We use the Brownian motion to express the harmonic measure, lengths of closed geodesics, length of orthogeodesics, zeta-regularized determinant of the Laplace-Beltrami operator of a hyperbolic surface. In particular, the lengths of closed geodesics are expressed using Brownian loop measure introduced by Lawler and Werner.

This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a hyperbolic surface and that of the same surface with an arbitrary number of additional cusps. This is mainly based on a joint work with Yuhao Xue (IHES).