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Minkowski sum of fractal percolation and random sets

Part of: Stochastic processes Special processes Markov processes

Published online by Cambridge University Press:  29 July 2025

TIANYI BAI ,
XINXIN CHEN  and
YUVAL PERES
TIANYI BAI
Affiliation:
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Svstems Science, Chinese Academy of Sciences, Beijing, China, e-mail: tianyi.bai73@amss.ac.cn
XINXIN CHEN
Affiliation:
School of Mathematical Sciences, Beijing Normal University No. 19, Xinjiekouwai St, Haidian District, Beijing, China. e-mail: xinxin.chen@bnu.edu.cn
YUVAL PERES
Affiliation:
Beijing Institute of Mathematical Sciences and Applications No. 544 Hefangkou Village, Huairou District, Beijing, China. e-mail: yperes@bimsa.cn
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Abstract

In this paper, we prove that the hitting probability of the Minkowski sum of fractal percolations can be characterised by capacity. Then we extend this result to Minkowski sums of general random sets in $\mathbb Z^d$, including ranges of random walks and critical branching random walks, whose hitting probabilities are described by Newtonian capacity individually.

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 22
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

This research was supported by the National Key R&D Program of China No. 2022YFA1006500.

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