arXiv 1003.60852010-03-31math.PRmath-phmath.MP

Bessel过程与双曲布朗运动在不同随机时刻停止

Bessel processes and hyperbolic Brownian motions stopped at different random times

Mirko D'Ovidio, Enzo Orsingher

备注：: 30 pages
发表信息：: Stochastic Processes and their Applications, 121, (2011), 441 - 465
DOI：: 10.1016/j.spa.2010.11.002

摘要：

研究了双曲空间上的迭代Bessel过程R^γ（t），t>0，γ>0及其对应过程，即双曲布朗运动B^{hp}（t），t>0，并导出了它们的概率律.给出了控制I_R（t）=_1R^γ（_2R^γ（t）），t>0和J_R（t）=_1R^γ（|_2R^γ（t）|^2），t>0时的解，并进行了讨论。研究了R^γ（T_t），t>0，B^{hp}（T_t），t>0（其中T_t= \inf{s: B(s)=t}）形式的过程，并导出了许多概率律，包括学生律、反正弦律（也包括它们的非对称形式）、独立偏正稳定随机变量之比的兰珀蒂分布等。对于过程R^γ（T^μ t），t>0（其中T^μ t = \inf{s: B^μ(s)=t}，B^μ是带漂移μ的布朗运动），得到了显式的概率律和控制方程.对于Poincaré半空间H^+_2，H^+_3上的双曲布朗运动，我们研究了B^{hp}（T_t），t>0和相应的控制方程.迭代过程可用于模拟理想化为贝塞尔过程（在欧几里德空间中）或双曲布朗运动（在非欧几里得空间中）的裂缝上的粒子运动。

英文摘要：

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).

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