Analytic continuation of time in Brownian motion. Stochastic distributions approach
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910798513700864 |
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| author | Abreu, Luis Daniel Alpay, Daniel Georgiou, Tryphon Jorgensen, Palle |
| author_facet | Abreu, Luis Daniel Alpay, Daniel Georgiou, Tryphon Jorgensen, Palle |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14676 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analytic continuation of time in Brownian motion. Stochastic distributions approach Abreu, Luis Daniel Alpay, Daniel Georgiou, Tryphon Jorgensen, Palle Probability 60J65, 60H, 46F, 46H 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 |
| title | Analytic continuation of time in Brownian motion. Stochastic distributions approach |
| topic | Probability 60J65, 60H, 46F, 46H |
| url | https://arxiv.org/abs/2501.14676 |