Area Distribution of Elastic Brownian Motion

TL;DR

This paper explores the area distributions of elastic Brownian motion using self-adjoint extensions of quantum Hamiltonians, highlighting boundary condition impacts and extending to Brownian bridges.

Contribution

It introduces a novel method employing self-adjoint extensions to analyze boundary conditions in elastic Brownian motion.

Findings

Derived excursion and meander area distributions for elastic Brownian motion.

Demonstrated the effectiveness of self-adjoint extension in boundary condition analysis.

Provided insights into Brownian bridge areas with boundary considerations.

Abstract

We calculate the excursion and meander area distributions of the elastic Brownian motion by using the self adjoint extension of the Hamiltonian of the free quantum particle on the half line. We also give some comments on the area of the Brownian motion bridge on the real line with the origin removed. We will stress on the power of self adjoint extension to investigate different possible boundary conditions for the stochastic processes.