Brownian motion in negative curvature

Abstract: Brownian motion in the hyperbolic space $H^n$ is rather well-known with a precise formula for the heat kernel, which is the probability density function of the Brownian motion. In this talk, we will talk about the asymptotic formula for the heat kernel in a connected simply connected negatively curved Riemannian manifold X whose metric is lifted from a compact manifold M. As time goes to infinity, we show that the heat kernel $p(t,x,y)$ is asymptotically $e^{-\lambda_0} t^{-3/2} C(x,y)$ where $\lambda_0$ is the bottom of the spectrum of the geometric Laplacian. The proof uses the uniform Harnack inequality on the boundary $\partial X$ as well as the uniform mixing of the geodesic flow on the quotient manifold M. (This is a joint work with François Ledrappier.)

dynamical systems

Audience: researchers in the topic

Bremen Online Dynamics Seminar

Series comments: Talks are approx. 55 min plus discussion. Talks are not recorded.

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