Dirichlet Heat Kernel Estimates for $\Delta ^{\alpha /2}+ \Delta ^{\beta /2}$ (2009)
Chen, Zhen-Qing, Kim, Panki, Song, Renming
For $d\geq 1$ and $0
Sharp Heat Kernel Estimates for Relativistic Stable Processes in Open Sets (2009)
Chen, Zhen-Qing, Kim, Panki, Song, Renming
In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes (or equivalently, for the heat kernels of the operators...
Boundary Harnack principle for $\Delta + \Delta^{\alpha/2}$ (2009)
Chen, Zhen-Qing, Kim, Panki, Song, Renming, Vondraček, Zoran
For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta...
Time-reversal and elliptic boundary value problems (2009)
Chen, Zhen-Qing, Zhang, Tusheng
In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second-order elliptic operators with singular...
Global Heat Kernel Estimates for Fractional Laplacians in Unbounded Open Sets (2009)
Chen, Zhen-Qing, Tokle, Joshua
In this paper, we derive global sharp heat kernel estimates for symmetric alpha-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of...
Regularity of harmonic functions for a class of singular stable-like processes (2009)
Bass, Richard F., Chen, Zhen-Qing
We consider the system of stochastic differential equations dX_t=A(X_{t-}) dZ_t, where Z_t^1, ..., Z^d_t are independent one-dimensional symmetric stable processes of order \alpha, and the...
Richard F. Bass, Krzysztof Burdzy, Zhen-qing Chen, Martin Hairer
distributions for diffusions with inert drift ∗
DISCRETE APPROXIMATIONS TO REFLECTED BROWNIAN MOTION 1 (2009)
In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in R n that includes...
© Institute of Mathematical Statistics, 2003 BROWNIAN MOTION WITH SINGULAR DRIFT (2009)
We consider the stochastic differential equation dXt = dWt + dAt, where Wt is d-dimensional Brownian motion with d ≥ 2 and the ith component of At is a process of bounded variation that stands in...
HOLOMORPHIC DIFFUSIONS AND BOUNDARY BEHAVIOR OF HARMONIC FUNCTIONS (2009)
Zhen-qing Chen, Richard Durrett, Gang Ma
We study a family of differential operators �L α �α ≥ 0 � in the unit ball D of C n with n ≥ 2 that generalize the classical Laplacian, α = 0, and the conformal Laplacian, α = 1/2 (that...
Heat kernel estimates for Dirichlet fractional Laplacian (2009)
Zhen-qing Chen, Panki Kim, Renming Song
In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such...
Pathwise uniqueness for a degenerate stochastic differential equation (2009)
F. Bass, Krzysztof Burdzy, Zhen-qing Chen
We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation dXt =|Xt | α dWt, where Wt is a one-dimensional Brownian motion and α...
Stationary distributions for diffusions with inert drift (forthcoming paper (2009)
Richard F. Bass, Krzysztof Burdzy, Zhen-qing Chen, Martin Hairer
Consider a reflecting diffusion in a domain in R d that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the...
Traces of symmetric Markov processes and their characterizations (2009)
Zhen-qing Chen, Masatoshi Fukushima, Jiangang Ying, Dedicated Memory, Martin L. Silverstein
Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the...
Symmetric Jump Processes and their Heat Kernel Estimates (2009)
We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently, a class of symmetric integro-differential operators). We focus...
Exit Time, Green Function and Semilinear Elliptic Equations (2009)
Atar, Rami; Technion; Atar@ee.technion.ac.il, Athreya, Siva; Indian Statistical Institute; Athreya@isibang.ac.in, Chen, Zhen-Qing; University Of Washington; Zchen@math.washington.edu
Let $D$ be a bounded Lipschitz domain in $R^n$ with $ngeq 2$ and $tau_D$ be the first exit time from $D$ by Brownian motion on $R^n$. In the first part of this paper, we are concerned with sharp...
ON THE ROBIN PROBLEM IN FRACTAL DOMAINS (2008)
Richard F. Bass, Krzysztof Burdzy, Zhen-qing Chen
Abstract. We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum is greater than 0, and some other...
PROBABILITY THEORY AND EXAMPLES. 2nd edition (2008)
Eric Blair, Zhen-qing Chen, Ted Cox, Jan Hannig, Andrew Hayen, ...
spell checking I have found many more misspelled words. These and other small points of grammar have not been added to the list.
On notions of harmonicity (2008)
In this paper, we address the equivalence of the analytic and probabilistic notions of harmonicity in the context of general symmetric Hunt processes on locally compact separable metric spaces....
On unique extension of time changed reflecting Brownian motions (2008)
Chen, Zhen-Qing, Fukushima, Masatoshi
Let $D$ be an unbounded domain in $\RR^d$ with $d\geq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $\overline D$ is transient....
