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We consider non-interacting particles (or lions) performing one-dimensionalrandom walks or L\'evy flights (with L\'evy index $1 < \mu \leq 2$) in thepresence of a constant drift $c$. Initially these random walkers are uniformlydistributed over the positive real line $z\geq 0$ with a density $\rho_0$. Atthe origin $z=0$ there is an immobile absorbing trap (or a lamb), such thatwhen a particle crosses the origin, it gets absorbed there. Our main focus ison (i) the flux of particles $\Phi_c(n)$ out of the system (the "Smoluchowskiproblem") 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 beexpressed in terms of the average maximum $\mathbb{E}[M_c(n)]$ of a singlerandom walk or L\'evy flight after $n$ steps. This allows us to obtain theprecise asymptotic behavior of both $\Phi_c(n)$ and $S_c(n)$ analytically forlarge $n$ 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 counterintuitiveresult 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 $\lambda$ isa $\mu$-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. Ouranalytical results are confirmed by numerical simulations.

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