Optimal mean first-passage time of a run-and-tumble particle in a class of one-dimensional confining potentials

We consider a run-and-tumble particle (RTP) in one dimension, subjected to a telegraphic noise with a constant rate $\gamma$, and in the presence of an external confining potential $V(x)=\alpha \mathrm{\mid }x{\mathrm{\mid }}^p$ with $p\ge 1$. We compute the mean first-passage time (MFPT) at the origin ${\tau }_{\gamma }(x_0)$ for an RTP starting at $x_0$. We obtain a closed form expression for ${\tau }_{\gamma }(x_0)$ for all $p\ge 1$, which becomes fully explicit in the case $p=1$, $p=2$ and in the limit $p\to \mathrm{\infty }$. For generic $p>1$ we find that there exists an optimal rate ${\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}}$ that minimizes the MFPT and we characterize in detail its dependence on $x_0$. We find that ${\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}}\propto 1\mathrm{/}{x}_0$ as $x_0\to 0$, while ${\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}}$ converges to a nontrivial constant as $x_0\to \mathrm{\infty }$. In contrast, for $p=1$, there is no finite optimum and ${\gamma }_{\mathrm{o}\mathrm{p}\mathrm{t}}\to \mathrm{\infty }$ in this case. These analytical results are confirmed by our numerical simulations.