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

We consider non-interacting particles (or lions) performing onedimensional random walks or Lévy flights (with Lévy index 1 < μ ≤ 2) in the presence of a constant drift c. Initially these random walkers are uniformly distributed over the positive real line z ≥ 0 with a density ρ0. 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 φc(n) 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 E[Mc(n)] of a single random walk or Lévy flight after n steps. This allows us to obtain the precise asymptotic behavior of both φc(n) and Sc(n) analytically for large n in the two problems, for any value of 1 < μ ≤ 2 and c ∈ ℝ. In particular, for c > 0 and 1 < μ < 2, we show that Sc>0(n→ ∞) vanishes as Sc<0(n → ∞) ≈ exp (-λn^2-μ), where λ is a μ-dependent positive constant, in contrast with the case of standard random walks (i.e. with μ = 2) for which Sc>0(n →∞) → K_RW > 0. Our analytical results are confirmed by numerical simulations.