Higher order PDE's and iterated Processes

Details Higher order PDE's and iterated Processes Higher order PDE's and iterated Processes

2005-08-15

arxiv.org/abs/math/0508262v2

Trans. Amer. Math. Soc. 360 (2008), 2681-2692.

Mathematics

Probability

17 pages

Scientific paper

10.1090/S0002-9947-07-04437-6

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.

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