Bessel processes and hyperbolic Brownian motions stopped at different random times

Abstract

Iterated Bessel processes R γ ( t ) , t > 0 , γ > 0 and their counterparts on hyperbolic spaces, i.e. hyperbolic Brownian motions B h p ( t ) , t > 0 are examined and their probability laws derived. The higher-order partial differential equations governing the distributions of I R ( t ) = R 1 γ 1 ( R 2 γ 2 ( t ) ) , t > 0 and J R ( t ) = R 1 γ 1 ( R 2 γ 2 ( t ) 2 ) , t > 0 are obtained and discussed. Processes of the form R γ ( T t ) , t > 0 , B h p ( T t ) , t > 0 where T t = inf { s ≥ 0 : B ( s ) = t } are examined and numerous probability laws derived, including the Student law, the arcsine laws (also their asymmetric versions), the Lamperti distribution of the ratio of independent positively skewed stable random variables and others. For the random variable R γ ( T t μ ) , t > 0 (where T t μ = inf { s ≥ 0 : B μ ( s ) = t } and B μ is a Brownian motion with drift μ ), the explicit probability law and the governing equation are obtained. For the hyperbolic Brownian motions on the Poincaré half-spaces H 2 + , H 3 + (of respective dimensions 2 , 3 ) we study B h p ( T t ) , t > 0 and the corresponding governing equation. Iterated processes are useful in modelling motions of particles on fractures idealized as Bessel processes (in Euclidean spaces) or as hyperbolic Brownian motions (in non-Euclidean spaces).