The Annals of Probability

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

Hassan Allouba and Weian Zheng

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Abstract

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

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

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1015345772

Digital Object Identifier
doi:10.1214/aop/1015345772

Mathematical Reviews number (MathSciNet)
MR1880242

Zentralblatt MATH identifier
1018.60066

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aop/1015345772


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References

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