# Smoluchowski flux and Lamb-Lion Problems for Random Walks and L\'evy Flights with a Constant Drift

## TL;DR

This paper analyzes the flux and survival probability of particles performing random walks or Lévy flights with drift towards an absorbing boundary, deriving asymptotic behaviors and revealing counterintuitive decay for certain Lévy flight parameters.

## Contribution

It provides exact asymptotic formulas for flux and survival probability for Lévy flights with drift, extending classical results to non-Gaussian processes with new insights.

## Key findings

- Survival probability decays as a stretched exponential for 1<μ<2 with positive drift.
- For μ=2 (standard random walk), survival probability approaches a positive constant.
- Analytical results are validated by numerical simulations.

## Abstract

We consider non-interacting particles (or lions) performing one-dimensional random walks or L\'evy flights (with L\'evy index $1 < \mu \leq 2$) in the presence of a constant drift $c$. Initially these random walkers are uniformly distributed over the positive real line $z\geq 0$ with a density $\rho_0$. 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 $\Phi_c(n)$ out of the system (the "Smoluchowski problem") and (ii) the survival probability $S_c(n)$ 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 $\mathbb{E}[M_c(n)]$ of a single random walk or L\'evy flight after $n$ steps. This allows us to obtain the precise asymptotic behavior of both $\Phi_c(n)$ and $S_c(n)$ analytically for large $n$ in the two problems, for any value of $1<\mu\leq 2$ and $c \in {\mathbb{R}}$. In particular, for $c>0$, we show the rather counterintuitive result that for $1< \mu < 2$, $S_{c>0}(n \to \infty)$ vanishes as $S_{c>0}(n \to \infty) \approx \exp\left(-\lambda \, n^{2-\mu}\right)$, where $\lambda$ is a $\mu$-dependent positive constant, while for standard random walks (i.e., with $\mu = 2$), $S_{c>0}(n \to \infty) \to K_{RW} > 0$, as expected. Our analytical results are confirmed by numerical simulations.