Pitman's discrete $2M-X$ theorem for arbitrary initial laws and continuous time limits

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).