Nonlocal Fisher information: lifting, local limit, and the Blachman-Stam inequality

TL;DR

This paper introduces a nonlocal Fisher information framework, establishes a Blachman-Stam inequality for fractional Fisher information, and demonstrates its convergence to classical Fisher information as the fractional parameter approaches one.

Contribution

It extends Fisher information to nonlocal settings, proves a Blachman-Stam inequality for fractional Fisher information, and links nonlocal and classical Fisher information through a limit process.

Findings

Nonlocal Fisher information admits a natural lifting.

A Blachman-Stam inequality is established for fractional Fisher information.

Fractional Fisher information converges to classical Fisher information as s approaches 1.

Abstract

We show that the nonlocal Fisher information - defined as the entropy dissipation of the Boltzmann entropy for nonlocal heat equations - admits a natural lifting in the sense of Guillen and Silvestre (2025). Important examples include the discrete Fisher information arising in Markov chains and the fractional Fisher information $i_s$ associated with the fractional Laplacian $(-\Delta)^{s}$ on $\mathbb{R}^d$, $s\in (0,1)$. We further establish a Blachman-Stam inequality (BSI) for the fractional Fisher information $i_s$, and prove that, for a large class of functions, $i_s$ converges to the classical Fisher information as $s\to 1$. Through this nonlocal-to-local limit, we recover the classical BSI and the lifting property of the classical Fisher information.