Chen, Zhen-Qing, Kumagai, Takashi
In this paper, we consider the following type of non-local (pseudo-differential) operators $\LL $ on $\R^d$: $$ \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x)...
Stationary distributions for diffusions with inert drift (2008)
Bass, Richard F., Burdzy, Krzysztof, Chen, Zhen-Qing, Hairer, Martin
Consider a reflecting diffusion in a domain in $R^d$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for...
ON THE ROBIN PROBLEM IN FRACTAL DOMAINS (2008)
Richard F. Bass, Krzysztof Burdzy, Zhen-qing Chen
Abstract. We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum of the solution to the mixed...
Schramm-Loewner Equations Driven by Symmetric Stable Processes (2008)
We consider shape, size and regularity of the hulls Kt of the chordal Schramm-Loewner evolution driven by a symmetric α-stable process. We obtain derivative estimates, show that the domains H \ Kt...
Zhen-qing Chen, Takashi Kumagai
Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping...
Heat Kernel Estimates for Dirichlet Fractional Laplacian (2008)
Zhen-qing Chen, Panki Kim, Renming Song
In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such...
Two-sided heat kernel estimates for censored stable-like processes (2008)
Zhen-qing Chen, Panki Kim, Renming Song
In this paper we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first...
Time-Reversal and Elliptic Boundary Value Problems (2008)
In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second order elliptic operators with singular...
Zhen-qing Chen, Takashi Kumagai
In this paper, we consider the following type of non-local (pseudo-differential) operators L on R d: Lu(x) = 1 2 d� i,j=1 ∂xi
On unique extension of time changed reflecting Brownian motions (2008)
Zhen-qing Chen, Masatoshi Fukushima
Let D be an unbounded domain in R d with d ≥ 3. We show that if D contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on D is transient. Next assume that RBM X...
On the Robin problem in fractal domains (2008)
Bass, Richard F., Burdzy, Krzysztof, Chen, Zhen-Qing
We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum of the solution to the mixed...
General gauge and conditional gauge theorems are established for a large class of (not necessary symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric...
COALESCENCE OF SYNCHRONOUS COUPLINGS (2007)
Krzysztof Burdzy, Zhen-qing Chen
re ected inside a planar domain D, i.e., a vector process (X t; Y t) such that X t and Y t are re ected Brownian motions in D, and for every (random) interval (t 1; t 2) such that neither
Feynman-Kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of Markov processes. General gauge and conditional gauge theorems are established for...
In this paper we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed...
Traps For Reflected Brownian Motion (2007)
Zhen-qing Chen, Donald E. Marshall
Consider an open set D 2, and a closed ball B D. Let E denote the expectation of the hitting time of B for reflected Brownian motion in D starting from x D. We say that D is a trap domain if sup x E...
The heat equation in time dependent domains with Neumann boundary conditions (2007)
Chris Burdzy, Zhen-Qing Chen, John Sylvester
We study the heat equation in domains in R^n with insulated fast moving boundaries. We prove existence and uniqueness theorems in the case that the boundary moves at speeds that are square integrable.
Schramm-Loewner Equations Driven by Symmetric Stable Processes (2007)
Chen, Zhen-Qing, Rohde, Steffen
We consider shape, size and regularity of the hulls of the chordal Schramm-Loewner evolution driven by a symmetric alpha-stable process. We obtain derivative estimates, show that the complements of...
Heat Kernel Estimates for Truncated Stable-like Processes and Weighted Poincaré Inequality (2007)
Zhen-qing Chen, Panki Kim, Takashi Kumagai
In this paper, we investigate pure jump symmetric processes in R d whose jumping kernel is comparable to the one for truncated rotationally symmetric α-stable process where jumps of size larger than...
Zhen-qing Chen, Masatoshi Fukushima
This paper is a continuation of the works by Fukushima-Tanaka [14] and Chen-Fukushima-Ying [8] on the study of one-point extendability of a pair of standard Markov processes in weak duality. In this...
Interacting Superprocesses with Discontinuous Spatial Motion and their Associated SPDEs (2007)
Zhen-qing Chen, Hao Wang, Jie Xiong
A class of interacting superprocesses arising from branching particle systems with continuous spatial motions, called superprocesses with dependent spatial motion (SDSMs), has been introduced and...
Extending Markov processes in weak duality by Poisson point processes of excursions (2007)
Zhen-qing Chen, Masatoshi Fukushima, Jiangang Ying
Let a be a non-isolated point of a topological space E. Suppose we are given standard processes X 0 and � X 0 on E0 = E \ {a} in weak duality with respect to a σ-finite measure m on E0 which are...
