## The Annals of Probability

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

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

Michael B. Marcus and Jay Rosen

#### 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

**Keywords**

Continuous additive functionals Markov Gaussian chaos

#### 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