ by using the Malliavin calculus, we prove the existence of the heat kernel $$\rho _t(x,y)$$ to the operator $${\mathcal {L}}^{(\alpha )}_{\sigma ,b}$$ as well as the continuity of $$x\mapsto \rho _t(x,\cdot )$$ in $$L^1({\mathbb {R}}^d)$$ as a density function for each $$t>0$$ . Moreover, when $$\sigma (x)=\sigma $$ is constant and $$B_j\in C^\infty _b$$ for each $$j\in {\mathbb {N}}$$ , under the following uniform Hörmander’s type condition: for some $$j_0\in {\mathbb {N}}$$ , $$\begin{aligned} \inf _{x\in {\mathbb {R}}^d}\inf _{|u|=1}\sum _{j=1}^{j_0}|u B_j(x)|^2>0, \end{aligned}$$ we also show the smoothness of $$(t,x,y)\mapsto \rho _t(x,y)$$ with $$\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)$$ for each $$t>0$$ .

## Stochastic Flows for Nonlinear SPDEs Driven by Linear Multiplicative Space-time White Noises

### Stochastic Analysis with Financial Applications (2011-01-01) 65: 83-97 , January 01, 2011

For a nonlinear stochastic partial differential equation driven by linear multiplicative space-time white noises, we prove that there exists a bicontinuous version of the solution with respect to the initial value and thetime variable.

## A stochastic representation for backward incompressible Navier-Stokes equations

### Probability Theory and Related Fields (2010-09-01) 148: 305-332 , September 01, 2010

By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer’s forward formulations in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.

## Heat kernels and analyticity of non-symmetric jump diffusion semigroups

### Probability Theory and Related Fields (2016-06-01) 165: 267-312 , June 01, 2016

Let $$d\geqslant 1$$ and $$\alpha \in (0, 2)$$ . Consider the following non-local and non-symmetric Lévy-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 Hölder 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 Lévy 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 Hölder 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

### Applied Mathematics and Optimization (2010-04-01) 61: 267-285 , April 01, 2010

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 (Röckner and Zhang in Probab. Theory Relat. Fields 145(1–2), 211–267, 2009).

## Well-posedness of fully nonlinear and nonlocal critical parabolic equations

### Journal of Evolution Equations (2013-03-01) 13: 135-162 , March 01, 2013

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.

## Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

### Probability Theory and Related Fields (2008-07-19) 145: 211-267 , July 19, 2008

In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly (Ann. Math. 164:993–1032, 2006), we prove the uniqueness of invariant measures for the corresponding transition semigroup.

## Bismut Type Formulae for Diffusion Semigroups on Riemannian Manifolds

### Potential Analysis (2006-09-01) 25: 121-130 , September 01, 2006

Applying the stochastic calculus of variations on frame bundles along tangent processes, we derive Bismut type formulae for the derivatives of diffusion semigroups on Riemannian manifold in both variables. We also obtain the Bismut formulae expressed in terms of the Ricci and torsion tensors for the connection with torsion.