  We introduce a class of interesting stochastic processes based on
Brownian-time processes. These are obtained by taking Markov processes and
replacing the time parameter with the modulus of Brownian motion. They
generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of
Le Gall, and they introduce new interesting examples. After defining
Brownian-time processes, we relate them to fourth order parabolic PDEs. We then
study their exit problem as they exit nice domains in $\Rd$, and connect it to
elliptic PDEs. We show that these processes have the peculiar property that
they solve fourth order parabolic PDEs, but their exit distribution - at least
in the standard Brownian-time process case - solves the usual second order
Dirichlet problem. We recover fourth order PDEs in the elliptic setting by
encoding the iterative nature of the Brownian-time process, through its exit
time, in a standard Brownian motion. We also show that it is possible to assign
a formal generator to these non-Markovian processes by giving such a generator
in the half-derivative sense.
