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

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 $$\frac{1}{2}\mathrm{\Delta }+\frac{\beta }{2}{\delta }_0(\cdot ),\beta >0,$$ we obtain sub-probability kernels along finite partitions on the punctured domain $$E_{\epsilon }=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>\epsilon \},$$ 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.