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^{hp}(t), t>0 are examined and their probability laws derived. The higher-order partial differential equations governing the distributions of I_R(t)=_1R^γ(_2R^γ(t)), t>0 and J_R(t) =_1R^γ(|_2R^γ(t)|^2), t>0 are obtained and discussed. Processes of the form R^γ(T_t), t>0, B^{hp}(T_t), t>0 where T_t=\inf{s: B(s)=t} are examined and numerous probability laws derived, including the Student law, the arcsin laws (also their asymmetric versions), the Lamperti distribution of the ratio of independent positively skewed stable random variables and others. For the process R^γ(T^μ_t), t>0 (where T^μ_t = \inf{s: B^μ(s)=t} and B^μis 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 we study B^{hp}(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).

30 pages

References in corpus (4)

• Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
• Iterated Brownian motion in an open set
• Space-time duality for fractional diffusion
• Cascades of Particles Moving at Finite Velocity in Hyperbolic Spaces

Cited by in corpus (4)

• Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes
• Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes
• Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
• Drifted Brownian motions governed by fractional tempered derivatives