Entropy inequalities for unbounded spin systems

Pra, Paolo Dai; Paganoni, Anna Maria; Posta, Gustavo

doi:10.1214/aop/1039548378

Cited by 51 publications

(94 citation statements)

References 12 publications

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“…which, by tensorization, recovers the following L 1 inequality of [11,7], and proved in [28] in the Poisson case. In the next proposition we state and prove this inequality in the multidimensional case, using the Clark representation formula, similarly to Theorem 13.3.…”

Section: Lemma 131supporting

confidence: 71%

“…Our approach is based on the intrinsic tools (gradient, divergence, Laplacian) of infinite-dimensional stochastic analysis. We refer to [4,3,17,20], for other versions of logarithmic Sobolev inequalities in discrete settings, and to [7,28] for the Poisson case. Section 14 contains a change of variable formula in discrete time, which is applied with the Clark formula in Section 15 to a derivation of the Black-Scholes formula in discrete time, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic analysis of Bernoulli processes

Privault¹

2008

Probab. Surveys

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These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.

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Section: Lemma 131supporting

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Stochastic analysis of Bernoulli processes

Privault¹

2008

Probab. Surveys

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“…In particular, the inequality (1.5) was considered by M. Ledoux [29] for product measures on the discrete cube and, as an application, for Poissonian limits. While preparing the present paper, we learnt that the inequality (1.5) was also introduced by Dai Pra, Paganoni and Posta (see [13] where ρ 0 is referred to as the "entropy constant") in the context of certain Gibbs measures on Z d . In particular, they showed examples of measures which fail to satisfy the classical inequality (1.7), while satisfying (1.5).…”

Section: Introductionmentioning

confidence: 85%

Modified Logarithmic Sobolev Inequalities in Discrete Settings

2006

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Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare' (spectral gap) inequality. We show that, in contrast with the spectral gap, for bounded degree expander graphs, various log-Sobolev constants go to zero with the size of the graph. We also derive a hypercontractivity formulation equivalent to our main modified log-Sobolev inequality. Along the way we survey various recent results that have been obtained in this topic by other researchers.

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“…e.g. [9]. In this paper we consider the general discrete setting of a probability space (E, E, µ), and a finite difference gradient d + defined as d…”

Section: Introductionmentioning

confidence: 99%

Functional inequalities for discrete gradients and application to the geometric distribution

Joulin

Privault

2004

ESAIM: PS

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Abstract. We present several functional inequalities for finite difference gradients, such as a Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities, associated deviation estimates, and an exponential integrability property. In the particular case of the geometric distribution on N we use an integration by parts formula to compute the optimal isoperimetric and Poincaré constants, and to obtain an improvement of our general logarithmic Sobolev inequality. By a limiting procedure we recover the corresponding inequalities for the exponential distribution. These results have applications to interacting spin systems under a geometric reference measure.Mathematics Subject Classification. 60E07, 60E15, 60K35.

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Entropy inequalities for unbounded spin systems

Cited by 51 publications

References 12 publications

Stochastic analysis of Bernoulli processes

Stochastic analysis of Bernoulli processes

Modified Logarithmic Sobolev Inequalities in Discrete Settings

Functional inequalities for discrete gradients and application to the geometric distribution

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