Symmetric $\alpha$-stable subordinators and Cauchy problems

TL;DR

This paper reviews the connection between iterated processes and PDEs, providing a new probabilistic proof of their equivalence, focusing on symmetric alpha-stable subordinators and Cauchy problems.

Contribution

It offers a novel probabilistic proof establishing the equivalence between higher order PDEs and fractional in time PDEs for symmetric alpha-stable subordinators.

Findings

Unified framework for higher order and fractional PDEs

Probabilistic proof of PDE equivalence

Application to symmetric alpha-stable subordinators

Abstract

We survey the results in Nane (E. Nane, Higher order PDE's and iterated processes, Trans. American Math. Soc. (to appear)) and Baeumer, Meerschaert, and Nane (B. Baeumer, M.M. Meerschaert and E. Nane, Brownian subordinators and fractional Cauchy problems: Submitted (2007)) which deal with PDE connection of some iterated processes, and obtain a new probabilistic proof of the equivalence of the higher order PDE's and fractional in time PDE's.