The Bethe Ansatz for Sticky Brownian Motions

TL;DR

This paper models a diffusion process with sticky interactions among Brownian particles, deriving exact solutions using the Bethe ansatz, and analyzes the flow of kernels for specific interactions.

Contribution

It introduces an exactly solvable model of sticky Brownian motions using the Bethe ansatz and analyzes the resulting flow of kernels.

Findings

Exact solutions for the Kolmogorov backward equation with sticky interactions

Characterization of the flow of kernels in the solvable case

Insights into the behavior of sticky Brownian particles

Abstract

We consider a diffusion in $\mathbb{R}^n$ whose coordinates each behave as one-dimensional Brownian motions, that behave independently when apart, but have a sticky interaction when they meet. The diffusion in $\mathbb{R}^n$ can be viewed as the $n$-point motion of a stochastic flow of kernels. We derive the Kolmogorov backwards equation and show that for a specific choice of interaction it can be solved exactly with the Bethe ansatz. We then use our formulae to study the behaviour of the flow of kernels for the exactly solvable choice of interaction.