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 $\mu$.