Gamma-convergence of fractional Gaussian perimeter

TL;DR

This paper proves that the fractional Gaussian perimeter converges to the classical Gaussian perimeter as the fractional parameter approaches one, using a definition based on Ornstein-Uhlenbeck operators, with dimension-independent constants.

Contribution

It establishes the $ ext{Gamma}$-convergence of fractional Gaussian perimeter to Gaussian perimeter, introducing a new approach via Bochner subordination for fractional powers of Ornstein-Uhlenbeck operators.

Findings

Fractional Gaussian perimeter $ ext{Gamma}$-converges to Gaussian perimeter as $s o 1^-$.

The convergence constant is independent of the dimension.

The approach uses Bochner subordination to define fractional powers of Ornstein-Uhlenbeck operators.

Abstract

We prove the $\Gamma$-convergence of the renormalised fractional Gaussian $s$-perimeter to the Gaussian perimeter as $s\to 1^-$. Our definition of fractional perimeter comes from that of the fractional powers of Ornstein-Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the $\Gamma$-limit does not depend on the dimension.