Stochastic motions of the two-dimensional many-body delta-Bose gas, III: Path integrals

TL;DR

This paper establishes Feynman-Kac formulas for the stochastic motions of the two-dimensional N-body delta-Bose gas, extending previous work by deriving new multiplicative functionals from analytic solutions and covering all N ≥ 2.

Contribution

It provides a rigorous construction of stochastic motions and Feynman-Kac formulas for the 2D N-body delta-Bose gas for all N ≥ 2, using new multiplicative functionals.

Findings

Proved Feynman-Kac-type formulas for N ≥ 3 using stochastic delta motions.

Derived new form of multiplicative functionals from analytic solutions.

Included the N=2 case with a minor modification of existing formulas.

Abstract

This paper is the third 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 prove the Feynman-Kac-type formulas by using the stochastic many-$\delta$ motions from [7] as the underlying diffusions. The associated multiplicative functionals show a new form and are derived from the analytic solutions of the two-dimensional $N$-body delta-Bose gas obtained in [4]. For completeness, the main theorem includes the formula for $N=2$, which is a minor modification of the Feynman--Kac-type formula proven in [5] for the relative motions.