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Hypercontractivity type property for generalized Mehler semigroups

Luciana Angiuli, Simone Ferrari

Abstract

We investigate the hypercontractivity property of generalized Mehler semigroups on the $L^p$-scale with respect to invariant measures. This property is first obtained in the purely theoretical setting of skew operators and, subsequently, deduced for generalized Mehler semigroups arising from linear stochastic differential equations perturbed by Lévy noise. When the associated invariant measure $μ$ lacks a purely Gaussian structure, jump components may prevent the validity of Nelson's classical $L^p$-$L^q$ estimates. However, a summability-improving property can be obtained in the setting of mixed-norm spaces $\mathcal{X}_{p,q}(E;γ,π)$ related to the factorization of the invariant measure $μ= γ* π$ into a Gaussian part $γ$ and an infinitely divisible non-Gaussian part $π$. As in the classical Gaussian case, some modified logarithmic Sobolev inequalities with respect to invariant measures can be inferred.

Hypercontractivity type property for generalized Mehler semigroups

Abstract

We investigate the hypercontractivity property of generalized Mehler semigroups on the -scale with respect to invariant measures. This property is first obtained in the purely theoretical setting of skew operators and, subsequently, deduced for generalized Mehler semigroups arising from linear stochastic differential equations perturbed by Lévy noise. When the associated invariant measure lacks a purely Gaussian structure, jump components may prevent the validity of Nelson's classical - estimates. However, a summability-improving property can be obtained in the setting of mixed-norm spaces related to the factorization of the invariant measure into a Gaussian part and an infinitely divisible non-Gaussian part . As in the classical Gaussian case, some modified logarithmic Sobolev inequalities with respect to invariant measures can be inferred.

Paper Structure

This paper contains 10 sections, 22 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Tensor products
  4. The mixed-norm spaces $\mathcal{X}_{p,q}(E;\mu_1, \mu_2)$
  5. Main results
  6. Gaussian measure
  7. Hypercontractivity type estimates
  8. An application to compactness
  9. Generalized Mehler semigroups
  10. Logarithmic Sobolev type inequalities

Key Result

Proposition 2.1

Let $f \in C^1(\mathbb{R})$, $g \in C(\mathbb{R})$, and $x_0 \in \mathbb{R}$. The following properties hold: Analogous results hold for $D_+$ with obvious modifications.

Theorems & Definitions (50)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark
  • Proposition 3.1
  • proof
  • Remark
  • Proposition 3.2
  • proof
  • Remark
  • ...and 40 more