Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.09706 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916042135044096 |
|---|---|
| author | Mian, Barkat |
| author_facet | Mian, Barkat |
| contents | We discuss a probabilistic approximation framework for the three-dimensional attractive point interaction on a finite time horizon. By iterating the Doob transforms of the explicit heat kernel associated with the singular Schrödinger operator formally given by \[ \frac12Δ\,+\, \fracβ{2}\, δ_0(\cdot), \qquad β>0, \] we obtain sub-probability kernels along finite partitions on the punctured domain \[ E_\varepsilon=\{x\in\mathbb R^3:\ |x|>\varepsilon\}, \] which yield a limiting sub-probability kernel via refinement along global dyadic partitions, and we extend this limit to a transition probability kernel on an enlarged space obtained by adjoining a cemetery state. These kernels determine a time-inhomogeneous Markov process on the set of dyadic times, and its step-function interpolations yield càdlàg processes with consistent finite-dimensional distributions and partial tightness properties. The present work may also be viewed as an alternative direct probabilistic approximation scheme for the three-dimensional zero-range homopolymer measure constructed in the work of Cranston, Koralov, Molchanov, and Vainberg, which is constructed as a weak limit of Gibbs measures associated with regularized Schrödinger operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09706 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A dyadic construction of a three-dimensional attractive point interaction Markov family Mian, Barkat Probability Mathematical Physics 60J35, 60J45, 35J10 G.3 We discuss a probabilistic approximation framework for the three-dimensional attractive point interaction on a finite time horizon. By iterating the Doob transforms of the explicit heat kernel associated with the singular Schrödinger operator formally given by \[ \frac12Δ\,+\, \fracβ{2}\, δ_0(\cdot), \qquad β>0, \] we obtain sub-probability kernels along finite partitions on the punctured domain \[ E_\varepsilon=\{x\in\mathbb R^3:\ |x|>\varepsilon\}, \] which yield a limiting sub-probability kernel via refinement along global dyadic partitions, and we extend this limit to a transition probability kernel on an enlarged space obtained by adjoining a cemetery state. These kernels determine a time-inhomogeneous Markov process on the set of dyadic times, and its step-function interpolations yield càdlàg processes with consistent finite-dimensional distributions and partial tightness properties. The present work may also be viewed as an alternative direct probabilistic approximation scheme for the three-dimensional zero-range homopolymer measure constructed in the work of Cranston, Koralov, Molchanov, and Vainberg, which is constructed as a weak limit of Gibbs measures associated with regularized Schrödinger operators. |
| title | A dyadic construction of a three-dimensional attractive point interaction Markov family |
| topic | Probability Mathematical Physics 60J35, 60J45, 35J10 G.3 |
| url | https://arxiv.org/abs/2605.09706 |