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We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type exp(−|x| α ) or more complex exp(−|x| α log β (2 + |x|)) (α ∈]1, 2[ and β ∈ R) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities.

The purpose of this paper is to present a probabilistic proof of the well-known result stating that the time needed by a continuous-time finite birth and death process for going from the left end to the right end of its state space is a sum of independent exponential variables whose parameters are the sign reversed eigenvalues of the underlying generator with a Dirichlet condition at the right end. The exponential variables appear as fastest strong quasi-stationary times for successive dual processes associated to the original absorbed process. As an aftermath, we get an interesting probabilistic representation of the time marginal laws of the process in terms of "local equilibria".

Consider an ergodic Markov operator M reversible with respect to a probability measure µ on a general measurable space. It is shown that if M is bounded from L 2 pµq to L p pµq, where p ą 2, then it admits a spectral gap. This result answers positively a conjecture raised by Høegh-Krohn and Simon [31] in a semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee, Gharan and Trevisan [23]. It provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general. In addition, the usual Cheeger inequality is extended to the higher eigenvalues in the compact Riemannian setting.