arXiv subjects
Publications and source records attributed to Zhen-Qing Chen.
In this paper, we show that reflecting Brownian motion in any bounded domain D can be approximated, as $k\to\infty$, by simple random walks on "maximal connected" subsets of $(2^{-k}\mathbb{Z}^d)\cap D$ whose filled-in interiors are inside of D.
In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C^{1,1} open sets D in R^d, of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided Dirichlet heat kernel estimates hold for all t>0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp two-sided estimates for the Green functions, in bounded C^{1,1} open sets, of such subordinate Brownian motions with Gaussian components.
In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a κ-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Levy process. When D is a C^{1, 1} open set and the Levy exponent of the process is given by Ψ(ξ)= ϕ(|ξ|^2) with ϕbeing a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are explicit in terms of Ψ, the distance function to the boundary of D and the jumping kernel of X, which give an affirmative answer to the conjecture posted in [Potential Anal., 36 (2012) 235-261]. Our results are the first sharp two-sided Dirichlet heat kernel estimates for a large class of symmetric Levy processes with general Levy exponents. We also derive an explicit lower bound estimate for symmetric Levy processes on R^d in terms of their Levy exponents.
Suppose that $d\geq2$ and $α\in(1,2)$. Let D be a bounded $C^{1,1}$ open set in $\mathbb{R}^d$ and b an $\mathbb{R}^d$-valued function on $\mathbb{R}^d$ whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for $\mathcal{L}^b=Δ^{α/2}+b\cdot\nabla$ in D with zero exterior condition. We also obtain the boundary Harnack principle for $\mathcal{L}^b$ in D with explicit decay rate.
In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes [i.e., for the heat kernels of the operators $m-(m^{2/α}-Δ)^{α/2}$] in $C^{1,1}$ open sets. Here $m>0$ and $α\in(0,2)$. The estimates are uniform in $m\in(0,M]$ for each fixed $M>0$. Letting $m\downarrow0$, we recover the Dirichlet heat kernel estimates for $Δ^{α/2}:=-(-Δ)^{α/2}$ in $C^{1,1}$ open sets obtained in [14]. Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in bounded $C^{1,1}$ open sets.
This erratum corrects the article arXiv:0806.2044 published in Ann. Probab. 36 (2008) 931--970
In the first part of this paper, we give a useful criterion for uniform integrability of exponential martingales in the context of Markov processes. The condition of this criterion is easy to verify and is, in general, much weaker than the commonly used Novikov's condition. In the second part of this paper, we present a new approach to the study of spectral bounds of Feynman-Kac semigroups for a large class of symmetric Markov processes. We first establish criteria for the $L^p$-independence of spectral bounds for Feynman-Kac semigroups generated by continuous additive functionals, using gaugeability results obtained by the author in \cite{C}. We then extend these analytic criteria for the $L^p$-independence of spectral bounds to non-local Feynman-Kac semigroups via pure jump Girsanov transforms. For this, the uniform integrability of the exponential martingales established in the first part of this paper plays an important role. We use it to show that Kato classes introduced in \cite{C} can only become larger under pure jump Girsanov transforms with symmetric jumping functions.
Let $X$ be a symmetric strong Markov process on a Luzin space. In this paper, we present criteria of the $L^p$-independence of spectral bounds for generalized non-local Feynman-Kac semigroups of $X$ that involve continuous additive functionals of $X$ having zero quadratic variations and discontinuous additive functionals of $X$.
We consider a family of hard core objects moving as independent Brownian motions confined to a vessel by reflection. These are subject to gravitational forces modeled by drifts. The stationary distribution for the process has many interesting implications, including an illustration of the Archimedes' principle. The analysis rests on constructing reflecting Brownian motion with drift in a general open connected domain and studying its stationary distribution. In dimension two we utilize known results about sphere packing.
In this paper we show that Dirichlet heat kernel estimates for a class of (not necessarily symmetric) Markov processes are stable under non-local Feynman-Kac perturbations. This class of processes includes, among others, (reflected) symmetric stable-like processes on closed $d$-sets in $\bR^d$, killed symmetric stable processes, censored stable processes in $C^{1, 1}$ open sets as well as stable processes with drifts in bounded $C^{1, 1}$ open sets.
In this paper, sharp two-sided estimates for the transition densities of relativistic $α$-stable processes with mass $m\in (0, 1]$ in $C^{1,1}$ exterior open sets are established for all time $t>0$. These transition densities are also the Dirichlet heat kernels of $m-(m^{2/α}-Δ)^{α/2}$ with $m\in (0, 1]$ in $C^{1,1}$ exterior open sets. The estimates are uniform in $m$ in the sense that the constants are independent of $m\in (0, 1]$. As a corollary of our main result, we establish sharp two-sided Green function estimates for relativistic $α$-stable processes with mass $m\in (0, 1]$ in $C^{1,1}$ exterior open sets.
Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space-time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.
Suppose that $d\ge 1$ and $α\in (0, 2)$. In this paper, by using probabilistic methods, we establish sharp two-sided pointwise estimates for the Dirichlet heat kernels of $\{Δ+ a^αΔ^{α/2}; \ a\in (0, 1]\}$ on half-space-like $C^{1, 1}$ domains in ${\mathbb R}^d$ for all time $t>0$. The large time estimates for half-space-like domains are very different from those for bounded domains. Our estimates are uniform in $a \in (0, 1]$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$. Thus it yields the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we obtain uniform sharp two-sided estimates for the Green functions of $\{Δ+ a^αΔ^{α/2}; \ a\in (0, 1]\}$ in half-space-like $C^{1, 1}$ domains in ${\mathbb R}^d$.
In this paper we give general criteria on tightness and weak convergence of discrete Markov chains to symmetric jump processes on metric measure spaces under mild conditions. As an application, we investigate discrete approximation for a large class of symmetric jump processes. We also discuss some application of our results to the scaling limit of random walk in random conductance.
In this paper we develop an $L_2$-theory for stochastic partial differential equations driven by Lévy processes. The coefficients of the equations are random functions depending on time and space variables, and no smoothness assumption of the coefficients is assumed.
In this paper we present an $L^p$-theory for the stochastic partial differential equations (SPDEs in abbreciation) driven by Lé{}vy processes. Existence and uniqueness of solutions in Sobolev spaces are obtained. The coefficients of SPDEs under consideration are random functions depending on time and space variables.
We consider a family of pseudo differential operators $\{Δ+ a^αΔ^{α/2}; a\in (0, 1]\}$ on $\bR^d$ for every $d\geq 1$ that evolves continuously from $Δ$ to $Δ+ Δ^{α/2}$, where $α\in (0, 2)$. It gives rise to a family of Lévy processes $\{X^a, a\in (0, 1]\}$ in $\bR^d$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $α$-stable process with weight $a$. We establish sharp two-sided estimates for the heat kernel of $Δ+ a^α Δ^{α/2}$ with zero exterior condition in a family of open subsets, including bounded $C^{1, 1}$ (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric $α$-stable process with weight $a$ in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in $a$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$ so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we recover the two-sided sharp uniform Green function estimates of $X^a$ in bounded $C^{1,1}$ open sets in $\bR^d$, which were recently established in \cite{CKSV2} by using a completely different approach.
We establish the equivalence of the analytic and probabilistic notions of subharmonicity in the framework of general symmetric Hunt processes on locally compact separable metric spaces, extending an earlier work of the first named author on the equivalence of the analytic and probabilistic notions of harmonicity. As a corollary, we prove a strong maximum principle for locally bounded finely continuous subharmonic functions in the space of functions locally in the domain of the Dirichlet form under some natural conditions.