The Matsumoto and Yor process and infinite dimensional hyperbolic space

32The Matsumoto\,--Yor process is $\int_0^t \exp(2B_s-B_t)\, ds$, where $(B_t)$ is a Brownian motion. It is shown that it is the limit of the radial part of the Brownian motion at the bottom of the spectrum on the hyperbolic space of dimension $q$, when $q$ tends to infinity. Analogous processes on infinite series of non compact symmetric spaces and on regular trees are described