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

 

Higher order PDE's and iterated processes


Author: Erkan Nane
Journal: Trans. Amer. Math. Soc. 360 (2008), 2681-2692
MSC (2000): Primary 60J65, 60K99
Published electronically: December 20, 2007
MathSciNet review: 2373329
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Abstract: We introduce a class of stochastic processes based on symmetric $ \alpha$-stable processes, for $ \alpha \in (0,2]$. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric $ \alpha$-stable process. We call them $ \alpha$-time processes. They generalize Brownian time processes studied in Allouba and Zheng (2001), Allouba (2002), (2003), and they introduce new interesting examples. We establish the connection of $ \alpha$-time processes to some higher order PDE's for $ \alpha$ rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.


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Additional Information

Erkan Nane
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906
Address at time of publication: Department of Statistics and Probability, A 413 Wells Hall, Michigan State University, East Lansing, Michigan 48824
Email: enane@math.purdue.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04437-6
PII: S 0002-9947(07)04437-6
Keywords: Iterated Brownian motion, exit time, PDE connection, $\alpha$-stable process, $\alpha$-time process, subordinate killed Brownian motion.
Received by editor(s): May 8, 2006
Published electronically: December 20, 2007
Additional Notes: This work was supported in part by NSF Grant # 9700585-DMS.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.