The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point

We consider a stochastic differential equation on a domain D in n-dimensional real space, where the associated dynamical system is linear, and D contains a repulsive type stationary point at the origin O. We obtain an exit law for the first exit time of the solution process from a ball of arbitrary radius centered at the origin, which involves additive scaling as in Day [6]. The form of the scaling constant is worked out and shown to depend on the structure of the Jordan form of the linear drift. We then obtain an extension of this exit law to the first exit time of the solution process from the general domain D by considering the exit in two stages: first from the origin O to the boundary of the ball, for which the aforementioned exit law applies, and then from the boundary of the ball to the boundary of D. In this way we are able to determine for which Jordan forms we can obtain a limiting distribution for the first exit time to the boundary of D as the noise approaches 0. In particular, we observe there are cases for which the exit time distribution diverges as the noise approaches 0. Acknowledgements No graduate student can arrive at this point in his or her education without the support of a great number of people, and I am no exception. I would like to thank first of all the faculty of the Mathematics Department of Virginia Tech, for accepting me as a graduate student and teaching me during the course of this work; the successful completion of this degree is a reflection upon their outstanding pedagogy. In particular, I wish to thank Dr