Nonlocal Fisher information: lifting, local limit, and the Blachman-Stam inequality
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911538291408896 |
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| author | Merz, Fabian Zacher, Rico |
| author_facet | Merz, Fabian Zacher, Rico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22079 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonlocal Fisher information: lifting, local limit, and the Blachman-Stam inequality Merz, Fabian Zacher, Rico Analysis of PDEs Information Theory Probability 94A17 (primary), 35R11, 60J27 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. |
| title | Nonlocal Fisher information: lifting, local limit, and the Blachman-Stam inequality |
| topic | Analysis of PDEs Information Theory Probability 94A17 (primary), 35R11, 60J27 |
| url | https://arxiv.org/abs/2603.22079 |