Stochastic processes relating to particles distributed in a continuous infinity of states

Abstract

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E , and this distribution varies with another parameter t (which may be continuous or discrete; if t represents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If the E -space were discrete and the states represented by E 1 , E 2 , …, then it would be possible to define a function representing the probability that there are ν 1 particles in E 1 , ν 2 particles in E 2 , …, at t . The variation of π with t is governed by the transitions defined for the process; ν 1 , ν 2 , … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variables with the help of the π function which yields also the correlation between the stochastic variables ν i .