Fractional Boundary Value Problems and elastic sticky Brownian motions, II: Non-local dynamic boundary conditions on smooth domains

TL;DR

This paper investigates fractional boundary value problems related to sticky diffusion processes on bounded domains, characterizing their boundary behaviors and introducing fractional dynamic boundary conditions that model trap effects.

Contribution

It introduces fractional boundary conditions for sticky diffusions, providing a novel description of boundary trapping effects using fractional calculus and time-change techniques.

Findings

Sticky diffusions can spend infinite mean time on boundaries with fractional conditions.

Fractional boundary conditions model trap effects at the boundary.

The approach links fractional PDEs with boundary trapping phenomena.

Abstract

Sticky diffusion processes on bounded domains spend finite time (and finite mean time) on the lower-dimensional space given by the boundary. Once the process hits the boundary, then it starts again after a random amount of time. While on the boundary it can stay or move according to dynamics that are different from those in the interior. Such processes may be characterized by a time-derivative appearing in the boundary condition for the governing problem. We use time changes obtained by right-inverses of suitable processes in order to describe fractional sticky conditions and the associated boundary behaviours. We obtain that fractional boundary value problems (involving fractional dynamic boundary conditions) lead to sticky diffusions spending an infinite mean time (and finite time) on a lower-dimensional boundary. Such a behaviour can be associated with a trap effect in the macroscopic point of view.