Large systems of symmetrized trapped Brownian Bridges and Schrodinger processes

TL;DR

This paper analyzes large systems of symmetrized Brownian bridges and Schrödinger processes, connecting their asymptotic behavior with optimal transport and large deviations, and providing formulas for quantum trace asymptotics.

Contribution

It establishes the large N behavior of symmetrized Brownian systems, linking it to Schrödinger's optimal transport and Donsker-Varadhan rate functions, with explicit asymptotic formulas.

Findings

Connection between symmetrized Brownian bridges and Schrödinger's optimal transport

Large deviations rate function matches Donsker-Varadhan rate function

Explicit asymptotic formula for symmetrized trace of quantum Hamiltonian

Abstract

Consider a large system of $N$ Brownian motions in $\R ^d$ fixed on a time interval $[0,\beta]$ with symmetrized initial and terminal conditions, under the influence of a trap potential. Such systems describe systems of bosons at positive temperatures confined in a spatial domain. We describe the large $N$ behavior of the averaged path (that is, their empirical path measure) and its connection with a well known optimal transport problem formulated by Erwin Schr\"odinger. We also explore the asymptotic behavior of the Brownian motions in terms of Large Deviations. In particular, the rate function that governs the mean of occupation measures turns out to be the well-known Donsker-Varadhan rate function. We therefore prove a simple formula for the large $N$ asymptotic of the symmetrized trace of $e^{-\beta \Hcal_N}$, where $\Hcal_N$ is an $N$ particle Hamilton operator in a trap