We study the dynamics of a one-dimensional run-and-tumble particle subjected to confining potentials of the type $V\left(x\right)={\alpha \left|x\right|}^p$, with $p>0$. The noise that drives the particle dynamics is telegraphic and alternates between $\pm 1$ values. We show that the stationary probability density $P\left(x\right)$ has a rich behavior in the $(p,\alpha )$ plane. For $p>1$, the distribution has a finite support in $[x_{-},x_{+}]$ and there is a critical line ${\alpha }_c\left(p\right)$ that separates an activelike phase for $\alpha >{\alpha }_c\left(p\right)$ where $P\left(x\right)$ diverges at $x_{\pm }$, from a passivelike phase for $\alpha <{\alpha }_c\left(p\right)$ where $P\left(x\right)$ vanishes at $x_{\pm }$. For $p<1$, the stationary density $P\left(x\right)$ collapses to a delta function at the origin, $P\left(x\right)=\delta \left(x\right)$. In the marginal case $p=1$, we show that, for $\alpha <{\alpha }_c$, the stationary density $P\left(x\right)$ is a symmetric exponential, while for $\alpha >{\alpha }_c$, it again is a delta function $P\left(x\right)=\delta \left(x\right)$. For the harmonic case $p=2$, we obtain exactly the full time-dependent distribution $P(x,t)$, which allows us to study how the system relaxes to its stationary state. In addition, for this $p=2$ case, we also study analytically the full distribution of the first-passage time to the origin. Numerical simulations are in complete agreement with our analytical predictions.

• Received 19 November 2018
• Revised 5 February 2019

DOI:https://doi.org/10.1103/PhysRevE.99.032132