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On a stochastic version of transfer operators

Ksenia Fedosova

Montag, 7. Juni 2021 00:00Uhr

About thirty years ago, the classical statistical mechanics inspired a method that allows to obtain some information on the automorphic forms. The method, called the transfer operator approach, involves a construction of a so-called transfer operator from a certain discretisation of the geodesic flow on the manifold. For a modular surface, this transfer operator is ultimately connected to a Gauss map. One can show that the 1-eigenfunctions of this operator correspond via a certain integral transform to the eigenfunctions of the Laplace operator.

In this talk, we try to construct an analogue of the transfer operator, using the Brownian paths on the manifold instead of the geodesics. We obtain an operator, whose 1-eigenfunctions turn out to be the boundary forms of eigenfunctions of the Laplace operator. We investigate some of its properties and hopefully show the connection with quantum modular forms.