Occupation time of a run-and-tumble particle with resetting
File(s)
Published version
Author(s)
Bressloff, Paul C
Type
Journal Article
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
generated by a Poisson process with rate r. The velocity state is reset to ±v with fixed probabilities ρ1 and
ρ−1 = 1 − ρ1, 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 ρ1. First, we show that the
corresponding LDP for a Brownian particle with resetting is recovered in the fast switching limit α → ∞. We
then consider the case of a finite switching rate. In particular, we investigate how a directional bias in the resetting
protocol (ρ1 = 0.5) skews the LDP rate function so that its minimum is shifted away from the expected fractional
occupation time of one-half. The degree of shift increases with r and decreases with α.
Under the resetting protocol, the position of the particle is reset to the origin at a random sequence of times
generated by a Poisson process with rate r. The velocity state is reset to ±v with fixed probabilities ρ1 and
ρ−1 = 1 − ρ1, 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 ρ1. First, we show that the
corresponding LDP for a Brownian particle with resetting is recovered in the fast switching limit α → ∞. We
then consider the case of a finite switching rate. In particular, we investigate how a directional bias in the resetting
protocol (ρ1 = 0.5) skews the LDP rate function so that its minimum is shifted away from the expected fractional
occupation time of one-half. The degree of shift increases with r and decreases with α.
Date Issued
2020-10
Date Acceptance
2020-10-14
Citation
Physical Review E, 2020, 102 (4)
ISSN
2470-0045
Publisher
American Physical Society (APS)
Journal / Book Title
Physical Review E
Volume
102
Issue
4
Copyright Statement
©2020 American Physical Society
Identifier
http://dx.doi.org/10.1103/physreve.102.042135
Publication Status
Published
Article Number
042135
Date Publish Online
2020-10-28