Date of Award

Spring 5-2018

Embargo Period

11-30-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Advisor(s)

Misha Shkolnikov

Abstract

In this thesis, we tackle two problems. In the first problem, we study fluctuations of a system of diffusions interacting through the ranks when the number of diffusions goes to infinity. It is known that the empirical cumulative distribution function of such diffusions converges to a non-random limiting cumulative distribution function which satisfies the porous medium PDE. We show that the fluctuations of the empirical cumulative distribution function around its limit are governed by a suitable SPDE. In the second problem, we introduce common noise that has a rank preserving structure into systems of diffusions interacting through the ranks and study the behaviour of such diffusion processes as the number of diffusions goes to infinity. We show that the limiting distribution function is no longer deterministic and furthermore, it satisfies a suitable SPDE. iii

Available for download on Friday, November 30, 2018

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