Representation formula for the entropy and functional inequalities

By Joseph Lehec. 2010

107 citations recorded by the source corpus.

We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell's formula for the Laplace transform. As an application, we give unified and short proofs of a number of functional inequalities.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5056769731708ded385e3a1370e391c2db36e4ea

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Quantitative logarithmic Sobolev inequalities and stability estimates (2014, methodology): “For the LSI, one could follow the proof of the LSI of [32] to obtain a representation of the deficit in terms of the quadratic variation of a certain stochastic process, but deducing from this representation a lower bound in terms of a useful distance seems like a challenging problem.”
The Wasserstein space of stochastic processes (2021, background): “the survey of Leonard [55]), martingale optimal transport, the weak transport problem (see [36, 6] among many others), the theory of linear transfers ([25]), and mathematical finance / martingale optimal transport, we refer to [10, 15, 2, 12].”
Displacement convexity of entropy and related inequalities on graphs (2012, background): “Lehec [25] in the context of Brascamp-Lieb inequalities.”
Semi Log-Concave Markov Diffusions (2013, methodology): “More recently, using similar arguments, Lehec [35] has studied gaussian functional inequalities and Fontbona and Jourdain [26] obtained a pathwise version of the Γ2 theory.”
Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations (2016, methodology): “As shown in Lehec [2013], among the drifts v t under which we have An easy consequence of this is that v t has to be a martingale (up to the fact that it becomes zero in coordinates that reach {− 1 , +1 ).”
Thin Shell Implies Spectral Gap Up to Polylog via a Stochastic Localization Scheme (2012, result): “7 It is interesting to compare this construction with the construction by Lehec in [Leh].”
Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit (2019, methodology): “…x ) φ d ( x ) for a sufﬁciently smooth f : R d → R + and the Itô SDE where Then X 1 ∼ q , i.e., one can use (8) to obtain an exact sample from q , and this construction is information-theoretically optimal (see, e.g., Dai Pra [1991], Lehec [2013], Eldan and Lee [2018], or Tzen and Raginsky [2019]).”