Searcharxiv⌕ Search

arXiv subjects

Xicheng Zhang

Publications and source records attributed to Xicheng Zhang.

At least 73 records · Page 4Linked to original sources
arXiv2017

Ergodicity of stochastic differential equations with jumps and singular coefficients

We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov's {\it apriori} estimates for SDEs.

math.PR↗
arXiv2017

Heat Kernels for Non-symmetric Non-local Operators

We survey the recent progress in the study of heat kernels for a class of non-symmetric non-local operators. We focus on the existence and sharp two-sided estimates of the heat kernels and their connection to jump diffusions.

math.PR↗
arXiv2017

Stochastic Hamiltonian flows with singular coefficients

In this paper we study the following stochastic Hamiltonian system in ${\mathbb R}^{2d}$ (a second order stochastic differential equation), $$ d \dot X_t=b(X_t,\dot X_t)d t+σ(X_t,\dot X_t)d W_t,\ \ (X_0,\dot X_0)=(x,v)\in{\mathbb R}^{2d}, $$ where $b(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d$ and $σ(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d\otimes{\mathbb R}^d$ are two Borel measurable functions. We show that if $σ$ is bounded and uniformly non-degenerate, and $b\in H^{2/3,0}_p$ and $\nablaσ\in L^p$ for some $p>2(2d+1)$, where $H^{α,β}_p$ is the Bessel potential space with differentiability indices $α$ in $x$ and $β$ in $v$, then the above stochastic equation admits a unique strong solution so that $(x,v)\mapsto Z_t(x,v):=(X_t,\dot X_t)(x,v)$ forms a stochastic homeomorphism flow, and $(x,v)\mapsto Z_t(x,v)$ is weakly differentiable with ess.$\sup_{x,v}E\left(\sup_{t\in[0,T]}|\nabla Z_t(x,v)|^q\right)<\infty$ for all $q\geq 1$ and $T\geq 0$. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli \cite{Fi} and Trevisan \cite{Tre}.

math.PR↗
arXiv2016

Heat kernels for non-symmetric diffusion operators with jumps

For $d\geq 2$, we establish the existence and uniqueness of heat kernels for a large class of time-dependent second order diffusion operator with jumps, which is the sum of time-dependent of a second order elliptic differential operators non-divergence form and a non-local $α$-stable-type operator with bounded time-dependent coefficient. Moreover, we obtain sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernels under some mild conditions. Our approach is mainly analytic but also uses some probabilistic techniques.

math.AP↗
arXiv2016

$L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators

For $p\in(1,\infty)$, let $u(t,x,v)$ and $f(t,x,v)$ be in $L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)$ and satisfy the following nonlocal kinetic Fokker-Plank equation on $\mathbb{R}^{1+2d}$ in the weak sense: $$ \partial_t u+v\cdot\nabla_x u=Δ^{α/{2}}_v u+f, $$ where $α\in(0,2)$ and $Δ^{α/{2}}_v$ is the usual fractional Laplacian applied to $v$-variable. We show that there is a constant $C=C(p,α,d)>0$ such that for any $f(t, x, v)\in L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)=L^p(\mathbb{R}^{1+2d})$, $$ \|Δ_x^{α/{(2(1+α))}}u\|_p+\|Δ_v^{α/{2}}u\|_p\leq C\|f\|_p, $$ where $\|\cdot\|_p$ is the usual $L^p$-norm in $L^p(\mathbb{R}^{1+2d}; d z)$. In fact, in this paper the above inequality is established for a large class of time-dependent non-local kinetic Fokker-Plank equations on $\mathbb{R}^{1+2d}$, with $U_t v$ and $\mathscr{L}^{ν_t}_{σ_t}$ in place of $v\cdot \nabla_x$ and $Δ^{α/2}_v$. See Theorem 3.3 for details.

math.AP↗
arXiv2016

Stochastic integrals and BDG's inequalities in Orlicz-type spaces

In this paper we extend an inequality of Lenglart, Lépingle and Pratelli \cite[Lemma 1.1]{LLP} to general continuous adapted stochastic processes with values in topology spaces. By this inequality we show Burkholder-Davies-Gundy's inequality for stochastic integrals in Orlicz-type spaces (a class of quasi-Banach spaces) with respect to cylindrical Brownian motions.

