Mathematical Tools for Physicists: Stochastic processes

James R. Cruise, Ostap O. Hryniv, and Andrew R. Wade

Chapter 1, pp. 3–38 in Mathematical Tools for Physicists, 2nd ed., Ed. M. Grinfeld, Wiley, New York, November 2014.

Summary

The theory of stochastic processes is concerned with mathematical models of systems whose behavior is time dependent and incorporates randomness or uncertainty. The statistical structure of the system can be determined from the physical application in mind: a rich class consists of Markov processes whose future evolution depends on the past only through the present state of the system.We give a tour through some important examples of stochastic processes, primarily (but not exclusively) in a discrete setting where technicalities are minimized. Our tour is facilitated by a selection of mathematical tools of broad applicability, with a focus on generating functions and renewal ideas.We investigate in detail some illustrative specimens chosen for their relevance to applications in the physical sciences and for the mathematical ideas they elucidate.