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

  • Mathieu Mure-Ravaud UC Davis
  • M. Levent Kavvas University of California Davis, Civil and Environmental Engineering Department
  • Ali Ercan University of California Davis, Civil and Environmental Engineering Department

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