On exponential functionals of isotropic Gaussian fields and related hypergeometric equations

Prelegent(ci)

Maciej Wiśniewolski

Afiliacja

Uniwersytet Warszawski

Język referatu

angielski

Termin

20 listopada 2025 12:15

Pokój

p. 3160

Tytuł w języku polskim

On exponential functionals of isotropic Gaussian fields and related hypergeometric equations

Seminarium

Seminarium Zakładu Rachunku Prawdopodobieństwa

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I will discuss a class of exponential integral functionals of isotropic Gaussian fields on the unit interval whose expectations satisfy a specific second-order ordinary differential equation. In a certain sense, this equation can be reduced to the Gauss hypergeometric equation, leading to explicit representations in terms of hypergeometric functions. This extends to smooth isotropic kernels a phenomenon previously observed in the logarithmic case by Rémy and Zhu (2020). We show that, for a particular form of exponential integral functional associated with an isotropic Gaussian field, the second-order ODE with functional coefficients can be reduced to a PDE with constant coefficients. Under natural boundary conditions, the solution is unique and analytic, and the coefficients of the PDE admit a probabilistic representation via a specially constructed Poisson point process.

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