Analytic continuation of time in Brownian motion. Stochastic distributions approach

2025/01/24 by Luís Daniel Abreu, Daniel Alpay, Abreu, Luis Daniel +5
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #46F #46H #60H #60J65 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2501.14676

openalex publication_date 2025/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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

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