Some Basics of Stochastic Processes

Abstract

Stochastic processes are introduced as ‘time’ dependent processes depending on randomness where ‘time’ is a totally ordered set. Each stochastic process is coupled to a pdf which describes the probability of a realization of the process at ‘time’ $t$. The general characterization of these processes is discussed in detail. The focus moves on to a specific class of stochastic processes, the Markov processes. It has the remarkable property that a future realization of the process depends solely on its current realization (Markov property). A huge class of processes in physics and related sciences are Markovian in nature. The Wiener process and the Poisson process are discussed as typical examples. This is followed by the introduction of Markov-chains together with their classification. The role of detailed balance in stochastics and physics is elucidated. This explains why Markov-chains are the back-bone of Monte-Carlo sampling techniques. In a last point continuous-time Markov-chains are described.