Mathematics – Probability
Scientific paper
2011-10-11
Mathematics
Probability
Scientific paper
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) $\lambda\to\infty$.
No associations
LandOfFree
Semi-classical limit of the generalized second lowest eigenvalue of Dirichlet Laplacians on small domains in path spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Semi-classical limit of the generalized second lowest eigenvalue of Dirichlet Laplacians on small domains in path spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Semi-classical limit of the generalized second lowest eigenvalue of Dirichlet Laplacians on small domains in path spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-471721