Theory of Second and Higher Order Stochastic Processes

Details Theory of Second and Higher Order Stochastic Processes Theory of Second and Higher Order Stochastic Processes

1995-03-31

arxiv.org/abs/cond-mat/9503171v1

Physics

Condensed Matter

35 pages, PlainTex, accepted for publication in Phys Rev. E.

Scientific paper

10.1103/PhysRevE.51.5298

This paper presents a general approach to linear stochastic processes driven by various random noises. Mathematically, such processes are described by linear stochastic differential equations of arbitrary order (the simplest non-trivial example is $\ddot x = R(t)$, where $R(t)$ is not a Gaussian white noise). The stochastic process is discretized into $n$ time-steps, all possible realizations are summed up and the continuum limit is taken. This procedure often yields closed form formulas for the joint probability distributions. Completely worked out examples include all Gaussian random forces and a large class of Markovian (non-Gaussian) forces. This approach is also useful for deriving Fokker-Planck equations for the probability distribution functions. This is worked out for Gaussian noises and for the Markovian dichotomous noise.

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