Theory of Second and Higher Order Stochastic Processes

TL;DR

This paper develops a comprehensive framework for analyzing linear stochastic processes driven by various types of random noise, providing closed-form solutions and methods for deriving probability distributions and Fokker-Planck equations.

Contribution

It introduces a general approach to higher-order stochastic differential equations with non-Gaussian and Markovian noise, including derivations of probability distributions and Fokker-Planck equations.

Findings

Closed-form formulas for joint probability distributions.

Worked out examples for Gaussian and Markovian non-Gaussian forces.

Method for deriving Fokker-Planck equations for different noise types.

Abstract

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.