Occupation time of a run-and-tumble particle with resetting

Abstract

We study the positive occupation time of a run-and-tumble particle (RTP) subject to stochastic resetting. Under the resetting protocol, the position of the particle is reset to the origin at a random sequence of times that is generated by a Poisson process with rate $r$. The velocity state is reset to $\pm v$ with fixed probabilities $ρ_1$ and $ρ_{-1}=1-ρ_1$, where $v$ is the speed. We exploit the fact that the moment generating functions with and without resetting are related by a renewal equation, and the latter generating function can be calculated by solving a corresponding Feynman-Kac equation. This allows us to numerically locate in Laplace space the largest real pole of the moment generating function with resetting, and thus derive a large deviation principle (LDP) for the occupation time probability density using the Gartner-Ellis theorem. We explore how the LDP depends on the switching rate $α$ of the velocity state, the resetting rate $r$ and the probability $ρ_1$. In particular, we show that the corresponding LDP for a Brownian particle with resetting is recovered in the fast switching limit $α\rightarrow \infty$. On the other hand, the behavior in the slow switching limit depends on $ρ_1$ in the resetting protocol.