Abstract
The aim of this pedagogical paper is to show how some renowned inequalities may be obtained via a simple argument: entropy projection from the path space onto finite-dimensional coordinates spaces. Some applications are given: ergodic behaviour, perturbation.
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References
Aida, S.: ‘Uniform positivity improving property, Sobolev inequalities and spectral gaps’, J. Funct. Anal. 158 (1998), 152–185.
Aida, S.: ‘An estimate of the gap of spectrum of Schrödinger operators which generate hyperbounded semigroups’, J. Funct. Anal. 185 (2001), 474–526.
Aida, S., Masuda, T. and Shigekawa, I.: ‘Logarithmic Sobolev inequalities and exponential integrability’, J. Funct. Anal. 126 (1994), 83–101.
Aida, S. and Shigekawa, I.: ‘Logarithmic Sobolev inequalities and spectral gaps: Perturbation theory’, J. Funct. Anal. 126 (1994), 448–475.
Bakry, D.: ‘L'hypercontractivité et son utilisation en théorie des semi groupes’, in Ecole d'été de probabilités de Saint-Flour, Lecture Notes in Math. 1581, 1994, pp. 1–114.
Carlen, E.: ‘Superadditivity of Fisher's information and Logarithmic Sobolev inequalities’, J. Funct. Anal. 101 (1991), 194–211.
Cattiaux, P.: ‘Hypercontractivity for perturbed diffusion semi groups’, Preprint, 2003.
Cattiaux, P. and Gamboa, F.: ‘Large deviations and variational theorems for marginal problems’, Bernoulli 5 (1999), 81–108.
Cattiaux, P. and Léonard, C.: ‘Minimization of the Kullback information for general Markov processes’, in Séminaire de Probas XXX, Lecture Notes in Math. 1626, 1996, pp. 283–311.
Deuschel, J.D. and Stroock, D.W.: Large Deviations, Academic Press, New York, 1989.
Föllmer, H.: ‘Random fields and diffusion processes’, in Ecole d'été de probabilités de Saint-Flour, Lecture Notes in Math. 1362, 1988, pp. 101–204.
Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Studies in Mathematics 19, Walter de Gruyter, Berlin, 1994.
Gong, F.Z. and Wu, L.M.: ‘Spectral gap of positive operators and its applications’, C. R. Acad. Sci. Paris Sér. I 331 (2000), 983–988.
Gross, L.: ‘Logarithmic Sobolev inequalities and contractivity properties of semi-groups in Dirichlet forms’, in Lecture Notes in Math. 1563, 1993, pp. 54–88.
Guionnet, A. and Zegarlinski, B.: Cours I.H.P., Paris, 1999.
Hino, M.: ‘Exponential decay of positivity preserving semigroups on L p’, Osaka J. Math. 37 (2000), 603–624.
Kavian, O., Kerkyacharian, G. and Roynette, B.: ‘Quelques remarques sur l'ultracontractivité’, J. Funct. Anal. 111 (1993), 155–196.
Ledoux, M.: ‘Concentration of measure and logarithmic Sobolev inequalities’, in Séminaire de Probas XXXIII, Lecture Notes in Math. 1709, 1999, pp. 120–216.
Liggett, T.M.: ‘\(\mathbb{L}^2 \) 2 rates of convergence for attractive reversible nearest particle systems’, Ann. Probab. 19 (1991), 935–959.
Mathieu, P.: ‘Convergence to equilibrium for spin glasses’, to appear in Comm. Math. Phys.
Mathieu, P.: ‘Hitting times and spectral gap inequalities’, Ann. Inst. Henri Poincaré. Probab. Statist. 33 (1997), 437–465.
Mathieu, P.: ‘Quand l'inégalité Log-Sobolev implique l'inégalité de trou spectral’, in Séminaire de Probas XXXII, Lecture Notes in Math. 1686, 1998, pp. 30–35.
Röckner, M. and Wang, F.Y.: ‘Weak Poincaré inequalities and L 2 convergence rates of Markov semigroups’, J. Funct. Anal. 185 (2001), 564–603.
Royer, G.: Une initiation aux inégalités de Sobolev logarithmiques, S.M.F., Paris, 1999.
Wu, L.M.: ‘Uniforme positive improvingness, tail norm condition and spectral gap’, Preprint, 2001.
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Cattiaux, P. A Pathwise Approach of Some Classical Inequalities. Potential Analysis 20, 361–394 (2004). https://doi.org/10.1023/B:POTA.0000009847.84908.6f
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DOI: https://doi.org/10.1023/B:POTA.0000009847.84908.6f