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

Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes

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 two different kinds of harmonicity with respect to a discontinuous symmetric stable process. One kind are functions harmonic in $D$ with respect to the whole process $X$, and the other are functions harmonic in $D$ with respect to the process $X^D$ killed upon leaving $D$. In this paper we show that for bounded Lipschitz domains, the Martin boundary with respect to the killed stable process $X^D$ can be identified with the Euclidean boundary. We further give integral representations for both kinds of positive harmonic functions. Also given is the conditional gauge theorem conditioned according to Martin kernels and the limiting behaviors of the $h$-conditional stable process, where $h$ is a positive harmonic function of $X^D$. In the case when $D$ is a bounded $C^{1, 1}$ domain, sharp estimate on the Martin kernel of $D$ is obtained

Sharp heat kernel estimates for relativistic stable processes in open sets

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 $m-(m^{2/\alpha}-\Delta)^{\alpha/2}$] in $C^{1,1}$ open sets. Here $m>0$ and $\alpha\in(0,2)$. 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 $\Delta^{\alpha/2}:=-(-\Delta)^{\alpha/2}$ 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.Comment: Published in at http://dx.doi.org/10.1214/10-AOP611 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Strong law of large numbers for supercritical superprocesses under second moment condition

Suppose that $X=\{X_t, t\ge 0\}$ is a supercritical superprocess on a locally compact separable metric space $(E, m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$\psi(x,\lambda)=-a(x)\lambda+b(x)\lambda^2+\int_{(0,+\infty)}(e^{-\lambda y}-1+\lambda y)n(x,dy), \quad x\in E, \quad\lambda> 0,$$ where $a\in \mathcal{B}_b(E)$, $b\in \mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\infty)$ satisfying $$\sup_{x\in E}\int_0^\infty y^2 n(x,dy)<\infty.$$ Put $T_tf(x)=\mathbb{P}_{\delta_x}$. Let $\lambda_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and $\phi_0$ and $\hat{\phi}_0$ be the eigenfunctions of $L$ and $\hat{L}$ (the dural of $L$) respectively associated with $\lambda_0$. Under some conditions on the spatial motion and the $\phi_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$\lim_{t\rightarrow\infty}e^{-\lambda_0 t}< f, X_t> = W_\infty\int_E\hat{\phi}_0(y)f(y)m(dy),\quad \mathbb{P}_{\mu}{-a.s.},$$ for any finite initial measure $\mu$ on $E$ with compact support, where $W_\infty$ is the martingale limit defined by $W_\infty:=\lim_{t\to\infty}e^{-\lambda_0t}$. Moreover, the exceptional set in the above limit does not depend on the initial measure $\mu$ and the function $f$

Intrinsic Ultracontractivity and Conditional Gauge for Symmetric Stable Processes

AbstractIt is shown in this paper that the conditional gauge theorem holds for symmetricα-stable processes on boundedC1,1domains inRnwhere 0<α<2 andn⩾2. Two of the major tools used to prove this conditional gauge theorem are logarithmic Sobolev inequality and intrinsic ultracontractivity