Markov processes and random fields

Abstract

DYNKIN 1. Introduction. 1.1. Suppose that a particle moves in a space E under the influence of random factors. Its position x t at time t is a random variable, that is a measurable function on a space Q where a probability measure P is given. The family X = {*,} is called a stochastic process in the state space E. It is important to evaluate the future behaviour of the particle using, in the best possible way, the information available at the present time. A stochastic process X is Markovian if, for a given value of x tQ , the prognosis of the future does not depend on the evolution before / 0 . A more symmetric form of the same property is: the families x r t > t 0 and x r t < t 0 are conditionally independent given x tQ . During the past decades Markov processes became a powerful tool in partial differential equations and potential theory with important applications to physics.