Singular Perturbation Theory and Large, 
                 Rare Fluctuations of Noise-Driven Systems 

                                   by

                               Robert Maier

               INSCC 110, 3:30pm  Monday, 31 August, 1998


                                Abstract

Randomly perturbed, finite dimensional dynamical systems are of interest in
statistical physics, chemical physics, and several fields of
engineering. Frequently the random perturbations are weak, and fluctuations
away from stable states only rarely occur. By applying singular
perturbation theory to the associated Kolmogorov equation for the 
diffusion of probability, it is possible to work out the most probable 
way that such fluctuations occur, in the weak-noise limit.  
There is a flow field of `most probable outgoing trajectories', 
analogous to the rays of geometrical optics, extending away
from each stable state.  These rays can cross, forming caustics. 
The interpretation of this phenomenon will be explained.  A full
analysis of noise-induced exit from a domain of attraction requires the
construction of an inner approximation near the boundary of the domain.  
What these inner approximations look like, and their consequences for 
the limiting distribution of exit points, will be explained too.
                             


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