The inverse first-passage time problem as hydrodynamic limit of a particle system

Given a distribution on the positive real line, the two-sided inverse first-passage time problem for Brownian motion asks for a function, such that the first-passage time of this function by a reflected Brownian motion has the given distribution. We present a particle system without branching but with selection at timepoints depending on the given distribution on the positive real line, whose hydrodynamic limit is the distribution of a Brownian motion conditioned to not have passed the solution of the inverse first-passage time problem. As application we extract a simple Monte-Carlo method to simulate these solutions.