Random processes in random environments

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.

[1]  Walter L. Smith,et al.  On Branching Processes in Random Environments , 1969 .

[2]  Samuel Karlin,et al.  On Branching Processes with Random Environments: I: Extinction Probabilities , 1971 .

[3]  Multitype branching processes in random environments , 1971 .

[4]  Samuel Karlin,et al.  Branching Processes with Random Environments, II: Limit Theorems , 1971 .

[5]  J. Gillespie.  Polymorphism in random environments , 1973 .

[6]  S. Karlin,et al.  Temporal fluctuations in selection intensities: Case of small population size , 1974 .

[7]  S. Karlin,et al.  Random temporal variation in selection intensities: case of large population size. , 1974, Theoretical population biology.

[8]  N. Kaplan.  Some Results about Multidimensional Branching Processes with Random Environments , 1974 .

[9]  M. Frank Norman,et al.  An ergodic theorem for evolution in a random environment , 1975, Journal of Applied Probability.

[10]  S. Karlin,et al.  Random temporal variation in selection intensities: One-locus two-allele model , 1975 .

[11]  Samuel Karlin,et al.  Random temporal variation in selection intensities acting on infinite diploid populations: diffusion method analysis. , 1975, Theoretical population biology.

[12]  Frederick Solomon.  Random Walks in a Random Environment , 1975 .

[13]  Harry Kesten,et al.  A limit law for random walk in a random environment , 1975 .

[14]  M. Turelli.  Random environments and stochastic calculus. , 1977, Theoretical population biology.

[15]  David Tanny.  Limit Theorems for Branching Processes in a Random Environment , 1977 .

[16]  George Papanicolaou,et al.  A limit theorem for turbulent diffusion , 1979 .

[17]  P Polansky,et al.  Invariant distributions for multi-population models in random environments. , 1979, Theoretical population biology.

[18]  J H Gillespie,et al.  Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology. , 1980, Theoretical population biology.

[19]  Yukio Ogura,et al.  Recurrence properties of Lotka-Volterra models with random fluctuations , 1981 .

[20]  S. Kalikow,et al.  GENERALIZED RANDOM WALK IN A RANDOM ENVIRONMENT , 1981 .