Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Fabian Merz, Rico Zacher

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 $(-Δ)^{s}$ 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.

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

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 associated with the fractional Laplacian on , . We further establish a Blachman-Stam inequality (BSI) for the fractional Fisher information , and prove that, for a large class of functions, converges to the classical Fisher information as . Through this nonlocal-to-local limit, we recover the classical BSI and the lifting property of the classical Fisher information.
Paper Structure (12 sections, 8 theorems, 127 equations)

This paper contains 12 sections, 8 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Lifting the classical Fisher information and kinetic theory
  3. Nonlocal Fisher information
  4. Main results
  5. Lifting property of nonlocal Fisher information
  6. Fisher information and $\Gamma$-calculus
  7. Nonlocal-to-local limit of the fractional Fisher information
  8. Nonlocal Blachman-Stam inequality
  9. Appendix
  10. Lifting property of entropy
  11. Proof of Remark \ref{['KernProp']} (i)
  12. Proof of Proposition \ref{['Sumformulas']}

Key Result

Proposition 2.1

For $i=1,2$, let $(M_{i},d_{i})$ be a metric space and $k_i$ a kernel on $M_i$. Let $(x,y)\in M_1\times M_2$ and $f:\, M_{1} \times M_{2} \rightarrow \mathbb{R}$ be measurable. Then the following statements hold.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Proposition 2.1
  • Lemma 2.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Proposition 3.1
  • proof
  • ...and 13 more