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We consider a one dimensional random-walk-like process, whose steps arecentered Gaussians with variances which are determined according to thesequence of arrivals of a Poisson process on the line. This process isdecorated by independent random variables which are added at each step, but donot get accumulated. We study the probability that such process, conditioned toform a bridge, stays below a curve which grows at most polynomially fast awayfrom the boundaries, with exponent less than one half. Both bounds andasymptotics are derived. These estimates are used in the manuscript "thestructure of extreme level sets in branching Brownian motion" by the sameauthors, to which this manuscript is a supplement.

Decorated Random Walk Restricted to Stay Below a Curve (Supplement Material)