Discrete approximations to reflected Brownian motion

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 $\mathbb{R}^n$ that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on $D\cap2^{-k}\mathbb{Z}^n$ moving at the rate $2^{-2k}$ with stationary initial distribution converge weakly in the space $\mathbf{D}([0,1],\mathbb{R}^n)$, equipped with the Skorokhod topology, to the law of the stationary reflected Brownian motion on $D$. We further show that the following ``myopic conditioning'' algorithm generates, in the limit, a reflected Brownian motion on any bounded domain $D$. For every integer $k\geq1$, let $\{X^k_{j2^{-k}},j=0,1,2,...\}$ be a discrete time Markov chain with one-step transition probabilities being the same as those for the Brownian motion in $D$ conditioned not to exit $D$ before time $2^{-k}$. We prove that the laws of $X^k$ converge to that of the reflected Brownian motion on $D$. These approximation schemes give not only new ways of constructing reflected Brownian motion but also implementable algorithms to simulate reflected Brownian motion.Comment: Published in at http://dx.doi.org/10.1214/009117907000000240 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains

For a symmetric $\alpha$-stable process $X$ on \RR^n with $0<\alpha <2$, $n\geq 2$ and a domain D \subset \RR^n, let $L^D$ be the infinitesimal generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class function $q$, it is shown that $L^D+q$ is intrinsic ultracontractive on a H\"older domain $D$ of order 0. This is then used to establish the conditional gauge theorem for $X$ on bounded Lipschitz domains in \RR^n. It is also shown that the conditional lifetimes for symmetric stable process in a H\"older domain of order 0 are uniformly bounded

On unique extension of time changed reflecting Brownian motions

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. Next assume that RBM $X$ on $\overline D$ is transient and let $Y$ be its time change by Revuz measure ${\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $\overline D$. We further show that if there is some $r>0$ so that $D\setminus \overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has a unique Martin boundary point at infinity.Comment: To appear in Ann. Inst. Henri Poincare Probab. Statis

Global Dirichlet Heat Kernel Estimates for Symmetric L\'evy Processes in Half-space

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) L\'evy processes on half spaces for all $t>0$. These L\'evy processes may or may not have Gaussian component. When L\'evy density is comparable to a decreasing function with damping exponent $\beta$,our estimate is explicit in terms of the distance to the boundary, the L\'evy exponent and the damping exponent $\beta$ of L\'evy density.Comment: 30 page