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

TL;DR

This paper introduces Brownian-time processes, a class of stochastic processes related to fourth order PDEs, and explores their properties, connections to elliptic PDEs, and a novel half-derivative generator for non-Markovian dynamics.

Contribution

The paper defines Brownian-time processes, relates them to fourth order PDEs, and introduces a formal half-derivative generator for these non-Markovian processes.

Findings

Brownian-time processes solve fourth order parabolic PDEs.

Their exit distributions solve second order Dirichlet problems.

A half-derivative generator can be formally assigned to these processes.

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.