arXiv subjects
Publications and source records attributed to Xicheng Zhang.
In this paper we establish the existence and uniqueness of heat kernels to a large class of time-inhomogenous non-symmetric nonlocal operators with Dini's continuous kernels. Moreover, quantitative estimates including two-sided estimates, gradient estimate and fractional derivative estimate of the heat kernels are obtained.
In this paper we show the existence and uniqueness for a class of density dependent SDEs with bounded measurable drift, where the existence part is based on Euler's approximation for density dependent SDEs and the uniqueness is based on the associated nonlinear Fokker-Planck equation. As an application, we obtain the well-posedness of a nonlinear Fokker-Planck equation.
This paper is devoted to studying the Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which is not well-defined in the classical sense and shall be understood by using paracontrolled distribution method introduced in \cite{GIP15}. By a new characterization of weighted Hölder space and Zvonkin's transformation we prove some new a priori estimates, and therefore, establish the global well-posedness for singular HJB equations. As an application, the global well-posedness for KPZ equations on the real line in polynomial weighted Hölder spaces is obtained without using Cole-Hopf's transformation. In particular, we solve the conjecture posed in \cite[Remark 1.1]{PR18}.
We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: $$ {\rm d} X_t=σ(t, X_{t-}){\rm d} Z_t+b (t, X_t){\rm d} t,\ \ X_0=x\in{\mathbb R}^d, $$ where $( Z_t)_{t\geq 0}$ is a $d$-dimensional cylindrical $α$-stable process with $α\in (0, 2)$, $σ(t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ 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 $β$-Hölder continuous in $x$ uniformly in $t$ with $β\in[0,1]$ satisfying $α+β>1$. 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.
In this paper we prove a discretized version of Krylov's estimate for discretized Itô's processes. As applications, we study the weak and strong convergences for Euler's approximation of mean-field SDEs with measurable discontinuous and linear growth coefficients. Moreover, we also show the propagation of chaos for Euler's approximation of mean-field SDEs.
Consider the following distribution dependent SDE: $$ {\mathrm d} X_t=σ_t(X_t,μ_{X_t}){\mathrm d} W_t+b_t(X_t,μ_{X_t}){\mathrm d} t, $$ where $μ_{X_t}$ stands for the distribution of $X_t$. In this paper for non-degenerate $σ$, we show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in $μ$ about $b$ and $σ$. In particular, we extend the results of Krylov-Röckner \cite{Kr-Ro} to the distribution dependent case.
We prove the superposition principle for probability measure-valued solutions to non-local Fokker-Planck equations, which in turn yields the equivalence between martingale problems for SDEs with jumps and such non-local PDEs with rough coefficients. As an application, we obtain a probabilistic representation for weak solutions of fractional porous media equations.
In this paper we show the weak differentiability of the unique strong solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's derivative formula for the following stochastic differential equation in $\mathbb R^d$: $$ {\rm d} X_t=b(t,X_t){\rm d} t+σ(t,X_t){\rm d} W_t,\ \ X_0=x\in\mathbb R^d, $$ where $σ$ is bounded, uniformly continuous and nondegenerate, $\nablaσ\in \widetilde{\mathbb L}^{p_1}_{q_1}$ and $b\in \widetilde{\mathbb L}^{p_2}_{q_2}$ for some $p_i,q_i\in[2,\infty)$ with $\frac{d}{p_i}+\frac{2}{q_i}<1$, $i=1,2$, where $\widetilde{\mathbb L}^{p_i}_{q_i}, i=1,2$ are some localized spaces. Moreover, in the endpoint case $b\in \widetilde{\mathbb L}^{d; {\rm uni}}_\infty$, we also show the weak well-posedness.
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 has a density $ρ_{s,x}(t,y)$ belonging to $\mathbb{H}^{1,p}_q$ provided $p,q\in[1,2)$ with $\frac{3}{p}+\frac{2}{q}>4$: $$ \mathrm{d} X_{s,t}={\mathbf u} (s,X_{s,t})\mathrm{d} t+\sqrt{2ν}\mathrm{d} W_t,\ \ X_{s,s}=x,\ \ t\geq s, $$ where $W$ is a three dimensional standard Brownian motion, $ν>0$ 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.
