Invariance of Brownian motion associated with exponential functionals

TL;DR

This paper introduces a new invariance property of Brownian motion involving anticipative path transformations with exponential functionals, expanding understanding of its distributional symmetries.

Contribution

It proves a novel invariance of Brownian motion compatible with time reversal, described via anticipative transformations involving exponential functionals.

Findings

New invariance property of Brownian motion established

Invariance involves anticipative path transformations with exponential functionals

Provides related results on distributional symmetries

Abstract

It is well known that Brownian motion enjoys several distributional invariances such as the scaling property and the time reversal. In this paper, we prove another invariance of Brownian motion that is compatible with the time reversal. The invariance, which seems to be new to our best knowledge, is described in terms of an anticipative path transformation involving exponential functionals as anticipating factors. Some related results are also provided.