  We consider a one dimensional random-walk-like process, whose steps are
centered Gaussians with variances which are determined according to the
sequence of arrivals of a Poisson process on the line. This process is
decorated by independent random variables which are added at each step, but do
not get accumulated. We study the probability that such process, conditioned to
form a bridge, stays below a curve which grows at most polynomially fast away
from the boundaries, with exponent less than one half. Both bounds and
asymptotics are derived. These estimates are used in the manuscript "the
structure of extreme level sets in branching Brownian motion" by the same
authors, to which this manuscript is a supplement.
