Elastic drifted Brownian motions and non-local boundary conditions

TL;DR

This paper explores the relationship between elastic drifted Brownian motions and tempered subordinators, linking non-local boundary conditions with multiplicative functionals, and providing new representations useful for applications.

Contribution

It establishes a novel connection between elastic drifted Brownian motions and inverse tempered subordinators, linking non-local boundary conditions with multiplicative functionals.

Findings

Connection between elastic drifted Brownian motions and tempered subordinators.

Representation of functionals using simple non-decreasing processes.

Link between multiplicative functionals and fractional boundary conditions.

Abstract

We provide a deep connection between elastic drifted Brownian motions and inverses to tempered subordinators. Based on this connection, we establish a link between multiplicative functionals and dynamical boundary conditions given in terms of non-local equations in time. Indeed, we show that the multiplicative functional associated to the elastic Brownian motion with drift is equivalent to a multiplicative functional associated with fractional boundary conditions of tempered type. By exploiting such connection we write some functionals in terms of a simple (positive and non-decreasing) process. In our view, such a representation is useful in many applications.