A Poisson process is a process satisfying the following properties: 1. The numbers of changes in nonoverlapping intervals are independent for all intervals. 2. The probability of exactly one change in a sufficiently small interval is , where is the probability of one change and is the number of trials. 3. The probability of two or more changes in a sufficiently small interval is essentially 0. In the limit of the number of trials becoming large, the resulting distribution iscalled a Poisson distribution.
Ergodic theory can be described as the statistical and qualitative behavior of measurable group and semigroup actions on measure spaces. The group is most commonly N, R, R-+, and Z.Ergodic theory had its origins in the work of Boltzmann in statistical mechanics problems where time- and space-distribution averages are equal. Steinhaus (1999, pp. 237-239) gives a practical application to ergodic theory to keeping one's feet dry ("in most cases," "stormy weather excepted") when walking along a shoreline without having to constantly turn one's head to anticipate incoming waves. The mathematical origins of ergodic theory are due to von Neumann, Birkhoff, and Koopman in the 1930s. It has since grown to be a huge subject and has applications not only to statistical mechanics, but also to number theory, differential geometry, functional analysis, etc. There are also many internal problems (e.g., ergodic theory..
An endomorphism is called ergodic if it is true that implies or 1, where . Examples of ergodic endomorphisms include the map mod 1 on the unit interval with Lebesgue measure, certain automorphisms of the torus, and "Bernoulli shifts" (and more generally "Markov shifts").Given a map and a sigma-algebra, there may be many ergodic measures. If there is only one ergodic measure, then is called uniquely ergodic. An example of a uniquely ergodic transformation is the map mod 1 on the unit interval when is irrational. Here, the unique ergodic measure is Lebesgue measure.