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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Brownian subordinators and fractional Cauchy problems


Authors: Boris Baeumer, Mark M. Meerschaert and Erkan Nane
Journal: Trans. Amer. Math. Soc. 361 (2009), 3915-3930
MSC (2000): Primary 60J65, 60J60, 26A33
Published electronically: January 28, 2009
MathSciNet review: 2491905
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Abstract: 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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Additional Information

Boris Baeumer
Affiliation: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand
Email: bbaeumer@maths.otago.ac.nz

Mark M. Meerschaert
Affiliation: Department of Probability and Statistics, Michigan State University, East Lansing, Michigan 48823
Email: mcubed@stt.msu.edu

Erkan Nane
Affiliation: Department of Probability and Statistics, Michigan State University, East Lansing, Michigan 48823
Address at time of publication: Department of Mathematics and Statistics, Auburn University, 340 Parker Hall, Auburn, Alabama 36849
Email: nane@stt.msu.edu, nane@auburn.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04678-9
PII: S 0002-9947(09)04678-9
Keywords: Fractional diffusion, L\'{e}vy process, Cauchy problem, iterated Brownian motion, Brownian subordinator, Caputo derivative
Received by editor(s): June 26, 2007
Received by editor(s) in revised form: November 13, 2007
Published electronically: January 28, 2009
Additional Notes: The second author was partially supported by NSF grant DMS-0417869.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.