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

TL;DR

This paper investigates the occupation time distribution of a run-and-tumble particle with stochastic resetting, deriving a large deviation principle and analyzing how it depends on system parameters, including switching and resetting rates.

Contribution

It introduces a renewal equation approach to relate the moment generating functions with and without resetting, enabling the derivation of a large deviation principle for the occupation time.

Findings

Large deviation principle for occupation time derived

Behavior converges to Brownian case in fast switching limit

Occupation time distribution depends on resetting and switching rates

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 $\rho_1$ and $\rho_{-1}=1-\rho_1$, 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 $\alpha$ of the velocity state, the resetting rate $r$ and the probability $\rho_1$. In particular, we show that the corresponding LDP for a Brownian particle with resetting is recovered in the fast switching limit $\alpha\rightarrow \infty$. On the other hand, the behavior in the slow switching limit depends on $\rho_1$ in the resetting protocol.