Nonlocal Fisher information: lifting, local limit, and the Blachman-Stam inequality
This work provides foundational mathematical tools for analyzing entropy dissipation in nonlocal heat equations, which is incremental but important for theoretical studies in partial differential equations and information theory.
The paper tackles the problem of extending the Fisher information concept to nonlocal settings, such as fractional Laplacians, by introducing a lifting property and proving a Blachman-Stam inequality for fractional Fisher information, with results showing convergence to classical Fisher information as s approaches 1.
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 $(-Î)^{s}$ 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.