Smoluchowski flux and lamb-lion problems for random walks and Lévy flights with a constant drift

Abstract

We consider non-interacting particles (or lions) performing one-dimensional random walks or Lévy flights (with Lévy index ) in the presence of a constant drift c. Initially these random walkers are uniformly distributed over the positive real line with a density . At the origin z = 0 there is an immobile absorbing trap (or a lamb), such that when a particle crosses the origin, it gets absorbed there. Our main focus is on (i) the flux of particles out of the system (the ‘Smoluchowski problem’) and (ii) the survival probability Sc(n) of the trap or lamb (the ‘lamb-lion problem’) until step n. We show that both observables can be expressed in terms of the average maximum of a single random walk or Lévy flight after n steps. This allows us to obtain the precise asymptotic behavior of both and Sc(n) analytically for large n in the two problems, for any value of and . In particular, for c > 0 and , we show that vanishes as , where is a -dependent positive constant, in contrast with the case of standard random walks (i.e. with ) for which . Our analytical results are confirmed by numerical simulations.