  In this paper we investigate deterministic diffusion in systems which are
spatially extended in certain directions but are restricted in size and open in
other directions, consequently particles can escape. We introduce besides the
diffusion coefficient D on the chaotic repeller a coefficient ${\hat D}$ which
measures the broadening of the distribution of trajectories during the
transient chaotic motion. Both coefficients are explicitly computed for
one-dimensional models, and they are found to be different in most cases. We
show furthermore that a jump develops in both of the coefficients for most of
the initial distributions when we approach the critical borderline where the
escape rate equals the Liapunov exponent of a periodic orbit.
