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

Abstract

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 Schr\"odinger operator formally given by \[ \frac12\Delta \,+\, \frac{\beta}{2}\, \delta_0(\cdot), \qquad \beta>0, \] 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 c\`adl\`ag 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 Schr\"odinger operators.