Non-Gaussian stochastic models: Iterated fractional Brownian motion and directed polymers
Abstract
We investigate two non-Gaussian models: one in which the effects of stochastic dependence are pronounced, even in the large-time asymptotics, and the other in which the behavior of the process is dominated by alternative considerations. The first half of this dissertation is concerned with the iterated fractional Brownian motion, a composition of independent fractional Brownian motions. We present a uniform modulus of continuity result, as well as a Chung-type law of the iterated logarithm. Our calculations rely on large and small deviations estimates. We also study the limiting situation, where the number of compositions is allowed to go to infinity. In the second half of this dissertation, we attempt to furnish a complete picture of the effects of a non-Gaussian medium on the configurations of a directed polymer. We consider a continuous time symmetric random walk on the d-dimensional integer lattice as the polymer, and by imposing only weak stationarity requirements on the medium, we provide sharp bounds for the free energy.
Degree
Ph.D.
Advisors
Viens, Purdue University.
Subject Area
Mathematics
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