  We consider the problem of leakage or effusion of an ensemble of independent
stochastic processes from a region where they are initially randomly
distributed. The case of Brownian motion, initially confined to the left half
line with uniform density and leaking into the positive half line is an example
which has been extensively studied in the literature. Here we derive new
results for the average number and variance of the number of leaked particles
for arbitrary Gaussian processes initially confined to the negative half line
and also derive its joint two-time probability distribution, both for the
annealed and the quenched initial conditions. For the annealed case, we show
that the two-time joint distribution is a bivariate Poisson distribution. We
also discuss the role of correlations in the initial particle positions on the
statistics of the number of particles on the positive half line. We show that
the strong memory effects in the variance of the particle number on the
positive real axis for Brownian particles, seen in recent studies, persist for
arbitrary Gaussian processes and also at the level of two-time correlation
functions.
