Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula

TL;DR

This paper explores the deep connection between Brownian-time processes and fourth order PDEs, introducing new probabilistic methods and formulas that relate to complex PDEs like Cahn-Hilliard and Kuramoto-Sivashinsky.

Contribution

It extends the understanding of BTPs by linking them to new classes of fourth order PDEs and developing a Feynman-Kac type formula specific to BTPs, with implications for advanced PDEs.

Findings

Connected BTPs to new fourth order PDEs involving initial functions

Developed a Feynman-Kac formula for BTPs

Proposed a probabilistic approach to complex PDEs like Cahn-Hilliard

Abstract

We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes, and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth order PDEs in which the initial function plays an important role in the PDE itself, not only as initial data. We connect two such types of interesting and new PDEs to BTPs. The first is obtained by running the BTP and then integrating along its path, and the second type of PDEs is related to what we call the Feynman-Kac formula for BTPs. A special case of the second type is a step towards a probabilistic solution to linearized Cahn-Hilliard and Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.