Generalization of discrete-time geometric bounds to convergence rate of Markov processes on ℝn

Geometric rates of convergence for reversible discrete-time Markov chains are closely related to the spectral gap of the corresponding operator. Quantitative geometric bounds on the spectral gap have been developed using the Cheeger's inequality and some path arguments. We extend the discrete-time results to homogeneous continuous-time reversible Markov processes. The limit path bounds and the limit Cheeger's bounds are introduced. Two quantitative examples of 1-dimensional diffusions are studied for the limit Cheeger's bounds and an n-dimensional diffusion is studied for the limit path bounds.