Pathwise structure of the three-dimensional attractive one-point interaction diffusion

Barkat Mian

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arXiv · 2606.08008

Pathwise structure of the three-dimensional attractive one-point interaction diffusion

Barkat Mian

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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, corresponding to the singular Schr\"odinger Hamiltonian \[ \frac12\Delta+\frac{\beta}{2}\delta_0, \qquad \beta>0. \] We identify 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.

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Barkat Mian. 2026-06-06. Pathwise structure of the three-dimensional attractive one-point interaction diffusion. https://arxiv.org/abs/2606.08008

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