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Analytic continuation of time in Brownian motion. Stochastic distributions approach

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

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

Analytic continuation of time in Brownian motion. Stochastic distributions approach

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
Paper Structure (10 sections, 17 theorems, 91 equations)

This paper contains 10 sections, 17 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The white noise space
  4. Stochastic test functions and stochastic distributions
  5. The complex Brownian motion $B_z$
  6. The complex white noise $N_z$
  7. Stochastic integral
  8. Real versus imaginary time
  9. A more general family of processes
  10. An extension of Itô’s formula to the case of complex time: The quadratic case

Key Result

Proposition 2.1

Let $|z|\le R$. Then

Theorems & Definitions (35)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Definition 3.1
  • Proposition 3.2
  • ...and 25 more