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Random walks and Brownian motion on cubical complexes

Author

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  • Nye, Tom M.W.

Abstract

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition kernels of the random walks converge to that for Brownian motion. The proof involves pulling back onto the complex the distribution of Brownian sample paths on a single cube, combined with a distribution on walks between cubes. The main application lies in analysing sets of evolutionary trees: several tree spaces are cubical complexes and we briefly describe our results and applications in this context.

Suggested Citation

  • Nye, Tom M.W., 2020. "Random walks and Brownian motion on cubical complexes," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2185-2199.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2185-2199
    DOI: 10.1016/j.spa.2019.06.013
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