On the group-theoretical formulation for the time evolution of stochastic processes

The temporal evolution of stochastic processes is described using a one-parameter group with time as the parameter. The generator of the group is a first order differential operator. This generator determines the time rate of change of physical variables. If these variables are the probabilities of the different possible states of the system then their equation of motion is exactly equivalent to the master equation. However, the very same equation of motion is equally valid for other variables such as expectation values or, in general, functions of the probabilities. The introduction of an equation of motion which is linear in the generators of the group provides a very convenient algebraic tool. For example, constants of the motion and symmetries of dissipative processes can readily be discussed. For many physical situations, a smaller subgroup suffices to describe the time evolution. The formalism is illustrated for two models of energy relaxation by binary collisions in the gas phase. While the models are valid for complementary physical situations, they have a common (two generators) group structure.