Minkowski sum of fractal percolation and random sets

TL;DR

This paper characterizes the hitting probabilities of Minkowski sums of fractal and random sets using capacity, extending the results to various random sets in integer lattices.

Contribution

It provides a capacity-based characterization of hitting probabilities for Minkowski sums of fractal and general random sets, including random walks and branching processes.

Findings

Hitting probabilities are characterized by capacity for Minkowski sums of fractal percolations.

Extension of results to Minkowski sums of general random sets in , including random walks.

Hitting probabilities for these sets are described by Newtonian capacity.

Abstract

In this paper, we prove that hitting probability of Minkowski sum of fractal percolations can be characterized by capacity. Then we extend this result to Minkowski sum 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.