Asymptotic Theory for Three Infinite Dimensional Diffusion Processes

Abstract

<p>This thesis is centered around three infinite dimensional diffusion processes:</p> <p>(i). the infinitely-many-neutral-alleles diffusion model [Ethier and Kurtz, 1981],</p> <p>(ii). the two-parameter infinite dimensional diffusion model [Petrov, 2009] and [Feng and Sun, 2010],</p> <p>(iii). the infinitely-many-alleles diffusion with symmetric dominance [Ethier and Kurtz, 1998].</p> <p>The partition structures, the ergodic inequalities and the asymptotic theory of these three models are discussed. In particular, the asymptotic theory turns out to be the major contribution of this thesis.</p> <p>In Chapter 2, a slightly altered version of Kingman's one-to-one correspondence theorem on partition structures is provided, which in turn becomes a handy tool for obtaining the asymptotic result on the partition structures associated with models (i) and (ii).</p> <p>In Chapter 3, the three diffusion models are briefly introduced. New representations of the transition densities of models (i) and (ii) are obtained simply by rearranging the previous representations obtained in [Ethier, 1992] and [Feng et al., 2011] respectively. These two new representations have their own advantages, by making use of which the corresponding ergodic inequalities easily follow. Furthermore, thanks to the functional inequalities in [Feng et al., 2011], the ergodic inequality for model (iii) becomes available as well.</p> <p>In Chapter 4, the asymptotic properties of models (i) and (ii) are thoroughly studied. Various asymptotic results are obtained, such as the weak limits of models (i) and (ii) at different time scales when the mutation rate approaches infinity, and the large deviation principle for models (i) and (ii) at a fixed time, and that of the transient partition structures of models (i) and (ii). Of all these results, the weak limit and the large deviation principle of the transient partition structures are of particular interest.</p> <p>In Chapter 5, the asymptotic results on the stationary distribution and the transient distribution of model (iii) are both obtained. The weak limit of the infinitely-many- alleles diffusion with symmetric overdominance at fixed time t serves as an alternative answer to Gillespie's conjecture [Gillespie, 1999]. The weak limit of the stationary distribution of the infinitely-many-alleles diffusion with symmetric overdominance provides a complete solution to the remaining problem in [Feng, 2009].</p>