arXiv subjects
Publications and source records attributed to B. Zegarlinski.
We investigate in a systematic way hypercontractivity property in Orlicz spaces for Markov semi-groups related to homogeneous and non homogeneous diffusions in $\mathbb{R}^{n}$. We provide an explicit construction of a family of Orlicz functions for which we prove that the associated hypercontractivity property is equivalent to a suitable functional inequality.
We extend the Gamma_2 calculus of Bakry and Emery to include a Carre du champ operator with multiplicative term, providing results which allow to analyse inhomogeneous diffusions.
We start by considering infinite dimensional Markovian dynamics in R^m generated by operators of hypocoercive type and for such models we obtain short and long time pointwise estimates for all the derivatives, of any order and in any direction, along the semigroup. We then look at infinite dimensional models (in (Rm)^{Z ^d}) produced by the interaction of infinitely many finite dimensional dissipative dynamics of the type indicated above. For these infinite dimensional models we study finite speed of propagation of information, well-posedness of the semigroup, time behaviour of the derivatives and strong ergodicity problem.
In this paper we show several applications of the general theory developed in \cite{MV_I}, where we studied smoothing and ergodicity for infinite dimensional Markovian systems with hypocoercive type generator.
We develop an effective strategy for proving strong ergodicity of (nonsymmetric) Markov semigroups associated to Hörmander type generators when the underlying configuration space is infinite dimensional.
We analyse certain degenerate infinite dimensional sub-elliptic generators, and obtain estimates on the long-time behaviour of the corresponding Markov semigroups that describe a certain model of heat conduction. In particular, we establish ergodicity of the system for a family of invariant measures, and show that the optimal rate of convergence to equilibrium is polynomial. Consequently, there is no spectral gap, but a Liggett-Nash type inequality is shown to hold.
In this paper we study applications of U-bounds to coercive and isoperimetric problems for probability measures on finite and infinite products of H-type groups.
We study coercive inequalities on finite dimensional metric spaces with probability measures which do not have volume doubling property. This class of inequalities includes Poincaré and Log-Sobolev inequality. Our main result is proof of Log-Sobolev inequality on Heisenberg group equipped with either heat kernel measure or "gaussian" density build from optimal control distance. As intermediate results we prove so called U-bounds.