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

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<SUB>1</SUB>, E<SUB>2</SUB>, …, then it would be possible to define a function representing the probability that there are &#957;<SUB>1</SUB> particles in E<SUB>1</SUB>, &#957;<SUB>2</SUB> particles in E<SUB>2</SUB>, …, at t. The variation of p with t is governed by the transitions defined for the process; &#957;<SUB>1</SUB>, &#957;<SUB>2</SUB>, … 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 p function which yields also the correlation between the stochastic variables &#957;i