This research will develop and apply new singular perturbation methods to boundary value problems for nonlinear ordinary differential equations and for partial integro-differential-difference equations. The results will enable us to calculate the stationary distribution of fluctuations about, and the rate of transitions from (deterministically) stable states of dynamical systems subject to noise. Stochastic integro-differential-difference equations and their associated forward and backward Kolmogorov equations will also be analyzed. The research will have direct applications to a variety of problems in science and engineering. Among them are computer performance evaluation, customer service queue arrangements, chemical physics, noise induced transitions between stable configurations of a dynamical system, etc.
