Brownian motion and geodesic dynamics on hyperbolic surfaces are connected in many classical ways. Harmonic measure on the ideal boundary, for instance, can be described either through the asymptotic behavior of Brownian motion or through the endpoint of a geodesic ray with uniformly chosen initial direction. Similarly, recurrence of Brownian motion is intimately tied to the ergodicity of the geodesic flow.
I will describe further manifestations of this connection. Brownian loop and excursion measures provide probabilistic expressions for the lengths of closed geodesics and orthogeodesics, as well as for the zeta-regularized determinant of the Laplace–Beltrami operator. Combining these formulas with conformal invariance of two dimensional Brownian motion yields a new identity relating the length spectrum of a hyperbolic surface to the length spectra obtained after introducing any countably many additional cusps.
