Analytic continuation of time in Brownian motion. Stochastic distributions approach

Luis Daniel Abreu; Daniel Alpay; Tryphon Georgiou; Palle Jorgensen

doi:10.48550/arxiv.2501.14676

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Analytic continuation of time in Brownian motion. Stochastic distributions approach

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Analytic continuation of time in Brownian motion. Stochastic distributions approach

Luis Daniel Abreu, Daniel Alpay, Tryphon Georgiou and Palle Jorgensen

ArXiV.org

Cornell University

01/24/2025

DOI: 10.48550/arxiv.2501.14676

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https://doi.org/10.48550/arxiv.2501.14676View

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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

Mathematics - Probability

Details

Title: Subtitle: Analytic continuation of time in Brownian motion. Stochastic distributions approach
Creators: Luis Daniel Abreu - University of Vienna
Daniel Alpay - Chapman University
Tryphon Georgiou - University of California, Irvine
Palle Jorgensen - University of Iowa
Resource Type: Preprint
Publication Details: ArXiV.org
DOI: 10.48550/arxiv.2501.14676
ISSN: 2331-8422
Publisher: Cornell University; Ithaca, New York
Language: English
Date posted: 01/24/2025
Academic Unit: Mathematics
Record Identifier: 9984781272702771

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