Bi-Kolmogorov type operators and weighted Rellich's inequalities

TL;DR

This paper establishes weighted Rellich's inequalities with optimal constants for symmetric Kolmogorov operators, analyzes their generated semigroups, and applies results to the bi-Ornstein-Uhlenbeck operator, advancing understanding of these operators' spectral and asymptotic properties.

Contribution

It introduces new weighted Rellich's inequalities for Kolmogorov operators and explores their semigroup generation and asymptotic behavior, including applications to bi-Ornstein-Uhlenbeck operators.

Findings

Proved weighted Rellich's inequalities with optimal constants.

Demonstrated that certain operators generate analytic semigroups of contractions.

Described the asymptotic behavior and positivity properties of the semigroups.

Abstract

In this paper we consider the symmetric Kolmogorov operator $L=\Delta +\frac{\nabla \mu}{\mu}\cdot \nabla$ on $L^2(\mathbb R^N,d\mu)$, where $\mu$ is the density of a probability measure on $\mathbb R^N$. Under general conditions on $\mu$ we prove first weighted Rellich's inequalities with optimal constants and deduce that the operators $L$ and $-L^2$ with domain $H^2(\mathbb R^N,d\mu)$ and $H^4(\mathbb R^N,d\mu)$ respectively, generate analytic semigroups of contractions on $L^2(\mathbb R^N,d\mu)$. We observe that $d\mu$ is the unique invariant measure for the semigroup generated by $-L^2$ and as a consequence we describe the asymptotic behaviour of such semigroup and obtain some local positivity properties. As an application we study the bi-Ornstein-Uhlenbeck operator and its semigroup on $L^2(\mathbb R^N,d\mu)$.