To logconcavity and beyond

Ishige, Kazuhiro; Salani, Paolo; Takatsu, Asuka

doi:10.1142/s0219199719500093

Cited by 14 publications

(25 citation statements)

References 15 publications

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“…The preservation of semi-log-concavity in (2.6) is more subtle and can be found, for instance, in [28,51]. For the sake of completeness, we give a proof of (2.6) here.…”

Section: Preliminariesmentioning

confidence: 95%

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

Bez¹,

Nakamura²

2022

Preprint

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We prove that the regularising property of the Fokker-Planck equation brings about improved versions of Nelson's hypercontractivity and logarithmic Sobolev inequalities. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Our approach is based on a flow monotonicity scheme and our regularised inequalities are a consequence of an investigation into the interaction between two different flows. We apply our results to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton-Jacobi equation, for Talagrand's transportation cost inequality, the Poincaré inequality, and for Beckner's inequality. We also explore the possibility of broadening the scope of our approach to more general measure spaces, as well as an alternative route based on mass transportation.

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“…The preservation of semi-log-concavity in (2.6) is more subtle and can be found, for instance, in [28,51]. For the sake of completeness, we give a proof of (2.6) here.…”

Section: Preliminariesmentioning

confidence: 95%

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

Bez¹,

Nakamura²

2022

Preprint

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“…Remark 2 (1) The refinement of the ordinary logarithmic function, that is the case q = 1, was introduced by Ishige, Salani and the second named author [2], where they studied the preservation of concavity by the heat flow in Euclidean space. (2) For a positive function χ : (0, ∞) → (0, ∞) and a ∈ R \ {0}, the χ -logarithmic function ln χ : (0, ∞) → R and its refinement ln χ,a : (0, 1) → R are defined in the same way as those of χ q , respectively.…”

Section: Definitionsmentioning

confidence: 99%

“…Assume I q,a = ∅. A direct calculation provides that 2 is positive in t ∈ I q,a . In the case q = 0, the condition I 0,a = ∅ leads to a − 1 > 0, consequently…”

Section: Propertiesmentioning

confidence: 99%

Gauge Freedom of Entropies on q-Gaussian Measures

Matsuzoe

Takatsu

2021

Signals and Communication Technology

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A q-Gaussian measure is a generalization of a Gaussian measure. This generalization is obtained by replacing the exponential function with the power function of exponent 1/(1 − q) (q = 1). The limit case q = 1 recovers a Gaussian measure. On the set of all q-Gaussian densities over the real line with 1 ≤ q < 3, escort expectations determine information geometric structures such as an entropy and a relative entropy. The ordinary expectation of a random variable is the integral of the random variable with respect to its law. Escort expectations admit us to replace the law by any other measures. One of the most important escort expectations on the set of all q-Gaussian densities is the q-escort expectation since this escort expectation determines the Tsallis entropy and the Tsallis relative entropy. The phenomenon gauge freedom of entropies is that different escort expectations determine the same entropy, but different relative entropies. In this chapter, we first introduce a refinement of the q-logarithmic function. Then we demonstrate the phenomenon on an open set of all q-Gaussian densities over the real line by using the refined q-logarithmic functions. We write down the corresponding Riemannian metric. Keywords Information geometry • Gauge freedom of entropies • Refined q-logarithmic function • q-Gaussian measure

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“…A non-exhaustive list of references includes [32,29,30,7,6] for classical solutions and [17,2,27,33] for viscosity solutions. A generalized type, called power convexity/concavity, is investigated in [23,24,20,25] for various equations. One major idea applied in [29,2,20] is to show the corresponding convex envelope of a solution is a supersolution of the equation and then use the comparison principle to conclude the proof.…”

Section: Introductionmentioning

confidence: 99%

Quasiconvexity preserving property for fully nonlinear nonlocal parabolic equations

Kagaya¹,

Liu²,

Mitake³

2022

Preprint

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This paper is concerned with a general class of fully nonlinear parabolic equations with monotone nonlocal terms. We investigate the quasiconvexity preserving property of positive, spatially coercive viscosity solutions. We prove that if the initial value is quasiconvex, the viscosity solution to the Cauchy problem stays quasiconvex in space for all time. Our proof can be regarded as a limit version of that for power convexity preservation as the exponent tends to infinity. We also present several concrete examples to show applications of our result.

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To logconcavity and beyond

Abstract: In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity. In this paper, introducing a variation of concavity, we show that it preserves in fact a stronger property than logconcavity and we identify the strongest concavity preserved by the heat flow.

Cited by 14 publications

References 15 publications

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

Gauge Freedom of Entropies on q-Gaussian Measures

Quasiconvexity preserving property for fully nonlinear nonlocal parabolic equations

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