If these are called a and b, the process can be described by the following master equations:
Knowledge of an initial state decays exponentially. Therefore for a time in the remote future, the process will reach the following stationary values, denoted by subscript s:
One can also calculate a correlation function:
This random process finds wide application in model building:
- In physics, spin systems and fluorescence intermittency show dichotomous properties. But especially in single molecule experiments probability distributions featuring algebraic tails are used instead of the exponential distribution implied in all formulas above.
- In finance for describing stock prices
- Bondarenko, YV (2000). "Probabilistic Model for Description of Evolution of Financial Indices". Cybernetics and systems analysis 36: 738–742. doi:10.1023/A:1009437108439.
- Margolin, G; Barkai, E (2006). "Nonergodicity of a Time Series Obeying Lévy Statistics". Journal of Statistical Physics 122: 137–167. arXiv:cond-mat/0504454. Bibcode:2006JSP...122..137M. doi:10.1007/s10955-005-8076-9.