Translator Disclaimer
2020 Rayleigh Random Flights on the Poisson line SIRSN
Wilfrid S. Kendall
Electron. J. Probab. 25: 1-36 (2020). DOI: 10.1214/20-EJP526

Abstract

We study scale-invariant Rayleigh Random Flights (“RRF”) in random environments given by planar Scale-Invariant Random Spatial Networks (“SIRSN”) based on speed-marked Poisson line processes. A natural one-parameter family of such RRF (with scale-invariant dynamics) can be viewed as producing “randomly-broken local geodesics” on the SIRSN; we aim to shed some light on a conjecture that a (non-broken) geodesic on such a SIRSN will never come to a complete stop en route. (If true, then all such geodesics can be represented as doubly-infinite sequences of sequentially connected line segments. This would justify a natural procedure for computing geodesics.) The family of these RRF (“SIRSN-RRF”), is introduced via a novel axiomatic theory of abstract scattering representations for Markov chains (itself of independent interest). Palm conditioning (specifically the Mecke-Slivnyak theorem for Palm probabilities of Poisson point processes) and ideas from the ergodic theory of random walks in random environments are used to show that at a critical value of the parameter the speed of the scale-invariant SIRSN-RRF neither diverges to infinity nor tends to zero, thus supporting the conjecture.

Citation

Download Citation

Wilfrid S. Kendall. "Rayleigh Random Flights on the Poisson line SIRSN." Electron. J. Probab. 25 1 - 36, 2020. https://doi.org/10.1214/20-EJP526

Information

Received: 21 January 2020; Accepted: 28 September 2020; Published: 2020
First available in Project Euclid: 12 October 2020

MathSciNet: MR4161134
Digital Object Identifier: 10.1214/20-EJP526

Subjects:
Primary: 60D05
Secondary: 37A50, 60G50

JOURNAL ARTICLE
36 PAGES


SHARE
Vol.25 • 2020
Back to Top