## The Annals of Probability

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

#### Abstract

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 $\mu$.

#### Article information

Source
Ann. Probab., Volume 24, Number 3 (1996), 1130-1177.

Dates
First available in Project Euclid: 9 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1065725177

Digital Object Identifier
doi:10.1214/aop/1065725177

Mathematical Reviews number (MathSciNet)
MR1411490

Zentralblatt MATH identifier
0862.60065

Subjects
Primary: G0J55 G0G15

#### Citation

Marcus, Michael B.; Rosen, Jay. Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes. Ann. Probab. 24 (1996), no. 3, 1130--1177. doi:10.1214/aop/1065725177. https://projecteuclid.org/euclid.aop/1065725177