Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator

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 PDEs. We then study their exit problem as they exit nice domains in $\Rd$, and connect it to elliptic PDEs. We show that these processes have the peculiar property that they solve fourth order parabolic PDEs, but their exit distribution - at least in the standard Brownian-time process case - solves the usual second order Dirichlet problem. We recover fourth order PDEs 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.

15 pages, 3/9 papers from my 2000-2006 collection (preprint version)

Cited by in corpus (8)

• Fractional Cauchy problems on bounded domains
• Fractional diffusion equations and processes with randomly varying time
• Iterated Brownian motion in an open set
• Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula
• Large deviations for local time fractional Brownian motion and applications
• Laws of the iterated logarithm for a class of iterated processes
• A linearized Kuramoto-Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process
• Lifetime asymptotics of iterated Brownian motion in R^{n}