A dyadic construction of a three-dimensional attractive point interaction Markov family

TL;DR

This paper develops a probabilistic framework for a three-dimensional attractive point interaction, constructing Markov processes via iterative Doob-transforms and dyadic partitions, with future work planned for continuous limits.

Contribution

It introduces a novel dyadic construction of Markov families for 3D point interactions using iterative Doob-transforms and limiting procedures.

Findings

Constructed sub-probability kernels along dyadic partitions.

Extended kernels to transition probabilities on an enlarged space.

Established properties of the resulting cadlag Markov processes.

Abstract

We discuss a probabilistic framework associated with the three-dimensional attractive point interaction under a survival constraint on the punctured domain $E_\varepsilon=\{x\in\mathbb R^3: |x|>\varepsilon\}$. By iterating the Doob-transforms of the fundamental solution of the corresponding singular heat equation, we obtain sub-probability kernels along finite partitions 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 c\`adl\`ag processes with consistent finite-dimensional distributions and partial tightness properties. The analysis of the continuous-time limit of the interpolated processes, as well as the limiting procedure $\varepsilon\downarrow 0$, which recovers the process associated with the three-dimensional point interaction, is deferred to future work.