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Xicheng Zhang

Publications and source records attributed to Xicheng Zhang.

At least 91 records · Page 5Linked to original sources
arXiv2014

Regularity of density for SDEs driven by degenerate Lévy noises

By using Bismut's approach about the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive Lévy noises. Under full Hörmander's conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density. Under the uniform first order Lie's bracket condition, we also prove the smoothness of the density.

math.PR↗
arXiv2013

Harnack inequalities for SDEs driven by cylindrical $α$-stable processes

By using the coupling argument, we establish the Harnack and log-Harnack inequalites for stochastic differential equations with non-Lipschitz drifts and driven by additive anisotropic subordinated Brownian motions (in particular, cylindrical $α$-stable processes). Moreover, the gradient estimate is also derived when the drift is Lipschitz continuous.

math.PR↗
arXiv2013

Heat kernels and analyticity of non-symmetric jump diffusion semigroups

Let $d\geq 1$ and $α\in (0, 2)$. Consider the following non-local and non-symmetric Lévy-type operator on $\mR^d$: $$ \sL^κ_αf(x):=\mbox{p.v.}\int_{\mR^d}(f(x+z)-f(x))\frac{κ(x,z)}{|z|^{d+α}} \dif z, $$ where $0<κ_0\leq κ(x,z)\leq κ_1$, $κ(x,z)=κ(x,-z)$, and $|κ(x,z)-κ(y,z)|\leqκ_2|x-y|^β$ for some $β\in(0,1)$. Using Levi's method, we construct the fundamental solution (also called heat kernel) $p^κ_α(t, x, y)$ of $\sL^κ_α$, and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates of the heat kernel. We also show that $p^κ_α(t, x, y)$ is jointly Hölder continuous in $(t, x)$. The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of $\sL^κ_α$ gives rise a Feller process $\{X, \mP_x, x\in \mR^d\}$ on $\mR^d$. We determine the Lévy system of $X$ and show that $\mP_x$ solves the martingale problem for $(\sL^κ_α, C^2_b(\mR^d))$. Furthermore, we obtain the analyticity of the non-symmetric semigroup associated with $\sL^κ_α$ in $L^p$-spaces for every $p\in[1,\infty)$. A maximum principle for solutions of the parabolic equation $\partial_t u =\sL^κ_αu$ is also established.

math.AP↗
arXiv2013

Heat kernel estimates for critical fractional diffusion operator

In this work we construct the heat kernel of the 1/2-order Laplacian perturbed by the first-order gradient term in Hölder space and the zero-order potential term in generalized Kato's class, and obtain sharp two-sided estimates as well as the gradient estimate of the heat kernel.

math.AP↗
arXiv2012

Stochastic functional differential equations driven by Lévy processes and quasi-linear partial integro-differential equations

In this article we study a class of stochastic functional differential equations driven by Lévy processes (in particular, $α$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals. This corresponds to the local solvability to a class of quasi-linear partial integro-differential equations. Moreover, in the constant diffusion coefficient case, without any assumptions on the Lévy generator, we also show the existence of a unique maximal weak solution for a class of semi-linear partial integro-differential equation systems under bounded Lipschitz assumptions on the coefficients. Meanwhile, in the nondegenerate case (corresponding to $Δ^{α/2}$ with $α\in(1,2]$), based upon some gradient estimates, the existence of global solutions is established too. In particular, this provides a probabilistic treatment for the nonlinear partial integro-differential equations, such as the multi-dimensional fractal Burgers equations and the fractal scalar conservation law equations.

math.PR↗
arXiv2012

Exponential Ergodicity of stochastic Burgers equations driven by $α$-stable processes

In this work, we prove the strong Feller property and the exponential ergodicity of stochastic Burgers equations driven by $α/2$-subordinated cylindrical Brownian motions with $α\in(1,2)$. To prove the results, we truncate the nonlinearity and use the derivative formula for SDEs driven by $α$-stable noises established in Zhang (arXiv:1204.2630v2).

math.PR↗
arXiv2012

Heat Kernel for Fractional Diffusion Operators with Perturbations

Let $L$ be an elliptic differential operator on a complete connected Riemannian manifold $M$ such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let $L^{(å)}$ be the $å$-stable subordination of $L$ for $å\in (1,2).$ We found some classes $\mathbb K_å^{\gg,\bb} (\bb,\gg\in [0,å))$ of time-space functions containing the Kato class, such that for any measurable $b: [0,\infty)\times M\to TM$ and $c: [0,\infty)\times M\to M$ with $|b|, c\in \mathbb K_å^{1,1},$ the operator $$L_{b,c}^{(å)}(t,x):= L^{(å)}(x)+ +c(t,x),\ \ (t,x)\in [0,\infty)\times M$$ has a unique heat kernel $p_{b,c}^{(å)}(t,x;s,y), 0\le s 1$, where $\rr$ is the Riemannian distance. The estimate of $\nabla_yp^{(å)}_{b,c}$ and the Hölder continuity of $\nn_x p_{b,c}^{(å)}$ are also considered. The resulting estimates of the gradient and its Hölder continuity are new even in the standard case where $L=\DD$ on $\R^d$ and $b,c$ are time-independent.

math-ph↗
arXiv2012

Derivative formula and gradient estimate for SDEs driven by $α$-stable processes

In this paper we prove a derivative formula of Bismut-Elworthy-Li's type as well as gradient estimate for stochastic differential equations driven by $α$-stable noises, where $α\in(0,2)$. As an application, the strong Feller property for stochastic partial differential equations driven by subordinated cylindrical Brownian motions is presented.

math.PR↗
arXiv2012

$L^p$-maximal regularity of nonlocal parabolic equation and applications

By using Fourier's transform and Fefferman-Stein's theorem, we investigate the $L^p$-maximal regularity of nonlocal parabolic and elliptic equations with singular and non-symmetric Lévy operators, and obtain the unique strong solvability of the corresponding nonlocal parabolic and elliptic equations, where the probabilistic representation plays an important role. In particular, a characterization for the domain of pseudo-differential operators of Lévy type with singular kernels is given in terms of the Bessel potential spaces. As a byproduct, we show that a large class of non-symmetric Lévy operators generates an analytic semigroup in $L^p$-space. Moreover, as applications, we prove a Krylov's estimate for stochastic differential equation driven by Cauchy processes (i.e. critical diffusion processes), and also obtain the well-posedness to a class of quasi-linear first order parabolic equation with critical diffusion. In particular, critical Hamilton-Jacobi equation and multidimensional critical Burger's equation are uniquely solvable and the smooth solutions are obtained.

math.AP↗
arXiv2011

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

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 Röckner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.

math.PR↗
arXiv2011

Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

In this article we study the fractal Navier-Stokes equations by using stochastic Lagrangian particle path approach in Constantin and Iyer \cite{Co-Iy}. More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Lévy processes. Basing on this representation, a self-contained proof for the existence of local unique solution for the fractal Navier-Stokes equation with initial data in $\mW^{1,p}$ is provided, and in the case of two dimensions or large viscosity, the existence of global solution is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Lévy processes with time dependent and discontinuous drifts is proved.

math.PR↗