  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.
