Stochastic Processes

Abstract

In this chapter, we lay the foundation of stochastic processes. We start with a general definition of a stochastic process, and then specialize to its many variants: stationary, homogeneous, and Markovian stochastic processes. We next discuss stationary Markov processes, and how they give rise to homogeneous Markov processes. For the latter, we introduce the notion of stationary state and condition of detailed balance. Next, we discuss how one studies analytically a homogeneous Markov process via the so-called Fokker-Planck equation for the conditional probability distribution of the relevant random variables. As an example of the formalism developed, we discuss the case of the ubiquitous Brownian motion, focusing in particular on two complementary approaches à la Langevin and Einstein to characterize the motion. While the approach due to Langevin relies on a phenomenological stochastic differential equation of motion for the Brownian particle, the one due to Einstein is instead based on a differential equation for the probability distribution of the velocity of the Brownian particle. In the following two sections, we discuss generalizations of the Langevin equation, and its derivation from first principles for a model system.