In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates of integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in $\mathbb R^{2d}$ with Hölder coefficients: $$ \partial_tu=\mathscr L^{(α)}_{κ;{\rm v}} u+b\cdot\nabla u+f,\ u_0=0, $$ where $u=u(t,x,{\rm v})$ and $\mathscr L^{(α)}_{κ;{\rm v}}$ is a nonlocal $α$-stable-like operator with $α\in(1,2)$ and kernel function $κ$, which acts on the variable ${\rm v}$. As an application, we show the strong well-posedness to the following degenerate stochastic differential equation with Hölder drift $b$: $$ {\rm d}Z_t=b(t,Z_t){\rm d}t+(0,σ(t,Z_t){\rm d}L^{(α)}_t),\ \ Z_0=(x,{\rm v})\in\mathbb R^{2d}, $$ where $L^{(α)}_t$ is a $d$-dimensional rotationally invariant and symmetric $α$-stable process with $α\in(1,2)$, and $b:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R^{2d}$ is a $(γ,β)$-Hölder continuous function in $(x,{\rm v})$ with $γ\in\big(\frac{2+α}{2(1+α)},1\big)$ and $β\in\big(1-\fracα{2},1\big)$, $σ:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R^d\otimes\mathbb R^d$ is a Lipschitz function. Moreover, we also show that for almost all $ω$, the following random transport equation has a unique $C^1_b$-solution: $$ \partial_tu(t,x,ω)+(b(t,x)+L^{(α)}_t(ω))\cdot\nabla_x u(t,x,ω)=0,\ \ u(0,x)=φ(x), $$ where $φ\in C^1_b(\mathbb R^d)$ and $b:\mathbb R_+\times\mathbb R^d\to\mathbb R^d$ is a bounded continuous function in $(t,x)$ and $γ$-order Hölder continuous in $x$ uniformly in $t$ with $γ\in\big(\frac{2+α}{2(1+α)},1\big)$.
In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together with interpolation techniques and some short time asymptotic estimates of the semigroup. As an application, we show the existence of fundamental solutions for operator $\partial_t-\mathscr{K}$, where $\mathscr{K}$ is the nonlocal kinetic operator: $$ \mathscr{K} f(x,{\rm v}):={\rm p.v}\int_{\mathbb{R}^d}(f(x,{\rm v}+w)-f(x,{\rm v}))\frac{κ(x,{\rm v},w)}{|w|^{d+α}}{\rm d} w +{\rm v}\cdot\nabla_x f(x,{\rm v})+b(x,{\rm v})\cdot\nabla_{\rm v} f(x,{\rm v}). $$ Here $κ_0^{-1}\leq κ(x,{\rm v},w)\leqκ_0$ belongs to $C^\infty_b(\mathbb{R}^{3d})$ and is symmetric in $w$, p.v. stands for the Cauchy principal value, and $b\in C^\infty_b(\mathbb{R}^{2d};\mathbb{R}^d)$.
Consider the following Kolmogorov type hypoelliptic operator $$ \mathscr L_t:=\mbox{$\sum_{j=2}^n$}x_j\cdot\nabla_{x_{j-1}}+{\rm Tr} (a_t \cdot\nabla^2_{x_n}), $$ where $n\geq 2$, $x=(x_1,\cdots,x_n)\in(\mathbb R^d)^n =\mathbb R^{nd}$ and $a_t$ is a time-dependent constant symmetric $d\times d$-matrix that is uniformly elliptic and bounded.. Let $\{\mathcal T_{s,t}; t\geq s\}$ be the time-dependent semigroup associated with $\mathscr L_t$; that is, $\partial_s {\mathcal T}_{s, t} f = - {\mathscr L}_s {\mathcal T}_{s, t}f$. For any $p\in(1,\infty)$, we show that there is a constant $C=C(p,n,d)>0$ such that for any $f(t, x)\in L^p(\mathbb R \times \mathbb R^{nd})=L^p(\mathbb R^{1+nd})$ and every $λ\geq 0$, $$ \left\|Δ_{x_j}^{{1}/{(1+2(n-j)})}\int^{\infty}_0 e^{-λt} {\mathcal T}_{s, s+t }f(t+s, x)dt\right\|_p\leq C\|f\|_p,\quad j=1,\cdots, n, $$ where $\|\cdot\|_p$ is the usual $L^p$-norm in $L^p(\mathbb R^{1+nd}; d s\times d x)$. To show this type of estimates, we first study the propagation of regularity in $L^2$-space from variable $x_n$ to $x_1$ for the solution of the transport equation $\partial_t u+\sum_{j=2}^nx_j\cdot\nabla_{x_{j-1}} u=f$.
Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(α)}_κu+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \ u|_{\mathbb R_+\times D^c}=0,\ u(0,\cdot)|_{D}=φ, $$ where $α\in(0,2)$ and $\mathscr L^{(α)}_κ$ is an $α$-stable-like nonlocal operator with kernel function $κ(x,z)$ bounded from above and below by positive constants, and $b:\mathbb R^d\to\mathbb R^d$ is a bounded $C^β$-function with $α+β>1$, $f: \mathbb R_+\times D\to\mathbb R$ is a $C^γ$-function in $D$ uniformly in $t$ with $γ\in((1-α)\vee 0,β]$, $φ\in C^{α+γ}(D)$. Under some Hölder assumptions on $κ$, we show the existence of a unique classical solution $u\in L^\infty_{loc}(\mathbb R_+; C^{α+γ}_{loc}(D))\times C(\mathbb R_+; C_b(D))$ to the above problem. Moreover, we establish the following probabilistic representation for $u$ $$ u(t,x)=\mathbb E_x \Big(φ(X_{t}){\bf 1}_{τ_{D}>t}\Big)+\mathbb E_x\left(\int^{t\wedgeτ_{D}}_0f(t-s,X_s){\rm d} s\right),\ t\geq 0,\ x\in D, $$ where $((X_t)_{t\geq 0},\mathbb P_x; x\in\mathbb R^d)$ is the Markov process associated with the operator $\mathscr L^{(α)}_κ+b\cdot \nabla$, and $τ_D$ is the first exit time of $X$ from $D$. In the sub and critical case $α\in[1,2)$, the kernel function $κ$ can be rough in $z$. In the supercritical case $α\in(0,1)$, we classify the boundary points according to the sign of $b(z)\cdot\vec{n}(z)$, where $z\in\partial D$ and $\vec{n}(z)$ is the unit outward normal vector. Finally, we provide an example and simulate it by Monte-Carlo method to show our results.
In this note we show the optimal gradient estimate for heat kernels of stable-like operators by providing a counterexample.
In this paper we prove a new strong uniqueness result and a weak existence result for possibly {\it degenerate} multidimensional stochastic differential equations with Sobolev diffusion coefficients and rough drifts. In particular, examples with Hölder diffusion coefficients are provided to show our results.
In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=σ(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $σ$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $α\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-α},\infty)$ and $β\in[α,1]$, $q\in(\frac{d}β,\infty)$. Assume $\|({\mathbb I}-Δ)^{-α/2}b\|_p+\|(-Δ)^{β/2}σ\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of the heat kernel associated to the above SDE. Moreover, we study the ergodicity and global regularity of the invariant measures of the associated semigroup under some dissipative assumptions.
When studying non-symmetric nonlocal operators $$ {\cal L} f(x) = \int_{{\bf R}^d} \left( f(x+z)-f(x)-\nabla f(x)\cdot z 1_{\{|z|\leq 1\}} \right) \frac{κ(x, z)}{|z|^{d+α}} d z , $$ where $0<α<2$ and $κ(x, z)$ is a function on ${\bf R}^d\times {\bf R}^d$ that is bounded between two positive constants, it is customary to assume that $κ(x, z)$ is symmetric in $z$. In this paper, we study heat kernel of ${\cal L}$ and derive its two-sided sharp bounds without the symmetric assumption $κ(x,z)=κ(x,-z)$. In fact, we allow the kernel $κ$ to be time-dependent and also derive gradient estimate when $β\in(0\vee (1-α),1)$ as well as fractional derivative estimate of order $θ\in(0,(α+β)\wedge 2)$ for the heat kernel, where $β$ is the Hölder index of $x\mapstoκ(x,z)$. Moreover, when $α\in(1,2)$, the drift perturbation with drift in Kato's class is also considered. As an application, when $κ(x,z)=κ(z)$ does not depend on $x$, we show the boundedness of nonlocal Riesz's transorfmation: for any $p>2d/(d+2α)$, $$ \| {\cal L}^{1/2}f\|_p\asymp \|Γ(f)^{1/2}\|_p, $$ where $Γ(f):=\frac{1}{2}{\cal L} (f^2)-f {\cal L} f$ is the carré du champ operator associated with ${\cal L}$, and ${\cal L}^{1/2}$ is the square root operator of ${\cal L}$ defined by using Bochner's subordination. Here $\asymp$ means that both sides are comparable up to a constant multiple.
In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\bf R}^d$: $$ d X_{t}= σ_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^d, $$ where $Z$ is a $d$-dimensioanl nondegenerate $α$-stable-like process with $α\in(0,2)$ (including cylindrical case), and uniform in $t\geq 0$, $x\mapsto σ_t(x): {\bf R}^d\to {\bf R}^d\otimes {\bf R}^d$ is Lipchitz and uniformly elliptic and $x\mapsto b_t (x)$ is $β$-order Hölder continuous with $β\in(1-α/2,1)$. Under these assumptions, we show the above SDE has a unique strong solution for every starting point $x \in {\bf R}^d$. When $σ_t (x)={\bf I}_{d\times d}$, the $d\times d$ identity matrix, our result in particular gives an affirmative answer to the open problem of Priola (2015).