  This paper addresses the question of how Brownian-like motion can arise from
the solution of a deterministic differential delay equation. To study this we
analytically study the bifurcation properties of an apparently simple
differential delay equation and then numerically investigate the probabilistic
properties of chaotic solutions of the same equation. Our results show that
solutions of the deterministic equation with randomly selected initial
conditions display a Gaussian-like density for long time, but the densities are
supported on an interval of finite measure. Using these chaotic solutions as
velocities, we are able to produce Brownian-like motions, which show
statistical properties akin to those of a classical Brownian motion over both
short and long time scales. Several conjectures are formulated for the
probabilistic properties of the solution of the differential delay equation.
Numerical studies suggest that these conjectures could be "universal" for
similar types of "chaotic" dynamics, but we have been unable to prove this.
