First-passage time of run-and-tumble particles with non-instantaneous resetting

We study the statistics of the first-passage time of a single run and tumble particle (RTP) in one spatial dimension, with or without resetting, to a fixed target located at $L>0$. First, we compute the first-passage time distribution of a free RTP, without resetting nor in a confining potential, but averaged over the initial position drawn from an arbitrary distribution $p(x)$. Recent experiments used a non-instantaneous resetting protocol that motivated us to study in particular the case where $p(x)$ corresponds to the stationary non-Boltzmann distribution of an RTP in the presence of a harmonic trap. This distribution $p(x)$ is characterized by a parameter $\nu>0$, which depends on the microscopic parameters of the RTP dynamics. We show that the first-passage time distribution of the free RTP, drawn from this initial distribution, develops interesting singular behaviours, depending on the parameter $\nu$. We then switch on resetting, mimicked by thermal relaxation of the RTP in the presence of a harmonic trap. Resetting leads to a finite mean first-passage time (MFPT) and we study this as a function of the resetting rate for different values of the parameters $\nu$ and $b = L/c$ where $c$ is the right edge of the initial distribution $p(x)$. In the diffusive limit of the RTP dynamics, we find a rich phase diagram in the $(b,\nu)$ plane, with an interesting re-entrance phase transition. Away from the diffusive limit, qualitatively similar rich behaviours emerge for the full RTP dynamics.