Arithmetic progressions in the trace of Brownian motion in space
Itai Benjamini, Gady Kozma
TL;DR
This paper proves that the trace of 3D Brownian motion almost surely contains arithmetic progressions of length 5 but not of length 6, revealing specific combinatorial structures in stochastic paths.
Contribution
It establishes the almost sure existence of length-5 arithmetic progressions and the absence of length-6 in 3D Brownian motion traces, a novel result in stochastic analysis.
Findings
Existence of length-5 arithmetic progressions in the trace
Non-existence of length-6 arithmetic progressions
Provides insight into the combinatorial structure of Brownian paths
Abstract
It is shown that the trace of dimensional Brownian motion contains arithmetic progressions of length and no arithmetic progressions of length a.s.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Topology and Set Theory · Stochastic processes and statistical mechanics