# Estimating the probability density function of nonlinear stochastic processes by use of asymptotic expansions in the Kubo number

## Abstract

In this article, a method to find evolution equations for the probability density function (pdf)of a stochastic process governed by a nonlinear stochastic ordinary differential equation is

presented. According to Van Kampen's lemma, the pdf of a stochastic process is equal to the

ensemble average of the phase-space density. The conservation of the phase-space density is

dictated by a stochastic Liouville equation, which is a linear stochastic partial differential

equation. Thus, finding the governing equation for the evolution of the pdf of the nonlinear

stochastic process amounts to finding the ensemble average form of a linear stochastic partial

differential equation. In this article, the cumulant expansion approach is used and combined

with an asymptotic expansion of the phase-space density in the Kubo number in order to develop

recursive linear deterministic partial differential equations for the pdf of the stochastic

process. Finally, as an illustration, the method is applied to a weakly diffusive nonlinear

wave problem. Although this problem is initially described by a nonlinear stochastic partial

differential equation, it can be recast into a nonlinear stochastic ordinary differential

equation using the method of characteristics.

## Published

2020-02-25

## Issue

## Section

Articles

## How to Cite

*Nonlinear Studies*,

*27*(1). https://www.nonlinearstudies.com/index.php/nonlinear/article/view/1896