On Random Walks with a General Moving Barrier

Abstract

Random walks with a general, nonlinear barrier have found recent applications ranging from reionization topol- ogy to refinements in the excursion set theory of halos. Here we derive the first-crossing distribution of random walks with a moving barrier ofarbitrary shape. Such a distribution is shown to satisfy an integral equation that can be solved by a simple matrix inversion, without the need for Monte Carlo simulations, making this useful for ex- ploring a large parameter space. We discuss examples in which common analytic approximations fail, a failure that can be remedied using the method described here.