43 research outputs found
Coercive Inequalities on Metric Measure Spaces
We study coercive inequalities on finite dimensional metric spaces with
probability measures which do not have volume doubling property. This class of
inequalities includes Poincar\'e 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
From U-bounds to isoperimetry with applications to H-type groups
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.Comment: 40 pages, with addition
Markov semigroups with hypocoercive-type generator in Infinite Dimensions: Ergodicity and Smoothing
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
Ergodic property of Markovian semigroups on standard forms of von Neumann algebras
We give sufficient conditions for ergodicity of the Markovian semigroups
associated to Dirichlet forms on standard forms of von Neumann algebras
constructed by the method proposed in Refs. [Par1,Par2]. We apply our result to
show that the diffusion type Markovian semigroups for quantum spin systems are
ergodic in the region of high temperatures where the uniqueness of the
KMS-state holds.Comment: 25 page
Thermodynamic Limit for Spin Glasses. Beyond the Annealed Bound
Using a correlation inequality of Contucci and Lebowitz for spin glasses, we
demonstrate existence of the thermodynamic limit for short-ranged spin glasses,
under weaker hypotheses than previously available, namely without the
assumption of the annealed bound.Comment: 8 page