Semi-classical limit of the generalized second lowest eigenvalue of Dirichlet Laplacians on small domains in path spaces

TL;DR

This paper investigates the asymptotic behavior of the second lowest eigenvalue of a Dirichlet Laplacian on path space over a Riemannian manifold, revealing its convergence to the Hessian's lowest eigenvalue in the semi-classical limit.

Contribution

It establishes the limit of the scaled second eigenvalue of the Ornstein-Uhlenbeck operator on small path space neighborhoods, linking it to the Hessian of the energy function.

Findings

Eigenvalue scaled by variance parameter converges to Hessian eigenvalue

Results apply to neighborhoods of unique minimal geodesics

Provides semi-classical analysis on path space eigenvalues

Abstract

Let $M$ be a complete Riemannian manifold. Let $P_{x,y}(M)$ be the space of continuous paths on $M$ with fixed starting point $x$ and ending point $y$. Assume that $x$ and $y$ is close enough such that the minimal geodesic $c_{xy}$ between $x$ and $y$ is unique. Let $-L_{\lambda}$ be the Ornstein-Uhlenbeck operator with the Dirichlet boundary condition on a small neighborhood of the geodesic $c_{xy}$ in $P_{x,y}(M)$. The underlying measure $\bar{\nu}^{\lambda}_{x,y}$ of the $L^2$-space is the normalized probability measure of the restriction of the pinned Brownian motion measure on the neighborhood of $c_{xy}$ and $\lambda^{-1}$ is the variance parameter of the Brownian motion. We show that the generalized second lowest eigenvalue of $-L_{\lambda}$ divided by $\lambda$ converges to the lowest eigenvalue of the Hessian of the energy function of the $H^1$-paths at $c_{xy}$ under the small variance limit (semi-classical limit) $\lambda\to\infty$.