Beneath the kinetic interpretation of noise

TL;DR

This paper investigates the specific conditions under which the kinetic interpretation of noise aligns with diffusion theory, revealing structural constraints and exploring implications for stochastic modeling and anomalous diffusion.

Contribution

It identifies precise structural conditions for the kinetic interpretation of noise to match diffusion theory, highlighting its non-generic nature and raising new questions in stochastic analysis.

Findings

Kinetic interpretation requires restrictive diffusion tensor structures.

Conditions for noise interpretation are highly constrained and non-generic.

Raises questions on stochastic integrals, heterogeneous media, and anomalous diffusion models.

Abstract

Diffusion theory establishes a fundamental connection between stochastic differential equations and partial differential equations. The solution of a partial differential equation known as the Fokker-Planck equation describes the probability density of the stochastic process that solves a corresponding stochastic differential equation. The kinetic interpretation of noise refers to a prospective notion of stochastic integration that would connect a stochastic differential equation with a Fokker-Planck equation consistent with the Fick law of diffusion, without introducing correction terms in the drift. This work is devoted to identifying the precise conditions under which such a correspondence can occur. One of these conditions is a structural constraint on the diffusion tensor, which severely restricts its possible form and thereby renders the kinetic interpretation of noise a non-generic situation. This point is illustrated through a series of examples. Furthermore, the analysis raises additional questions, including the possibility of defining a stochastic integral inspired by numerical algorithms, the behavior of stochastic transport equations in heterogeneous media, and the development of alternative models for anomalous diffusion. All these topics are addressed using stochastic analytical tools similar to those employed to study the main problem: the existence of the kinetic interpretation of noise.