  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
("SIRSNRRF"), 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 SIRSNRRF neither diverges to infinity nor
tends to zero, thus supporting the conjecture.

Probability