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July 1996 Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes
Michael B. Marcus, Jay Rosen
Ann. Probab. 24(3): 1130-1177 (July 1996). DOI: 10.1214/aop/1065725177


Let X be a strongly symmetric Hunt process with $\alpha$-potential density $u^\alpha(x,y). Let

$$ {\mathcal G}_{\alpha}^2 = \left\{\mu | \int\int(u^\alpha (x,y))^2 d\mu(x)\; d\mu (y)<\infty\right\}$$

and let $L_t^\mu$ denote the continuous additive functional with Revuz measure $\mu$. For a set of positive measures $M \subset G_\alpha^2$, subject to some additional regularity conditions, we consider families of continuous (in time) additive functionals $L = {L-t^\mu, (t, \mu) \in R^+ \times M} of X and a second-order Gaussian chaos $H_\alpha = {H_\alpha(\mu), \mu \in M}$ 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 Lévy 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 ${L_t^\mu, (t,\mu) \in R^+ \times M}$, where M is the set of all translates of $\mu$.


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Michael B. Marcus. Jay Rosen. "Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes." Ann. Probab. 24 (3) 1130 - 1177, July 1996.


Published: July 1996
First available in Project Euclid: 9 October 2003

zbMATH: 0862.60065
MathSciNet: MR1411490
Digital Object Identifier: 10.1214/aop/1065725177

Primary: G0G15 , G0J55

Keywords: Continuous additive functionals , Gaussian chaos , Markov

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 3 • July 1996
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