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

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 $-L_λ$ 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 $\barν^λ_{x,y}$ 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 $λ^{-1}$ is the variance parameter of the Brownian motion. We show that the generalized second lowest eigenvalue of $-L_λ$ divided by $λ$ 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) $λ\to\infty$.