The Annals of Probability

Brownian-time processes: The PDE Connection and the half-derivative generator

Hassan Allouba and Weian Zheng

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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.

Article information

Ann. Probab., Volume 29, Number 4 (2001), 1780-1795.

First available in Project Euclid: 5 March 2002

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Zentralblatt MATH identifier

Primary: 60H30 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J65: Brownian motion [See also 58J65]
Secondary: 60J60

Brownian-time processes excursion-based Brownian-time processes iterated Brownian motion Markov snake half-derivative generator


Allouba, Hassan; Zheng, Weian. Brownian-time processes: The PDE Connection and the half-derivative generator. Ann. Probab. 29 (2001), no. 4, 1780--1795. doi:10.1214/aop/1015345772.

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