Miclo, Laurent An asymptotic condition for computing the logarithmic Sobolev constant on the line. (Une condition asymptotique pour le calcul de constantes de Sobolev logarithmiques sur la droite.) (French. English summary) Zbl 1220.46022 Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 1, 146-156 (2009). Summary: An explicit formula for the logarithmic Sobolev constant relative to real diffusions or to integer-valued birth and death processes is presented, under an asymptotic assumption for quantities naturally associated to Hardy inequalities in this context. Taking into account exact comparisons between entropy and appropriate variances, the proof goes back to Poincaré’s inequality situation. Cited in 1 Document MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 37A30 Ergodic theorems, spectral theory, Markov operators 60E15 Inequalities; stochastic orderings 94A17 Measures of information, entropy 26D15 Inequalities for sums, series and integrals Keywords:logarithmic Sobolev inequality; Poincaré’s inequality; Hardy inequality; integer-valued birth and death process × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] F. Barthe et C. Roberto. Sobolev inequalities for probability measures on the real line. Studia Math. 159 (2003) 481-497. · Zbl 1072.60008 · doi:10.4064/sm159-3-9 [2] S. G. Bobkov et F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1-28. · Zbl 0924.46027 · doi:10.1006/jfan.1998.3326 [3] C. Dellacherie et P.-A. Meyer. Probabilités et potentiel. Chapitres V à VIII . Hermann, Paris, revised edition. Théorie des martingales, 1980. [Martingale theory.] · Zbl 0464.60001 [4] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061-1083. JSTOR: · Zbl 0318.46049 · doi:10.2307/2373688 [5] L. Miclo. Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite ? Préprint, consultable sur http://hal.ccsd.cnrs.fr/ccsd-00017875, 2005. [6] L. Miclo. Une condition asymptotique pour le calcul de constantes de Sobolev logarithmiques sur la droite. Préprint de la première version, consultable sur http://hal.ccsd.cnrs.fr/ccsd-00097577, 2006. [7] B. Muckenhoupt. Hardy’s inequality with weights. Studia Math. 44 (1972) 31-38. · Zbl 0236.26015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.