Absolute continuity of symmetric diffusions

Abstract

Let $X$ and $Y$ be symmetric diffusion processes with a common state space, and let $P^m$ (resp.$Q^{\mu}$) be the law of $X$ (resp. $Y$) with its symmetry measure $m$ (resp.$\mu$) as initial distribution. We study the consequences of the absolute continuity condition $Q^{\mu} \ll_{\loc} P^m$. We show that under this condition there is a "smooth" version $\rho$ of the Radon-Nikodym derivative $d \mu/dm$ such that $1/2[\log \rho(X_t) - \log \rho(X_0)] = M_t + N_t, t < \sigma$, where $M$ is a continuous local martingale additive functional, $N$ is a zero-energy continuous additive functional and $\sigma$ is an explosion time. The Girsanov density $L_t := dQ^{\mu} |_{F_t}/dP^m|_{F_t}$ then admits the representation $L_t = \exp(M_t - 1/2 \langle M \rangle_t)1_{{t \leq \sigma}}$. The density $\rho$ also serves to link the Dirichlet forms of $X$ and $Y$ in a simple way. Our identification of $L$ relies on notions of even and oddfor additive functionals. These notions complement Fukushima’s decomposition and the forward-backward martingale decomposition of Lyons and Zheng.