39 citations · 503 across the 77 of their papers we have counts for
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Compound Poisson particle approximation for McKean-Vlasov SDEs
Xicheng Zhang
We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or -stable processes. Our ap…
Strong convergence of propagation of chaos for McKean-Vlasov SDEs with singular interactions
Zimo Hao, Michael Röckner, Xicheng Zhang
In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular -interactions as well as for the moderat…
Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations
Xicheng Zhang
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for…
Singular kinetic equations and applications
Zimo Hao, Xicheng Zhang, Rongchan Zhu +1
In this paper we study singular kinetic equations on by the paracontrolled distribution method introduced in \cite{GIP15}. We first develop paracontrolled calculu…
Weak solutions of McKean-Vlasov SDEs with supercritical drifts
Xicheng Zhang
Consider the following McKean-Vlasov SDE: where stands for the distribution of and…
Euler scheme for density dependent stochastic differential equations
Zimo Hao, Michael Röckner, Xicheng Zhang
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 f…
Singular HJB equations with applications to KPZ on the real line
Xicheng Zhang, Rongchan Zhu, Xiangchan Zhu
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 underst…
Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical -stable processes
Zhen-Qing Chen, Zimo Hao, Xicheng Zhang
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…
Superposition principle for non-local Fokker-Planck operators
Michael Rockner, Longjie Xie, Xicheng Zhang
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…
A discretized version of Krylov's estimate and its applications
Xicheng Zhang
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 approximat…
-theory of stochastic differential equations
Pengcheng Xia, Longjie Xie, Xicheng Zhang +1
In this paper we show the weak differentiability of the unique strong solution with respect to the starting point as well as Bismut-Elworthy-Li's derivative formula for the fol…
Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations
Xicheng Zhang, Guohuan Zhao
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 ${\mat…
Hörmander's hypoelliptic theorem for nonlocal operators
Zimo Hao, Xuhui Peng, Xicheng Zhang
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'…
Well-posedness of distribution dependent SDEs with singular drifts
Michael Röckner, Xicheng Zhang
Consider the following distribution dependent SDE: where stands for the distribution…
Optimal gradient estimates of heat kernels of stable-like operators
Kai Du, Xicheng Zhang
In this note we show the optimal gradient estimate for heat kernels of stable-like operators by providing a counterexample.
Strong Uniqueness of Degenerate SDEs with Hölder diffusion coefficients
Zhen Wang, Xicheng Zhang
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 diff…
Heat kernel and ergodicity of SDEs with distributional drifts
Xicheng Zhang, Guohuan Zhao
In this paper we consider the following SDE with distributional drift : where is a bounded conti…
Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts
Zhen-Qing Chen, Xicheng Zhang, Guohuan Zhao
In this paper, we study the following time-dependent stochastic differential equation (SDE) in : $$ d X_{t}= σ_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^…
Heat kernels for time-dependent non-symmetric stable-like operators
Zhen-Qing Chen, Xicheng Zhang
When studying non-symmetric nonlocal operators $$ {\cal L} f(x) = \int_{{\bf R}^d} \left( f(x+z)-f(x)-\nabla f(x)\cdot z 1_{\{|z|\leq 1\}} \right) \frac{κ(x, z)}{|z|^{d+α}} d z , $…
Ergodicity of stochastic differential equations with jumps and singular coefficients
Longjie Xie, Xicheng Zhang
We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study th…
Heat Kernels for Non-symmetric Non-local Operators
Zhen-Qing Chen, Xicheng Zhang
We survey the recent progress in the study of heat kernels for a class of non-symmetric non-local operators. We focus on the existence and sharp two-sided estimates of the heat ker…
Stochastic integrals and BDG's inequalities in Orlicz-type spaces
Yingchao Xie, Xicheng Zhang
In this paper we extend an inequality of Lenglart, Lépingle and Pratelli \cite[Lemma 1.1]{LLP} to general continuous adapted stochastic processes with values in topology spaces. By…
Stochastic Hamiltonian flows with singular coefficients
Xicheng Zhang
In this paper we study the following stochastic Hamiltonian system in (a second order stochastic differential equation), $$ d \dot X_t=b(X_t,\dot X_t)d t+σ(X_t,\…
Uniqueness of stable-like processes
Zhen-Qing Chen, Xicheng Zhang
In this work we consider the following -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_{|…