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

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.