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Functional inequalities for some generalised Mehler semigroups

L. Angiuli, S. Ferrari, D. Pallara

Abstract

We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure $σ$, we prove functional integral inequalities with respect to $σ$, such as logarithmic Sobolev and Poincaré type. Consequently, some integrability properties of exponential functions with respect to $σ$ are deduced.

Functional inequalities for some generalised Mehler semigroups

Abstract

We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure , we prove functional integral inequalities with respect to , such as logarithmic Sobolev and Poincaré type. Consequently, some integrability properties of exponential functions with respect to are deduced.

Paper Structure

This paper contains 10 sections, 19 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Invariant measure
  4. A logarithmic Sobolev type inequality and its consequences
  5. Exponential integrability of Lipschitz functions
  6. Examples
  7. Ornstein--Uhlenbeck operators with fractional diffusion
  8. An example in infinite dimension
  9. A modified version of a model introduced by Koponen
  10. Approximation of and by Lipschitz functions

Key Result

Theorem 3.2

Under Hypotheses hyp_App, basetopo and lino, $M_\infty$ is a Lévy measure and the measure $\sigma\leftrightarrow [b_\infty,0, M_\infty]$ is invariant for $P_t$. In addition, if Hypothesis base(iii) holds true, then $\sigma$ is unique and for any $f \in \mathcal{F}C^2_A(X)$. Moreover $\mu_t$ converges weakly-star to $\sigma$ as $t \to \infty$.

Theorems & Definitions (42)

  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • proof
  • Corollary 3.7
  • ...and 32 more