First passage times in stochastic models of physical systems and in filtering theory

Abstract

Langevin-type equations are used as models for diffusions in force fields in many physical situations. In particular, the mean first passage time of such a process over a potential barrier, or the time of transition from one stable state to another, is the physical quantity that determines chemical reactions rates (Arrhenius' law), diffusion tensors for atomic migration in crystals, ionic conductivity, the I — V characteristic in Josephson junction devices, relative stability or equilibrium and non-equilibrium steady states and so on. In filtering theory the phenomenon of cycle slip and threshold in phase locked loops can be modeled by a multi-stable system of differential equations driven by a random signal and by the noisy measurements with transitions between the stable states. An asymptotic method for the computation of mean first passage times and exit probabilities yields new formulas for various physical quantities. The method is based on singular perturbation and boundary layer theory and matched asymptotic expansions of solutions of boundary value problems for partial differential equations. It was recently developed by D. Ludwig [8], B. Matkowsky, Z. Schuss

Partially supported by A.F.O.S.R. Grant No. AFOSR 75-3620A and by Israel Academy of Science Grant No. 7407.