On exponential functionals of isotropic Gaussian fields and related hypergeometric equations

Speaker(s)

Maciej Wiśniewolski

Affiliation

Uniwersytet Warszawski

Language of the talk

English

Date

Nov. 20, 2025, 12:15 p.m.

Room

room 3160

Title in Polish

On exponential functionals of isotropic Gaussian fields and related hypergeometric equations

Seminar

Seminar of Probability Group

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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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