The probability density function for Brownian motion in real and complex hyperbolic spaces

In [3], Nobuyuki Ikeda and Hiroyuki Matsumoto compute the probability density function for Brownian Motion in the Real Hyperbolic Plane. There, taking expectations of eigen functions of the Real Hyperbolic Laplacian, they obtain an integral equation containing the density function. The density function is then found, essentially, by observing that the density function reduces to a function of the Hyperbolic distance and then taking a Fourier transform and inverting the resulting Abel transform. Their method is extended in this paper to Real and Complex Hyperbolic Spaces of arbitrary dimension, the computations for Complex Hyperbolic Space relying upon its Symmetric Space presentation.