RANDOM EVOLUTIONS, MARKOV CHAINS, AND SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS

Abstract

Several authors have considered Markov processes defined by the motion of a particle on a fixed line with a random velocity(1, 6, 8, 10) or a random diffusivity.(5, 12) A "random evolution" is a natural but apparently new generalization of this notion. In this note we hope to show that this concept leads to simple and powerful applications of probabilistic tools to initial-value problems of both parabolic and hyperbolic type. We obtain existence theorems, representation theorems, and asymptotic formulas, both old and new.