Brownian subordinators and fractional Cauchy problems
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- by Boris Baeumer, Mark M. Meerschaert and Erkan Nane PDF
- Trans. Amer. Math. Soc. 361 (2009), 3915-3930 Request permission
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.References
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Additional Information
- Boris Baeumer
- Affiliation: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand
- MR Author ID: 688464
- 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
- MR Author ID: 782700
- Email: nane@stt.msu.edu, nane@auburn.edu
- 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.
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 3915-3930
- MSC (2000): Primary 60J65, 60J60, 26A33
- DOI: https://doi.org/10.1090/S0002-9947-09-04678-9
- MathSciNet review: 2491905