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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
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.
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
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References:
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