Elastic drifted Brownian motions and non-local boundary conditions

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 functional associated with non-local boundary conditions of tempered type. By exploiting such connections we write some functionals of the drifted Brownian motion in terms of a simple (positive and non-decreasing) process, the inverse of a tempered subordinator. In our view, such a representation is useful in many applications and brings new light on dynamic boundary value problems.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This article explores the connection between elastic drifted Brownian motions and inverses to tempered subordinators, and establishes a link between multiplicative functionals and dynamical boundary conditions. By representing functionals of the drifted Brownian motion as the inverse of a tempered subordinator, the problem is simplified.