Discrete Approximations to Reflected Brownian Motion ∗ (2007)
Krzysztof Burdzy, Zhen-qing Chen
In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in R n that includes...
Boundary Harnack Principle for Positive Solutions of Semilinear Elliptic Equations (2007)
Rami Atar, Siva Athreya, Zhen-qing Chen
We study whether the Boundary Harnack Principle holds for positive solutions for the equation ∆u = u p in D with boundary value u = φ on ∂D. When D is a bounded Lipschitz domain and φ is...
An Almost Sure Scaling Limit Theorem for Dawson-Watanabe (2007)
Zhen-qing Chen, Yan-xia Ren, Hao Wang
We establish a scaling limit theorem for a large class of Dawson-Watanabe superprocesses whose underlying spatial motions are symmetric Hunt processes, where the convergence is in the sense of...
Discrete approximations to reflected Brownian motion (2006)
Burdzy, Krzysztof, Chen, Zhen-Qing
In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains $D$ in $\mathbb{R}^n$...
Traces of symmetric Markov processes and their characterizations (2006)
Chen, Zhen-Qing, Fukushima, Masatoshi, Ying, Jiangang
Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the...
Traces of symmetric Markov processes and their characterizations (2006)
Chen, Zhen-Qing, Fukushima, Masatoshi, Ying, Jiangang
Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the...
Pathwise uniqueness for a degenerate stochastic differential equation (2006)
Bass, Richard F., Burdzy, Krzysztof, Chen, Zhen-Qing
We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation \[dX_t=|X_t|^{\alpha} dW_t,\] where $W_t$ is a one-dimensional Brownian...
R.: Continuity of eigenvalues of subordinate processes in domains (2006)
Abstract. Let X ={Xt,t ≥ 0} be a symmetric Markov process in a state space E and D an open set of E. Let S (n) ={S (n) t,t ≥ 0} be a subordinator with Laplace exponent φn and S ={St,t ≥ 0} a...
Non-local Dirichlet forms and symmetric jump processes (2006)
Martin T. Barlow, Richard F. Bass, Zhen-qing Chen, Moritz Kassmann
We consider the non-local symmetric Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y) dx dy Rd Rd with F the closure with respect to E1 of the set of C1 functions on Rd with compact...
Benjamini, Itai, Burdzy, Krzysztof, Chen, Zhen-Qing
A pair of Markov processes is called a Markov coupling if both processes have the same transition probabilities and the pair is also a Markov process. We say that a coupling is ``shy'' if the...
On the Robin problem in fractal domains (2005)
Bass, Richard, Burdzy, Krzysztof, Chen, Zhen-Qing
We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum of the solution is greater than 0, and some...
Synchronous couplings of reflected Brownian motions in smooth domains (2005)
Burdzy, Krzysztof, Chen, Zhen-Qing, Jones, Peter
For every bounded planar domain $D$ with a smooth boundary, we define a `Lyapunov exponent' $\Lambda(D)$ using a fairly explicit formula. We consider two reflected Brownian motions in $D$, driven by...
On Feynman-Kac perturbation of symmetric Markov processes (2005)
Let X be an m-symmetric right process on Luzin space E and (E, F) be its associated quasiregular Dirichlet form. Let µ be a signed smooth measure of X and A µ be the continuous additive functional...
The heat equation and reflected Brownian motion in time-dependent domains (2004)
Burdzy, Krzysztof, Chen, Zhen-Qing, Sylvester, John
The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as "noncylindrical domains,'' and its connections with partial differential...
The Heat Equation And Reflected Brownian Motion In Time-Dependent Domains (2004)
Krzysztof Burdzy, Zhen-qing Chen, John Sylvester
this paper, for example, a Feynman--Kac type formula, are the basis for several effective quantitative and qualitative arguments in the second paper in this series, Burdzy, Chen and Sylvester (2003)....
Hardy inequality for censored stable processes (2003)
Chen, Zhen-Qing, Song, Renming
A Hardy inequality is established for censored stable processes on a large class of bounded domains including bounded Lipschitz domains in $\RR^n$ with $n\geq 2$.
Boundary Trace of Reflecting Brownian Motions (2003)
Benjamini, Itai, Chen, Zhen-Qing, Rohde, Steffen
A uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains is established. Exact Hausdorff dimensions for the boundary occupation time and the...
Brownian motion with singular drift (2003)
Bass, Richard F., Chen, Zhen-Qing
We consider the stochastic differential equation \[ dX_t=dW_t+dA_t, \] where $W_t$ is $d$-dimensional Brownian motion with $d\geq 2$ and the $i$th component of $A_t$ is a process of bounded variation...
Hardy inequality for censored stable processes (2003)
A Hardy inequality is established for censored stable processes on a large class of bounded domains including bounded Lipschitz domains in R n with n ≥ 2.
