A stochastic density matrix approach to approximation of probability distributions and its application to nonlinear systems

TL;DR

This paper introduces a stochastic density matrix method for approximating probability density functions, ensuring normalization and non-negativity, and applies it to nonlinear stochastic systems like the Fokker-Planck equation.

Contribution

It proposes a novel SDM approach for PDF approximation that maintains positivity and normalization, with applications to nonlinear stochastic differential equations.

Findings

SDM approximations satisfy normalization and non-negativity

The method can incorporate moment constraints

Application to nonlinear SDEs like the Smoluchowski equation

Abstract

This paper outlines an approach to the approximation of probability density functions by quadratic forms of weighted orthonormal basis functions with positive semi-definite Hermitian matrices of unit trace. Such matrices are called stochastic density matrices in order to reflect an analogy with the quantum mechanical density matrices. The SDM approximation of a PDF satisfies the normalization condition and is nonnegative everywhere in contrast to the truncated Gram-Charlier and Edgeworth expansions. For bases with an algebraic structure, such as the Hermite polynomial and Fourier bases, the SDM approximation can be chosen so as to satisfy given moment specifications and can be optimized using a quadratic proximity criterion. We apply the SDM approach to the Fokker-Planck-Kolmogorov PDF dynamics of Markov diffusion processes governed by nonlinear stochastic differential equations. This leads to an ordinary differential equation for the SDM dynamics of the approximating PDF. As an example, we consider the Smoluchowski SDE on a multidimensional torus.