  Systems whose time evolutions are entirely deterministic can nevertheless be
studied probabilistically, i.e. in terms of the evolution of probability
distributions rather than individual trajectories. This approach is central to
the dynamics of ensembles (statistical mechanics) and systems with uncertainty
in the initial conditions.
  Delay differential equations (DDEs) are a particular class of deterministic
systems, distinguished by an explicit dependence of the dynamics on past
states. A probabilistic approach to DDEs is lacking. The main problems we
consider in developing such an approach are (1) to characterize the evolution
of probability distributions for DDEs, i.e. develop an analog of the
Perron-Frobenius operator; (2) to characterize invariant probability
distributions for DDEs; and (3) to develop a framework for the application of
ergodic theory to delay equations, with a view to a probabilistic understanding
of DDEs whose time evolutions are chaotic. We develop a variety of approaches
to each of these problems, employing both analytical and numerical methods.
  In transient chaos, a system evolves erratically during a transient period
that is followed by asymptotically regular behavior. Transient chaos in delay
equations has not been reported or investigated before. We find numerical
evidence of transient chaos (fractal basins of attraction and long chaotic
transients) in some DDEs, including the Mackey-Glass equation. Transient chaos
in DDEs can be analyzed numerically using a modification of the
"stagger-and-step" algorithm applied to a discretized version of the DDE.
