The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $$\mathbb {M}$$
M
is a stochastic process that models a random perturbation of the geodesic flow. If $$\mathbb {M}$$
M
is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $$L^2$$
L
2
-spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.
The kinetic Brownian motion on the sphere bundle of a Riemannian manifold M is a stochastic process that models a random perturbation of the geodesic flow. If M is a orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the L 2 -spectrum of the infinitesimal generator of a time rescaled version of the process converges to the Laplace spectrum of the base manifold.
Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no L 2 -eigenvalues ≥ 1/4. In this article we prove a generalization of this result for the joint L 2 -eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces Γ\G/K of higher rank. We derive dynamical assumptions on the Γ-action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.