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October 2001 Brownian-time processes: The PDE Connection and the half-derivative generator
Hassan Allouba, Weian Zheng
Ann. Probab. 29(4): 1780-1795 (October 2001). DOI: 10.1214/aop/1015345772


We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic partial differential equations (PDE’s). We then study their exit problem as they exit nice domains in $\mathbb{R}^d$ , and connect it to elliptic PDE’s. We show that these processes have the peculiar property that they solve fourth order parabolic PDE’s, but their exit distribution—at least in the standard Brownian time process case—solves the usual second order Dirichlet problem. We recover fourth order PDE’s in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense.


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Hassan Allouba. Weian Zheng. "Brownian-time processes: The PDE Connection and the half-derivative generator." Ann. Probab. 29 (4) 1780 - 1795, October 2001.


Published: October 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1018.60066
MathSciNet: MR1880242
Digital Object Identifier: 10.1214/aop/1015345772

Primary: 60H30, 60J35, 60J45, 60J65
Secondary: 60J60

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 4 • October 2001
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