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

TL;DR

This paper develops the foundational stochastic models for the one-$\delta$ motions in the two-dimensional many-body delta-Bose gas, establishing distributional properties and SDEs crucial for understanding the system.

Contribution

It introduces the stochastic one-$\delta$ motions for the 2D delta-Bose gas and proves their key distributional properties and associated SDEs, laying groundwork for the full N-body case.

Findings

Established the distributional properties of the stochastic one-$\delta$ motions.

Derived and proved SDEs governing the one-$\delta$ motions.

Developed analytical formulas for probability distributions of these motions.

Abstract

This paper is the first 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; see [12,13,14] for the remaining of the series. The main results of this paper establish the foundation by studying the stochastic one-$\delta$ motions, which relate to the two-dimensional many-body delta-Bose gas by turning off all but one delta function, and we prove the central distributional properties and the SDEs. The proofs extend the method in [11] for the stochastic relative motions and develop and use analytical formulas of the probability distributions of the stochastic one-$\delta$ motions.