Blending Brownian motion and heat equation

TL;DR

This paper introduces a novel coupling method between Brownian motion, the heat equation, and the Langevin equation, demonstrating how to preserve statistical properties while regularizing microscopic dynamics.

Contribution

It proposes an original approach to couple microscopic Langevin dynamics with macroscopic PDEs, enabling regularized stochastic processes that retain key statistical features.

Findings

Successful coupling of Brownian motion with heat equation.

Numerical results confirm preservation of statistical properties.

Reduction of microscopic degrees of freedom with macroscale dynamics.

Abstract

In this short communication we present an original way to couple the Brownian motion and the heat equation. More in general, we suggest a way for coupling the Langevin equation for a particle, which describes a single realization of its trajectory, with the associated Fokker-Planck equation, which instead describes the evolution of the particle's probability density function. Numerical results show that it is indeed possible to obtain a regularized Brownian motion and a Brownianized heat equation still preserving the global statistical properties of the solutions. The results also suggest that the more macroscale leads the dynamics the more one can reduce the microscopic degrees of freedom.