Pathwise Structure of the Three-Dimensional Attractive One-Point Interaction Diffusion

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This talk will be broadcast at 13:00 BT / 14:0 CET / 15:00 EET, September 30th, 2026 on Zoom only.

Meeting-ID: 667 4776 1513

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Please note that we start 30 mins earlier than usually.

Abstract

We study the pathwise behavior of the three-dimensional attractive one-point interaction diffusion whose law was constructed by Cranston, Koralov, Molchanov, and Vainberg in [Random Oper. Stoch. Equ. 18 (2010)], corresponding to the singular Schrödinger Hamiltonian

$\frac{1}{2}\Delta+\frac{\beta}{2}\delta_0(\cdot)$, $\beta>0.$

We derive a local stochastic differential equation satisfied by the process away from the origin and use it to construct a natural submartingale whose increasing component in the Doob-Meyer decomposition is supported on the set of times at which the process visits the origin. In particular, we show that the process visits the origin with positive probability and that the law conditioned on avoiding the origin is three-dimensional Wiener measure. In this talk, I will also present a comparison of these results with those for the corresponding two-dimensional attractive one-point interaction diffusion, obtained in joint work with Jeremy Clark [Electron. J. Probab. 30 (2025)].

About Barkat

Barkat is a Research Assistant Professor in the Department of Mathematics at the University of Tennessee, Knoxville. He earned his Ph.D. in Mathematics from the University of Mississippi in May 2025 under the supervision of Jeremy Clark. Barkat works in probability theory and stochastic processes, with a specific interest in the study of Doob transformed singular diffusions corresponding to the heat equation with one-point interaction (d=2 and 3). He is also interested in singular SDEs and the stochastic heat equation with Lévy noise.