Brownian Motion on the Hyperbolic Plane and Selberg Trace Formula

Abstract

We will show that the relation of the heat kernels for the Schro dinger operators with uniform magnetic fields on the hyperbolic plane H 2 (the Maass Laplacians) and for the Schro dinger operators with Morse potentials on R is given by means of a one-dimensional Fourier transform in the framework of stochastic analysis, where the Brownian motion on H 2 plays an important role. By using this relation, we will give the explicit forms of the Green functions. As a typical related problem, we will discuss the Selberg trace formula. The close relation of the trace formula with the corresponding classical mechanics will also be discussed. Academic Press

Exponential Functionals of Lévy Processes

“…Paulsen [55], Example 3.1) as well as in connection with invariant diffusions on the hyperbolic half-plane H (see Bougerol [16], Ikeda and Matsumoto [40], Baldi et al [1], ...). Hence, it is no surprise that the distribution of (22) has been much studied; it has a density given by…”

Polyanalytic relativistic second Bargmann transforms

“…The operator ∆ σ in (1.5) can also be obtained [12] from the σ-weight Maass Laplacian y 2 ∂ 2 x + ∂ 2 y − iσy∂ x on the Poincaré upper half-plane [14]. The condition σ > 1 ensuring the existence of eigenvalues ǫ m (hyperbolic Landau levels) should implies that the magnetic field B = σΩ (z), where Ω stands for the Khäler 2-form on D, has to be strong enough to capture the particle in a closed orbit.…”

The α-Hypergeometric Stochastic Volatility Model

“…Denote by g(σ 2 t, v) = F (t, v) then is solves the partial differential equation The operator H involves a Morse potential, see Grosche (1988), page 228 in Grosche and Steiner (1998), Ikeda and Matsumoto (1999) and the surveys Matsumoto and Yor (2005a) and Matsumoto and Yor (2005b).…”