Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes

• 成果类型: Article
• 署名作者: Marcus, MB; Rosen, J
• 署名单位: City University of New York (CUNY) System; College of Staten Island (CUNY)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1996
• 页码: 1130-1177
• 关键词: local-times

摘要

Let X be a strongly symmetric Hunt process with alpha-potential density u(alpha)(x, y). Let G(alpha)(2) = {mu\integral integral(u(alpha)(x, y))(2) d mu(x)d mu(y) < infinity} and let L(t)(mu) denote the continuous additive functional with Revuz measure mu. For a set of positive measures M subset of G(alpha)(2), subject to some additional regularity conditions, we consider families of continuous (in time) additive functionals L = {L(t)(mu), (t, mu) is an element of R(+) X M} of X and a second-order Gaussian chaos H-alpha = {H-alpha(mu), mu is an element of M} which is associated with L by an isomorphism theorem of Dynkin. A general theorem is obtained which shows that, with some additional regularity conditions depending on X and M, if H-alpha has a continuous version on M almost surely, then so does L and, furthermore, that moduli of continuity for H-alpha are also moduli of continuity for L. Special attention is given to Levy processes in R(n) and T-n, the n-dimensional torus, with M taken to be the set of translates of a fixed measure. Many concrete examples are given, especially when X is Brownian motion in R(n) and T-n for n = 2 and 3. For certain measures mu on T-n and processes, including Brownian motion in T-3, necessary and sufficient conditions are given for the continuity of {L(t)(mu), (t, mu) is an element of R(+) X M} where M is the set of all translates of mu.