  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.
