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


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.


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Wilfrid S. Kendall. "Rayleigh Random Flights on the Poisson line SIRSN." Electron. J. Probab. 25 1 - 36, 2020.


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

Primary: 60D05
Secondary: 37A50 , 60G50

Keywords: abstract scattering representation , critical SIRSN-RRF , Crofton cell , delineated scattering process , Dirichlet forms , dynamical detailed balance , environment viewed from particle , ergodic theorem , fibre process , Kesten-Spitzer-Whitman range theorem , Mecke-Slivnyak theorem , Metropolis-Hastings acceptance ratio , neighbourhood recurrence , Palm conditioning , Poisson line process , RRF (Rayleigh Random Flight) , RWRE (Random Walk in a Random Environment) , SIRSN (Scale-invariant random spatial network) , SIRSN-RRF


Vol.25 • 2020
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