A note on Green function estimates for symmetric stable processes (2003)
Abstract. Suppose D is a bounded C 1,1 domain in R n and GD is the Green function of the killed symmetric α-stable process in D. In this note we establish a sharp upper bound estimate for GD in such...
Censored Stable Processes (2003)
Krzysztof Bogdan, Krzysztof Burdzy, Zhen-qing Chen
We present several constructions of a "censored stable process" in an open set D , i.e., a symmetric stable process which is not allowed to jump outside D.
General gauge and conditional gauge theorems (2002)
Chen, Zhen-Qing, Song, Renming
General gauge and conditional gauge theorems are established for a large class of (not necessarily symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric...
Girsanov and Feynman-Kac type transformations for symmetric Markov processes (2002)
Chen, Zhen-Qing, Zhang, Tusheng
Studied in this paper is the transformation of an arbitrary symmetric Markov process X by multiplicative functionals which are the exponential of continuous additive functionals of X having zero...
Conditional gauge theorem for non-local Feynman-Kac transforms
Girsanov and Feynman-Kac type transformations for symmetric Markov processes (2002)
Zhen-qing Chen, Tu-sheng Zhang
Abstract. Studied in this paper is the transformation of an arbitrary symmetric Markov process X by multiplicative functionals which are the exponential of continuous additive functionals of X having...
Gaugeability and conditional gaugeability (2002)
New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Green-tight...
Local time flow related to skew brownian motion (2001)
Burdzy, Krzysztof, Chen, Zhen-Qing
We define a local time flow of skew Brownian motions ,that is, a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but...
Hardy inequality for censored stable processes (2001)
A Hardy inequality is established for censored stable processes on a large class of bounded domains including bounded Lipschitz domains. 1
Censored stable processes (2001)
Krzysztof Bogdan, Krzysztof Burdzy, Zhen-qing Chen
We present several constructions of a \censored stable process " in an open set D R n, i.e., a symmetric stable process which is not allowed to jump outside D. We address the question of...
Local time flow related to skew Brownian motion (preprint (1999)
Krzysztof Burdzy, Zhen-qing Chen
Summary. We dene a local time ow of skew Brownian motions, i.e., a family of solutions to the stochastic dierential equation dening the skew Brownian motion, starting from dierent points but driven...
Local Time Flow Related To Skew Brownian Motion (1999)
Krzysztof Burdzy, Zhen-qing Chen
Introduction. We will present some results on a family of local time processes, including a new Ray-Knight-type theorem. The results and techniques are directly inspired by those in a paper of...
Local time flow related to skew Brownian motion (preprint (1999)
Krzysztof Burdzy, Zhen-qing Chen
We define a local time flow of skew Brownian motions, that is, a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but...
Chen, Zhen-Qing, Song, Renming
Martin boundaries and integral representations of positive functions which are harmonic in a bounded domain $D$ with respect to Brownian motion are well understood. Unlike the Brownian case, there...
Chen, Zhen-Qing, Song, Renming
For a symmetric $\alpha$-stable process $X$ on $\RR^n$ with $0
Weak Convergence of Reflected Brownian Motions (1998)
Burdzy, Krzysztof; University Of Washington; Burdzy@math.washington.edu, Chen, Zhen-Qing; Cornell University; Zchen@math.cornell.edu
We show that if a sequence of domains $D_k$ increases to a domain $D$ then the reflected Brownian motions in $D_k$'s converge to the reflected Brownian motion in $D$, under mild technical assumptions.
Martin boundaries and integral representations of positive functions which are harmonic in a bounded domain D with respect to Brownian motion are well understood. Unlike the Brownian case, there are...
For a symmetric ff-stable process X on R n with 0 ! ff ! 2, n 2 and a domain D ae R n , let L D be the infinitesimal generator of the subprocess of X killed upon leaving D. For a Kato class function...
Holomorphic diffusions and boundary behavior of harmonic functions (1997)
Chen, Zhen-Qing, Durrett, Richard, Ma, Gang
We study a family of differential operators ${L^{\alpha}, \alpha \geq 0}$ in the unit ball $D$ of $C^n$ with $n \geq 2$ that generalize the classical Laplacian, $\alpha = 0$, and the conformal...
On reflecting diffusion processes /--by Zhen-Qing Chen. (1992)
Thesis (Ph. D.)--Washington University, 1992. Dept. of Mathematics.
Heat Equation And Reflected Brownian Motion In Time Dependent Domains
Krzysztof Burdzy, Zhen-qing Chen, John Sylvester
This article is mostly devoted to questions of analytic nature; somewhat paradoxically in view of the original motivation, probability mostly plays here the role of a tool and not an end. The...