Modified log-Sobolev inequalities, mixing and hypercontractivity

Motivated by (the rate of information loss or) the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of the standard logarithmic Sobolev inequality in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare' (spectral gap) inequality. We also derive a hypercontractivity formulation equivalent to our main modified log-Sobolev inequality which might be of independent interest. Finally we show that, in contrast with the spectral gap, for bounded degree expander graphs various log-Sobolev-type constants go to zero with the size of the graph.