math.PR↗
arXiv2016

Uniqueness of stable-like processes

In this work we consider the following $α$-stable-like operator (a class of pseudo-differential operator) $$ {\mathscr L} f(x):=\int_{\mathbb R^d}[f(x+σ_x y)-f(x)-1_{α\in[1,2)}1_{|y|\leq 1}σ_x y\cdot\nabla f(x)]ν_x(d y), $$ where the Lévy measure $ν_x(d y)$ is comparable with a non-degenerate $α$-stable-type Lévy measure (possibly singular), and $σ_x$ is a bounded and nondegenerate matrix-valued function. Under Hölder assumption on $x\mapstoν_x(d y)$ and uniformly continuity assumption on $x\mapstoσ_x$, we show the well-posedness of martingale problem associated with the operator $\mathscr L$. Moreover, we also obtain the existence-uniqueness of strong solutions for the associated SDE when $σ$ belongs to the first order Sobolev space $\mathbb W^{1,p}(\mathbb R^d)$ provided $p>d(1+α\vee 1)$ and $ν_x=ν$ is a non-degenerate $α$-stable-type Lévy measure.

math.PR↗
arXiv2015

Stochastic differential equations with Sobolev diffusion and singular drift

In this paper we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to the coefficients, weak differentiability with respect to starting points, and the Malliavin differentiability with respect to sample paths. We also establish Bismut-Elworthy-Li's formula for the solutions. As an application, we use the stochastic Lagrangian representation of incompressible Navier-Stokes equations given by Constantin-Iyer \cite{Co-Iy} to prove the local well-posedness of NSEs in $\mR^d$ with initial values in the first order Sobolev space $\mW^1_p(\mR^d;\mR^d)$ provided $p>d$.

math.PR↗
arXiv2015

Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients

By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility of the associated diffusion semigroup.

math.PR↗
arXiv2015

Gradient estimates for SDEs Driven by Multiplicative Lévy Noise

Gradient estimates are derived, for the first time, for the semigroup associated to a class of stochastic differential equations driven by multiplicative Lévy noise. In particular, the estimates are sharp for $α$-stable type noises. To derive these estimates, a new derivative formula of Bismut-Elworthy-Li's type is established for the semigroup by using the Malliavin calculus and a finite-jump approximation argument.

math.PR↗
arXiv2015

Degenerate SDE with Hölder-Dini Drift and Non-Lipschitz Noise Coefficient

The existence-uniqueness and stability of strong solutions are proved for a class of degenerate stochastic differential equations, where the noise coeffcicient might be non-Lipschitz, and the drift is locally Dini continuous in the component with noise (i.e. the second component) and locally Hölder-Dini continuous of order $\ff 2 3$ in the first component. Moreover, the weak uniqueness is proved under weaker conditions on the noise coefficient. Furthermore, if the noise coefficient is $C^{1+\vv}$ for some $\vv>0$ and the drift is Hölder continuous of order $å\in (\ff 2 3,1)$ in the first component and order $\bb\in(0,1) $ in the second, the solution forms a $C^1$-stochastic diffeormorphism flow. To prove these results, we present some new characterizations of Hölder-Dini space by using the heat semigroup and slowly varying functions.

math.PR↗
arXiv2015

Stochastic flows for Lévy processes with Hölder drifts

In this paper we study the following stochastic differential equation (SDE) in ${\mathbb R}^d$: $$ \mathrm{d} X_t= \mathrm{d} Z_t + b(t, X_t)\mathrm{d} t, \quad X_0=x, $$ where $Z$ is a Lévy process. We show that for a large class of Lévy processes ${Z}$ and Hölder continuous drift $b$, the SDE above has a unique strong solution for every starting point $x\in{\mathbb R}^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. As a consequence, we show that, when ${Z}$ is an $α$-stable-type Lévy process with $α\in (0, 2)$ and $b$ is bounded and $β$-Hölder continuous with $β\in (1- α/{2},1)$, the SDE above has a unique strong solution. When $α\in (0, 1)$, this in particular solves an open problem from Priola \cite{Pr1}. Moreover, we obtain a Bismut type derivative formula for $\nabla {\mathbb E}_x f(X_t)$ when ${Z}$ is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with Hölder continuous $b$ and $f$: $$ \partial_t u+{\mathscr L} u+b\cdot \nabla u+f=0,\quad u(1, \cdot )=0, $$ where $\mathscr L$ is the generator of the Lévy process ${Z}$.

