Random processes in random environments

Abstract

We shall give a brief survey of some random processes in random environments. Roughly speaking, our examples arise from models in which a phenomenon is described by a Markov process or chain. The Markov process in question is specified by a (usually infinite) number of parameters. Instead of assuming the parameters to be constant, we assume that the parameters themselves are subject to random fluctuations. In such situations it is conceptually helpful to describe the model in two stages. In the first stage the values of the parameters are chosen. Each realization of the parameter values constitutes a choice of the “random environment”. Once the environment has been fixed, it stays fixed forever. The second stage describes the evolution of the random process given the environment. In all our examples, the conditional behavior of the process, given the environment, is that of an inhomogeneous Markov process. In sections 3, 4 and 6 this Markov process actually has no random element left in it, but is deterministic.KeywordsRandom WalkLimit TheoremMarkov ProcessRandom EnvironmentSelection IntensityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.