Dynkin's isomorphism theorem and the stochastic heat equation

Consider the stochastic heat equation ${\mathrm{\partial }}_{t}u=\text{sL}u+\dot{W}$, where $\text{sL}$ is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin’s isomorphism theorem, to the local times of the replica-symmetric process that corresponds to $\text{sL}$.In the case that $\text{sL}$ is the generator of a L'evy process on ${\mathbb{R}}^d$, our result gives a probabilistic explanation of the recent findings of Foondun et al.