Volume growth, Comparison theorem and Escape Rate of Diffusion Process

We study the escape rate of diffusion process with two approaches. We first give an upper rate function for the diffusion process associated with a symmetric, strongly local regular Dirichlet form. The upper rate function is in terms of the volume growth of the underlying state space. The method is due to Hsu and Qin [Ann. Probab., 38(4), 2010] where an upper rate function was given for Brownian motion on Riemannian manifold. In the second part, we prove a comparison theorem and give an upper rate function for diffusion process on Riemannian manifold in terms of the upper rate function for the solution process of a one dimensional stochastic differential equation.