  We extend Pitman's representation of a discrete analog of the Bessel 3d
process that starts at 0 to arbitrary initial laws. The representation is in
terms of maxima of simple random walks and can be interpreted as a random walk
conditioned to be above a random level. As an application we establish
continuous time limit, which in particular yields convergence to the
non-Gaussian component of a stationary measure of the KPZ fixed point on the
half-line as proposed in Barraquand and Le Doussal (2022).
