Self-interacting Brownian motion

TL;DR

This paper investigates how certain interactions, like Coulomb potential, influence the long-term behavior of Brownian bridges, showing that they asymptotically forget initial endpoint distances under specific conditions.

Contribution

It proves that Brownian bridges with pairwise interactions, including Coulomb potential, asymptotically lose memory of their endpoint distances as time grows large.

Findings

Processes forget initial endpoint distances when time tends to infinity.

Interaction effects diminish over long times under specified conditions.

Main example is Coulomb potential interaction.

Abstract

We prove a property of Brownian bridges whose certain time-equidistant sequences of points are pairwise coupled by an interaction. Roughly saying, if the total time span $t$ of the bridge tends to infinity while the distance of its end points is fixed or increases slower than $\sqrt{t}$, the process asymptotically forgets this distance, just as in the absence of interaction. The conclusion remains valid if the bridge interacts in a similar way also with another set of trajectories. The main example for the interaction is the Coulomb potential.