Coupling by change of measure, Harnack inequality and hypercontractivity (Q1710444)

Editorial remark: Accidentally, this article has been issued twice for reviewing. Therefore, we display both reviews, by adding the second one to the originally published review: The aim of this paper is to study the hypercontractivity of Markov semigroups. The author uses the coupling method to reach this goal. We recall that the coupling method is a powerful tool in analysis of stochastic processes and, in order to apply this method, it is necessary that two marginal processes are constructed under different probability measures. By using these new type couplings, the author establishes Harnack inequalities for Markov semigroups and applies the coupling and Harnack inequality to the study of hypercontractivity of Markov semigroups. Reviewer: Vincenzo Vespri (Firenze) In this paper, a way is shown how to establish Harnack inequalities for Markov semigroups using the new coupling method, coupling by change of measure, and this is applied to the study of hypercontractivity. In the first section of the paper, the coupling method and the definition of coupling by change of measure for two stochastic processes are analyzed, and a general result on Harnack type inequalities using couple by change of measure is given. In the second section, a general result on the hypercontractivity using coupling and Harnack inequality is obtained. In the third section, an application of this result on hypercontractivity is applied for the study of degenerate functional stochastic differential equations. The last section of the paper is devoted to an application of the result to the study of hypercontractivity for non-degenerate functional semi-linear stochastic partial differential equations. Reviewer: Ilie Valusescu (Bucuresti) For the entire collection see [Zbl 1402.35005].