Stochastic motions of the two-dimensional many-body delta-Bose gas, II: Many-$\delta$ motions

TL;DR

This paper develops stochastic models for the two-dimensional many-body delta-Bose gas, constructing processes where particles behave like Brownian motions conditioned on contact interactions, and introduces a new 'no-triple-contacts' property.

Contribution

It extends the construction of stochastic motions to general N-particle systems and proves the pathwise 'no-triple-contacts' phenomenon in this context.

Findings

Constructed stochastic many-$\

Proved the 'no-triple-contacts' property at the pathwise level.

Established Feynman-Kac-type formulas for the system.

Abstract

This paper is the second in a series devoted to constructing stochastic motions for the two-dimensional $N$-body delta-Bose gas for all integers $N\geq 3$ and establishing the associated Feynman-Kac-type formulas. The main results here construct and study the more general stochastic many-$\delta$ motions for $N$ particles. They have the interpretation of independent two-dimensional Brownian motions conditioned to attain the contact interactions that realize multiple two-body $\delta$-function potentials. For the construction, we transform the stochastic one-$\delta$ motions studied in [7] by Girsanov's theorem locally before a pair of particles with different initial conditions begins to contact each other. The strong Markov processes with lifetime thus obtained are concatenated by using the "no-triple-contacts" (NTC). This NTC phenomenon appears in the functional integral solutions of the two-dimensional many-body delta-Bose gas obtained earlier and is now proven at the pathwise level to a generalized degree.