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Two-dimensional delta-Bose gas: skew-product relative motions

Yu-Ting Chen

TL;DR

This work develops a stochastic path integral for the two-dimensional delta-Bose gas by deriving a non-exponential Feynman–Kac-type formula for the relative motion. The radial component is realized as BES$(0,β\downarrow)$ and the full relative motion as a complex skew-product diffusion, with an angular clock driven by circular Brownian motion. Two complementary proofs are provided: an excursion-based method and a lower-dimensional, SDE-based approach using BES$(-α,β\downarrow)$ and Macdonald-function drift ratios, which together establish strong well-posedness and a precise irregular drift structure. The results illuminate the non-Gaussian nature of the 2D delta interaction and connect to moments and critical phenomena in the two-dimensional stochastic heat equation, while also introducing a tractable lower-dimensional approximation framework that could extend to many-body settings.

Abstract

We prove a Feynman-Kac-type formula for the relative motion of the two-body delta-Bose gas in two dimensions. The multiplicative functional is not exponential, and the process is a skew-product diffusion uniquely extended in law, in the sense of Erickson [30], from ${\rm BES}(0,β{\downarrow})$ of Donati-Martin and Yor [27] as the radial part. We give two different proofs of the formula. The first uses the original excursion characterization of ${\rm BES}(0,β\downarrow)$, and the second is via the lower-dimensional Bessel processes at the expectation level. The latter proof contrasts the long-standing approach for delta-function interactions by adding mollifiers to the Laplacians since the present approximations are from "lower, fractional dimensions." Moreover, the second proof conducts a new study of ${\rm BES}(0,β\downarrow)$ as an SDE since we handle the drift via certain ratios of the Macdonald functions. The properties proven include the strong well-posedness and comparison of the SDE of ${\rm BES}(0,β\downarrow)$ for all initial conditions. In particular, this well-posedness contrasts the fact that the skew-product diffusion for the Feynman-Kac-type formula has a singular drift of $L^p_{\rm\tiny loc}$-integrability only for $ p\leq 2$.

Two-dimensional delta-Bose gas: skew-product relative motions

TL;DR

This work develops a stochastic path integral for the two-dimensional delta-Bose gas by deriving a non-exponential Feynman–Kac-type formula for the relative motion. The radial component is realized as BES and the full relative motion as a complex skew-product diffusion, with an angular clock driven by circular Brownian motion. Two complementary proofs are provided: an excursion-based method and a lower-dimensional, SDE-based approach using BES and Macdonald-function drift ratios, which together establish strong well-posedness and a precise irregular drift structure. The results illuminate the non-Gaussian nature of the 2D delta interaction and connect to moments and critical phenomena in the two-dimensional stochastic heat equation, while also introducing a tractable lower-dimensional approximation framework that could extend to many-body settings.

Abstract

We prove a Feynman-Kac-type formula for the relative motion of the two-body delta-Bose gas in two dimensions. The multiplicative functional is not exponential, and the process is a skew-product diffusion uniquely extended in law, in the sense of Erickson [30], from of Donati-Martin and Yor [27] as the radial part. We give two different proofs of the formula. The first uses the original excursion characterization of , and the second is via the lower-dimensional Bessel processes at the expectation level. The latter proof contrasts the long-standing approach for delta-function interactions by adding mollifiers to the Laplacians since the present approximations are from "lower, fractional dimensions." Moreover, the second proof conducts a new study of as an SDE since we handle the drift via certain ratios of the Macdonald functions. The properties proven include the strong well-posedness and comparison of the SDE of for all initial conditions. In particular, this well-posedness contrasts the fact that the skew-product diffusion for the Feynman-Kac-type formula has a singular drift of -integrability only for .
Paper Structure (29 sections, 35 theorems, 263 equations, 2 figures)

This paper contains 29 sections, 35 theorems, 263 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The two-dimensional many-body delta-Bose gas
  3. Statistical moments of the two-dimensional SHE at criticality
  4. Overview of main results
  5. Formulations of main results
  6. The stochastic relative position
  7. First stochastic path integral representation
  8. The stochastic relative motion
  9. The Feynman--Kac-type formula for the relative motion
  10. Second look at the stochastic relative motion: regularity of the drift
  11. Second look at the stochastic relative position
  12. The lower-dimensional approximations
  13. Excursions of the stochastic relative position
  14. The stochastic relative position: the SDE and the clock for time change
  15. Proof of Theorem \ref{['thm:2']} (1${^\circ}$) for $\alpha\in (0,1/2)$
  16. ...and 14 more sections

Key Result

Proposition 2.6

For any $\alpha \in (0,1/2)$, there exists a $\Bbb C$-valued continuous strong Markov process $\{Z_t\}_{t\geq 0}$ such that $\{\lvert Z_t \rvert\}\sim {\rm BES}(-\alpha)$, and Condition cond:sp with ${\rm BES}(0,\beta{\downarrow})$ replaced by ${\rm BES}(-\alpha)$ holds. The analogous existence resu

Figures (2)

  • Figure 2.1: Relationships among the main radial objects in this paper
  • Figure 2.2: A Mathematica plot of the drift coefficients $\mu^{\beta{\downarrow}}_\alpha$ in \ref{['SDE:DMY']} for various values of $\alpha$ and $\beta=1/2$ --- the plot for $\mu^{(1/2){\downarrow}}_{0}(x)$ illustrates a very sharp cusp so that the value at $x=0$ should be understood as $1$

Theorems & Definitions (45)

  • Remark 1.1
  • Remark 2.1
  • Remark 2.3: Many-body delta-Bose gas
  • Remark 2.4
  • Proposition 2.6
  • Proposition 2.8
  • Proposition 2.9
  • Theorem 2.10
  • Corollary 2.12
  • Proposition 2.13
  • ...and 35 more