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arxiv: 2604.12462 · v2 · submitted 2026-04-14 · 🧮 math.PR · math.FA

Stochastic analysis of Beckner's and related functional inequalities

Yuu Hariya This is my paper

Pith reviewed 2026-05-10 14:43 UTC · model grok-4.3

classification 🧮 math.PR math.FA
keywords inequalityinequalitiesbecknerfunctionalwhenentropyimprovementlogarithmic
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The pith

Stochastic analysis yields an improvement to Beckner's inequality for 4/3 ≤ p < 2 under Gaussian measure, including an entropy-expressed error at p=3/2, and a Holder-type relation among entropy, variance, and related functionals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Beckner's inequality sits between two classic math tools: the logarithmic Sobolev inequality, which controls how fast things mix, and Poincare's inequality, which bounds variance by gradients. The authors apply ideas from random processes to show that for certain exponents p in a middle range, the inequality can be made stricter when the underlying distribution is Gaussian. At one specific value p=3/2 the extra error term is written using the entropy functional. The same approach also produces a new relation that bounds products of entropy and variance-like terms in a Holder fashion.

Core claim

employing a stochastic method, we prove an improvement of Beckner's inequality under the Gaussian measure when 4/3≤p<2; in particular, when p=3/2, the error bound is expressed in terms of the entropy functional. A similar reasoning ... enables us to obtain a Hölder-type inequality that holds among the entropy, variance and related functionals.

Load-bearing premise

The derivation relies on the underlying probability measure being Gaussian and on the applicability of the chosen stochastic processes (likely Ornstein-Uhlenbeck or similar) to the functional inequality setting; the abstract does not specify the precise regularity or domain conditions needed for the stochastic representation to hold.

read the original abstract

Beckner's inequality is a family of inequalities that interpolates the two fundamental functional inequalities, the logarithmic Sobolev and Poincar\'e's inequalities. It is parametrized by exponent $p\in (1,2]$ and it implies the logarithmic Sobolev inequality as $p\to 1$ and agrees with Poincar\'e's inequality when $p=2$. In this paper, employing a stochastic method, we prove an improvement of Beckner's inequality under the Gaussian measure when $4/3\le p<2$; in particular, when $p=3/2$, the error bound is expressed in terms of the entropy functional. A similar reasoning to the derivation of the improvement also enables us to obtain a H\"older-type inequality that holds among the entropy, variance and related functionals.

Editorial analysis

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Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of the Gaussian measure and on the existence of suitable stochastic representations (likely via the Ornstein-Uhlenbeck semigroup or related diffusions) that allow the functional inequalities to be recast as expectations. No free parameters are introduced in the abstract; the result is stated for the standard Gaussian without additional scaling constants.

axioms (2)
  • domain assumption The underlying measure is the standard Gaussian on Euclidean space.
    The improvement is explicitly stated to hold under the Gaussian measure.
  • domain assumption Stochastic processes (diffusions) admit representations that convert the functional inequality into an expectation identity or inequality.
    The method is described as stochastic analysis, which presupposes such a representation exists and is valid for the given range of p.

pith-pipeline@v0.9.0 · 5418 in / 1524 out tokens · 37497 ms · 2026-05-10T14:43:09.354309+00:00 · methodology

discussion (0)

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