Conditioned Brownian motion, spectral gaps and rates to equilibrium for diffusions
This thesis deals with the interplay between the geometry of a domain and probabilistic and analytic quantities associated to that domain. In Chapter 2 it is proved that a large class of domains, called $\Gamma$-domains satisfies the property that the lifetime of a conditioned Brownian motion is bounded above independent of both the starting point and conditioning function. We give several examples of domains which are $\Gamma$-domains including domains above the graph of functions. Chapter 3 deals with the interplay between geometry and the spectrum of the Laplacian. Specifically, the gap between the first two eigenvalues is examined for convex domains with an eye to isoperimetric inequalities. The difference between Dirichlet and Neumann boundary conditions is examined and an intermediary condition known as Robin boundary conditions is lastly considered. The final chapter 4 deals with the importance of the spectral gaps as it relates to the mixing time or rate to equilibrium for diffusions. We use results from Chapter 3 among others to describe this rate in terms of the underlying geometry of the diffusions.
Banuelos, Purdue University.
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