Quantitative geometric rates of convergence for reversible Markov chains are closely related to the spectral gap of the corresponding operator, which is hard to calculate for general state spaces. This thesis describes a geometric argument to give dierent types of bounds for spectral gaps of Markov chains on bounded subsets of R n and to compare the rates of convergence of dierent Markov chains. We also extend the discrete-time results to continuous-time reversible Markov processes. The limit path bounds and the limit Cheeger's bounds are introduced. Two quantitative examples of 1-dimensional diusions are studied for the limit Cheeger's bounds and a n-dimensional diusion is studied for the limit path bounds. ii Acknowledgments It is a pleasure to thank my supervisor, Professor Jerey S. Rosenthal
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