FRACTIONAL DIFFUSION EQUATIONS AND PROCESSES WITH RANDOMLY VARYING TIME

• 成果类型: Article
• 署名作者: Orsingher, Enzo; Beghin, Luisa
• 署名单位: Sapienza University Rome
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP401
• 发表日期: 2009
• 页码: 206-249
• 关键词: kinetic-equations; pde connection; ORDER; LAW

摘要

In this paper the solutions u(nu) = u(nu) (x, t) to fractional diffusion equations of order 0 < v <= 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order nu = 1/2(n), n >= 1, we show that the solutions u(1/2n) correspond to the distribution of the n-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order nu = 2/3(n), n >= 1, is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that u(nu) coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u(nu) and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.