Generating constrained run-and-tumble trajectories

TL;DR

This paper introduces an exact method to generate constrained run-and-tumble trajectories, extending effective Langevin equations to non-Markovian processes driven by telegraphic noise, with applications to excursions and meanders.

Contribution

It develops a novel approach for generating constrained run-and-tumble trajectories using effective space-time dependent tumbling rates, applicable to non-Markovian stochastic processes.

Findings

Efficient numerical implementation demonstrated.

Method accurately generates trajectories with specified start and end conditions.

Extended to various constrained run-and-tumble particle types.

Abstract

We propose a method to exactly generate bridge run-and-tumble trajectories that are constrained to start at the origin with a given velocity and to return to the origin after a fixed time with another given velocity. The method extends the concept of effective Langevin equations, valid for Markovian stochastic processes such as Brownian motion, to a non-Markovian stochastic process driven by a telegraphic noise, with exponentially decaying correlations. We obtain effective space-time dependent tumbling rates that implicitly accounts for the bridge constraint. We extend the method to other types of constrained run-and-tumble particles such as excursions and meanders. The method is implemented numerically and is shown to be very efficient.