Higher order PDE's and iterated Processes

Abstract

We introduce a class of stochastic processes based on symmetric $\alpha$-stable processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric $\alpha$-stable process. We call them $\alpha$-time processes. They generalize Brownian time processes studied in \cite{allouba1, allouba2, allouba3}, and they introduce new interesting examples. We establish the connection of $\alpha -$time processes to some higher order PDE's for $\alpha$ rational. We also study the exit problem for $\alpha$-time processes as they exit regular domains and connect them to elliptic PDE's. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.

17 pages

References in corpus (5)

• Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
• Iterated Brownian motion in an open set
• Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula
• Iterated Brownian Motion in Parabola-Shaped Domains
• A linearized Kuramoto-Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process

Cited by in corpus (8)

• Time-Changed Poisson Processes
• Large deviations for local time fractional Brownian motion and applications
• Laws of the iterated logarithm for a class of iterated processes
• Fractional equations via convergence of forms
• Randomly Stopped Nonlinear Fractional Birth Processes
• Stochastic solutions of a class of Higher order Cauchy problems in $\rd$
• Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
• Composition of processes and related partial differential equations