ABSTRACT
This chapter consists of a brief mathematical introduction to the theory and applications of Markov diffusion processes. It discusses Brownian motion, white noise, stochastic differential equations, and Fokker-Planck equations. The chapter approaches the subject with the aim of developing techniques for analyzing models of complex interactions as stochastic dynamic systems and obtaining reduced descriptions of those models in certain limits. It also discusses adiabatic elimination of fast noise as a singular perturbation analysis of higher-dimensional stochastic processes. The chapter develops the mathematical framework for the analysis of continuous-time stochastic processes, focussing on the Wiener process, a.k.a. Brownian motion. It refers to stationary states with a nonvanishing probability as non-equilibrium stationary states. The chapter introduces the notion of detailed balance in the stationary state of a stochastic dynamic system, and discusses the concepts of equilibrium vs. non-equilibrium stationary states.
