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Transactions of the American Mathematical Society

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Brownian-time processes: The PDE connection II and the corresponding Feynman-Kac formula


Author: Hassan Allouba
Journal: Trans. Amer. Math. Soc. 354 (2002), 4627-4637
MSC (2000): Primary 60H30, 60J45, 60J35; Secondary 60J60, 60J65
Published electronically: June 4, 2002
MathSciNet review: 1926892
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Abstract: We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth-order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth-order PDEs in which the initial function plays an important role in the PDE itself, not only as initial data. We connect two such types of interesting and new PDEs to BTPs. The first is obtained by running the BTP and then integrating along its path, and the second type of PDEs is related to what we call the Feynman-Kac formula for BTPs. A special case of the second type is a step towards a probabilistic solution to linearized Cahn-Hilliard and Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.


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Additional Information

Hassan Allouba
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405-7106
Address at time of publication: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240-0001
Email: allouba@indiana.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03074-X
Keywords: Brownian-time processes, initially perturbed fourth order PDEs, Brownian-time Feynman-Kac formula, iterated Brownian motion
Received by editor(s): November 5, 2001
Received by editor(s) in revised form: April 9, 2002
Published electronically: June 4, 2002
Additional Notes: Supported in part by NSA grant MDA904-02-1-0083
Article copyright: © Copyright 2002 American Mathematical Society