Brownian Dynamics of Globules

TL;DR

This paper establishes the mathematical foundation for the stochastic dynamics of multiple globules with random radii, including existence, uniqueness, and geometric properties of solutions, with applications to chain-like molecules.

Contribution

It proves existence and uniqueness of solutions for a stochastic differential equation modeling globules with random radii and reflections, extending to other hard core models.

Findings

Proved strong solution existence and uniqueness.

Analyzed geometric boundary properties of the process.

Applied techniques to chain-like molecule models.

Abstract

We prove the existence and uniqueness of a strong solution of a stochastic differential equation with normal reflection representing the random motion of finitely many globules. Each globule is a sphere with time-dependent random radius and a center moving according to a diffusion process. The spheres are hard, hence non-intersecting, which induces in the equation a reflection term with a local (collision-)time. A smooth interaction is considered too and, in the particular case of a gradient system, the reversible measure of the dynamics is given. In the proofs, we analyze geometrical properties of the boundary of the set in which the process takes its values, in particular the so-called Uniform Exterior Sphere and Uniform Normal Cone properties. These techniques extend to other hard core models of objects with a time-dependent random characteristic: we present here an application to the random motion of a chain-like molecule.