Derivative formula for capacities

TL;DR

This paper derives a derivative formula for various capacities, analyzing the second order term in the asymptotic expansion of capacity for unions of sets as their distance increases, with applications to multiple capacity notions.

Contribution

It provides a new derivative formula for capacities, including Newtonian, Bessel-Riesz, and Branching capacities, extending understanding of their asymptotic behavior.

Findings

Identifies second order term in capacity expansion

Applies to Newtonian, Bessel-Riesz, and Branching capacities

Open problem for percolation capacity in related work

Abstract

We obtain a derivative formula for various notions of capacity. Namely we identify the second order term in the asymptotic expansion of the capacity of a union of two sets, as their distance goes to infinity. Our result applies to the usual Newtonian capacity in the setting of random walks on the Euclidean lattice, to the family of Bessel-Riesz capacities, and to the Branching capacity, which has been introduced recently by Zhu [9] in connection with critical Branching random walks. On the other hand, the result remains open for the notion of capacity in the setting of percolation, which is introduced in a companion paper, but serves as a motivation, as it would have some interesting consequences there.