On an imaginary exponential functional of Brownian motion

TL;DR

This paper explores an imaginary exponential functional of Brownian motion, revealing its connections to reaction-diffusion systems and disordered systems, and provides partial solutions through different analytical approaches.

Contribution

It introduces a new perspective on an imaginary exponential functional of Brownian motion and compares two analytical methods, highlighting their complementarities.

Findings

Identified regimes of interest for the functional

Established links with disordered systems and Kesten variables

Provided partial analytical results

Abstract

We investigate a random integral which provides a natural example of an imaginary exponential functional of Brownian motion. This functional shows up in the study of the binary annihilation process, within the Doi-Peliti formalism for reaction-diffusion systems. The main emphasis is put on the complementarity between the usual Langevin approach and another approach based on the similarity with Kesten variables and other one-dimensional disordered systems. Even though neither of these routes leads to the full solution of the problem, we have obtained a collection of results describing various regimes of interest.