Regularity Properties of the Donsker-varadhan Rate Functional for Non-reversible Diffusions and Random Evolutions

The Donsker-Varadhan rate functional I(μ) for Markov processes has a simple form in the case that the generator of the process is self-adjoint, or equivalently, that the Markov process is reversible. In particular, in the case of a reversible diffusion generated by a secondorder elliptic operator L on a compact manifold, this allows one to give a simple necessary and sufficient criterion on the measure μ in order that I(μ) < ∞, and it also shows that I(μ) is continuous as a function of the coefficients of L in the sup-norm topology. In this paper, we first show that the same criterion for finiteness holds for non-reversible diffusions, and then show that I(μ) is locally Lipschitz continuous as a function of the coefficients of the generator L in the sup-norm topology. We then prove similar results in the setting of random evolutions.