Analytic continuation of time in Brownian motion. Stochastic distributions approach

TL;DR

This paper develops a comprehensive analytic continuation framework for Gaussian processes, especially Brownian motion, using white noise space theory, Hermite functions bounds, and stochastic distributions, leading to new representations of stochastic integrals.

Contribution

It introduces a novel approach to analytic continuation of Brownian motion within white noise space theory, including explicit formulas and new realizations of stochastic integrals.

Findings

Explicit formula for analytically continued white noise process

New bounds for Hermite functions in the complex plane

A novel realization of Hilbert space-valued stochastic integrals

Abstract

With the use of Hida's white noise space theory space theory and spaces of stochastic distributions, we present a detailed analytic continuation theory for classes of Gaussian processes, with focus here on Brownian motion. For the latter, we prove and make use a priori bounds, in the complex plane, for the Hermite functions; as well as a new approach to stochastic distributions. This in turn allows us to present an explicit formula for an analytically continued white noise process, realized this way in complex domain. With the use of the Wick product, we then apply our complex white noise analysis in a derivation of a new realization of Hilbert space-valued stochastic integrals