The Matsumoto and Yor process and infinite dimensional hyperbolic space

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The Matsumoto--Yor process is ∫_0ᵗ (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.