Strong solutions to singular SDEs and application to the Lennard-Jones potential

TL;DR

This paper establishes the existence and uniqueness of strong solutions for a class of singular stochastic differential equations, including those modeling particles with Lennard-Jones potentials, using probabilistic regularization techniques.

Contribution

It introduces a novel probabilistic regularization method to prove well-posedness for SDEs with boundary-singular drifts, applied to Lennard-Jones particle systems.

Findings

Solutions exist globally without explosion under certain conditions

The approach handles singularities at boundaries and at the origin

Application to Lennard-Jones systems demonstrates practical relevance

Abstract

We prove existence and uniqueness of strong solutions to a large class of autonomous stochastic differential equations on an open domain, where the drift exhibits a singular behaviour at the boundary. The main result involves a drift composed of the gradient of a singular potential and an additional possibly singular force. In order to achieve the well-posedness of the model, we employ a probabilistic regularization approach. Under suitable conditions, it is shown that the explosion time of the solution process is infinite. The result is finally applied to the case of an interacting particle system subject to a Lennard-Jones potential, which is singular at the origin.