Analysis of unexpected exits using the Fokker - Planck equation

In this thesis exit problems are considered for stochastic dynamical systems with small random fluctuations. We study exit from a domain in the state space through a boundary, or a specified part of the boundary, that is unattainable in the underlying deterministic system. We analyze diffusion approximations of the dynamical systems. The processes are described with a Fokker-Planck equation in a continuous state space. Taking the diffusion parameter as the small parameter, we determine asymptotic expressions for the probability of exit and the (conditional) expected exit time.

We consider applications in groundwater flow and epidemiology. For a contaminant in an advective-dispersive groundwater flow asymptotic expressions are derived for the probability of arrival at a well and the expected arrival time. For a stochastic SIR -model describing the spread of an infectious disease in a population we determine asymptotic expressions for the following quantities: the probability that a major outbreak occurs upon the introduction of the disease into the population, the probability of extinction of the disease at the end of a major outbreak, and the expected extinction time of the disease for an initial state in the stable equilibrium. Finally, for an interval in a one- dimensional stochastic system we study the expected exit time at precisely that end of the interval where exit is not likely, including in our analysis initial states outside a boundary layer.