期刊
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
卷 361, 期 7, 页码 3915-3930出版社
AMER MATHEMATICAL SOC
DOI: 10.1090/S0002-9947-09-04678-9
关键词
Fractional diffusion; Levy process; Cauchy problem; iterated Brownian motion; Brownian subordinator; Caputo derivative
类别
资金
- NSF [DMS-0417869]
- Directorate For Geosciences
- Division Of Earth Sciences [0823965] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0803360] Funding Source: National Science Foundation
A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involves subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes and, consequently, an equivalence between these two families of partial differential equations.
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