Stochastic processes

If a classical particle moves in an external force field F(x) its motion is deterministic and it could be called a ‘deterministic process’. If the particle interacts with randomly distributed obstacles its motion is still deterministic, but the trajectory depends on the random characteristics of the obstacles, and maybe also on the random choice of the initial conditions. Such a motion is an example of a ‘stochastic process’, or rather a ‘random process’.

In a more abstract generalized definition, a stochastic process is a random variable Xξ that depends on an additional (deterministic) independent variable ξ, which can be discrete or continuous. In most cases it stands for an index ξ ∈ N or time t, space x or energy E. An almost trivial ubiquitous stochastic process is given by additive noise η(t) on a time-dependent signal s(t), i.e. Xt = s(t) + η(t). As a consequence, Xt is no longer continuous. The most apparent applications of stochastic processes are time series of any kind that depend on some random impact. A broad field of applications are time series occurring, for example, in business, finance, engineering, medical applications and of course in physics. Beyond time series analysis, stochastic processes are at the heart of diffusion Langevin dynamics, Feynman's path integrals [43], as well as Klauder's stochastic quantization [62], which represents an unconventional approach to quantum mechanics. Here we will give a concise introduction and present a few pedagogical examples.