On random walks with a general moving barrier

Random walks with a general, nonlinear barrier have found recent applications ranging from reionization topology to refinements in the excursion set theory of halos. Here we derive the first-crossing distribution of random walks with a moving barrier of arbitrary 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 exploring a large parameter space. We discuss examples in which common analytic approximations fail, a failure that can be remedied using the method described here.