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

Yu-Ting Chen

TL;DR

This work provides a probabilistic framework for the two-dimensional $N$-body delta-Bose gas with $N\ge3$ by constructing stochastic motions that encode contact interactions. It derives Feynman–Kac-type representations for analytic path integrals of the model using a ${\mathbb C}^N$-valued Markov process $\{\mathscr Z_t\}$, with interaction weights ${\boldsymbol \beta}$ and ${\boldsymbol w}$, and local-time functionals that capture singular boundary behavior. A key feature is the independence of the final expectations from the weights, reflecting a cancellation in the multiplicative functional and enabling a unified FK formula (which also covers $N=2$ as a minor variant). The results extend the stochastic path-integral program to two-dimensional many-body systems with renormalized delta interactions, providing tools to study semigroups and spectral properties probabilistically. Overall, the paper builds a rigorous link between analytic solutions and stochastic representations, offering new insights into singular interactions in quantum many-body dynamics.

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-$δ$ 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.

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

TL;DR

This work provides a probabilistic framework for the two-dimensional -body delta-Bose gas with by constructing stochastic motions that encode contact interactions. It derives Feynman–Kac-type representations for analytic path integrals of the model using a -valued Markov process , with interaction weights and , and local-time functionals that capture singular boundary behavior. A key feature is the independence of the final expectations from the weights, reflecting a cancellation in the multiplicative functional and enabling a unified FK formula (which also covers as a minor variant). The results extend the stochastic path-integral program to two-dimensional many-body systems with renormalized delta interactions, providing tools to study semigroups and spectral properties probabilistically. Overall, the paper builds a rigorous link between analytic solutions and stochastic representations, offering new insights into singular interactions in quantum many-body dynamics.

Abstract

This paper is the third in a series devoted to constructing stochastic motions for the two-dimensional -body delta-Bose gas for all integers and establishing the associated Feynman-Kac-type formulas. The main results here prove the Feynman-Kac-type formulas by using the stochastic many- 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 -body delta-Bose gas obtained in [4]. For completeness, the main theorem includes the formula for , which is a minor modification of the Feynman--Kac-type formula proven in [5] for the relative motions.
Paper Structure (7 sections, 7 theorems, 100 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Analytic path integrals
  3. Main results of this paper
  4. Stochastic path integrals
  5. Representations by the stochastic one-$\boldsymbol \delta$ motions
  6. Proof of the Feynman--Kac-type formulas
  7. Transformations of the intermediate parts

Key Result

Theorem 1.1

Let $N\geq 2$ be an integer, ${\boldsymbol \beta}\in (0,\infty)^{\mathcal{E}_N}$, and $\boldsymbol w\in (0,\infty)^{\mathcal{E}_N}$. For any $z_0\in {\mathbb C}^{\it N}$ with $z_0^{j\prime}-z_0^j\neq 0$ for all $\mathbf j=(j\prime,j)\in\mathcal{E}_N$, where $K_\nu(\cdot)$ is the Macdonald function of index $\nu$, and $\{L^\mathbf i_t\}$ is the local time of the stochastic relative motion$\{Z^{\

Figures (1)

  • Figure 1.1: The figure illustrates the graphical representation of $P^{{\boldsymbol \beta};\mathbf i_1,\mathbf i_2,\mathbf i_3,\mathbf i_4,\mathbf i_5}_{s_1,s_2,s_3,s_4,s_5,t}f(z_0)$ defined in C:DBG-3, with $N=4$, $\mathbf i_1=(2,1)$, $\mathbf i_2=(3,2)$, $\mathbf i_3=(2,1)$, $\mathbf i_4=(4,3)$ and $\mathbf i_5=(2,1)$.

Theorems & Definitions (11)

  • Theorem 1.1: Feynman--Kac-type formulas
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4: Independence of weights
  • Remark 2.5: Normalization of the local times
  • Lemma 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proposition 2.10
  • ...and 1 more