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We study the dynamics of a one-dimensional run and tumble particle subjectedto confining potentials of the type $V(x) = \alpha \, |x|^p$, with $p>0$. Thenoise that drives the particle dynamics is telegraphic and alternates between$\pm 1$ values. We show that the stationary probability density $P(x)$ has arich behavior in the $(p, \alpha)$-plane. For $p>1$, the distribution has afinite support in $[x_-,x_+]$ and there is a critical line $\alpha_c(p)$ thatseparates an active-like phase for $\alpha > \alpha_c(p)$ where $P(x)$ divergesat $x_\pm$, from a passive-like phase for $\alpha < \alpha_c(p)$ where $P(x)$vanishes at $x_\pm$. For $p<1$, the stationary density $P(x)$ collapses to adelta function at the origin, $P(x) = \delta(x)$. In the marginal case $p=1$,we show that, for $\alpha < \alpha_c$, the stationary density $P(x)$ is asymmetric exponential, while for $\alpha > \alpha_c$, it again is a deltafunction $P(x) = \delta(x)$. For the special cases $p=2$ and $p=1$, we obtainexactly the full time-dependent distribution $P(x,t)$, that allows us to studyhow the system relaxes to its stationary state. In addition, in these twocases, we also study analytically the full distribution of the first-passagetime to the origin. Numerical simulations are in complete agreement with ouranalytical predictions.

Run-and-tumble particle in one-dimensional confining potential: Steady state, relaxation and first passage properties