math.PR↗
arXiv2015

Degenerate SDEs in Hilbert Spaces with Rough Drifts

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 Hölder continuous of order larger than $\ff 2 3$ 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 space-time white noises.

math.PR↗
arXiv2014

Densities for SDEs driven by degenerate $α$-stable processes

In this work, by using the Malliavin calculus, under Hörmander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker-Planck (nonlocal) operator: \[\mathscr{L}^{(α)}_b:=Δ^{α/2}_{\mathrm{v}}+\mathrm {v}\cdot \nabla_x+b(x,\mathrm{v})\cdot \nabla_{\mathrm{v}},\qquad x,\mathrm{v}\in\mathbb{R}^d,\] where $α\in(0,2)$ and $b:\mathbb{R}^d\times\mathbb{R}^d\to \mathbb{R}^d$ is smooth and has bounded derivatives of all orders.

math.PR↗
arXiv2014

Nonlocal Hormander's hypoellipticity theorem

Consider the following nonlocal integro-differential operator: for $α\in(0,2)$, $$ \cal L^{(α)}_{σ,b} f(x):=\mbox{p.v.} \int_{\mathbb{R}^d-\{0\}}\frac{f(x+σ(x)z)-f(x)}{|z|^{d+α}}d z+b(x)\cdot\nabla f(x), $$ where $σ:\mathbb{R}^d\to\mathbb{R}^d\times\mathbb{R}^d$ and $b:\mathbb{R}^d\to\mathbb{R}^d$ are two $C^\infty_b$-functions, and p.v. stands for the Cauchy principal value. Let $B_1(x):=σ(x)$ 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 $n=n(x)\in\mathbb{N}$, $$ \mathrm{Rank}[B_1(x), B_2(x),\cdots, B_n(x)]=d, $$ by using the Malliavin calculus, we prove the existence of the heat kernel $ρ_t(x,y)$ to the operator $\cal L^{(α)}_{σ,b}$ as well as the continuity of $x\mapsto ρ_t(x,\cdot)$ in $L^1(\mathbb{R}^d)$ for each $t>0$. Moreover, when $σ(x)=σ$ is constant, under the following uniform Hörmander's type condition: for some $j_0\in\mathbb{N}$, $$ \inf_{x\in\mathbb{R}^d}\inf_{|u|=1}\sum_{j=1}^{j_0}|u B_j(x)|^2>0, $$ we also prove the smoothness of $(t,x,y)\mapstoρ_t(x,y)$ with $ρ_t(\cdot,\cdot)\in C^\infty_b(\mathbb{R}^d\times\mathbb{R}^d)$ for each $t>0$.

math.PR↗
arXiv2014

Fundamental solutions of nonlocal Hörmander's operators

Consider the following nonlocal integro-differential operator: for $α\in(0,2)$, $$ \cL^{(α)}_{σ,b} f(x):=\mbox{p.v.} \int_{|z|<δ}\frac{f(x+σ(x)z)-f(x)}{|z|^{d+α}}\dif z+b(x)\cdot\nabla f(x)+\sL f(x), $$ where $σ:\mR^d\to\mR^d\times\mR^d$ and $b:\mR^d\to\mR^d$ are two $C^\infty_b$-functions, $δ$ is a small positive number, p.v. stands for the Cauchy principal value, and $\sL$ is a bounded linear operator in Sobolev spaces. Let $B_1(x):=σ(x)$ and $B_{j+1}(x):=b(x)\cdot\nabla B_j(x)-\nabla b(x)\cdot B_j(x)$ for $j\in\mN$. Under the following uniform Hörmander's type condition: for some $j_0\in\mN$, $$ \inf_{x\in\mR^d}\inf_{|u|=1}\sum_{j=1}^{j_0}|u B_j(x)|^2>0, $$ by using Bismut's approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator $\cL^{(α)}_{σ,b}$. In particular, we answer a question proposed by Nualart \cite{Nu1} and Varadhan \cite{Va}.

math.PR↗
arXiv2014

Strong Feller properties for degenerate SDEs with jumps

Under full Hörmander'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.

math.PR↗