# A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators

## TL;DR

This paper introduces a spectral theoretical framework using intertwining relations to analyze hypocoercivity, providing explicit decay rates for degenerate hypoelliptic and non-local operators in Hilbert spaces.

## Contribution

It develops a novel spectral approach based on intertwining to transfer spectral information, enabling explicit hypocoercivity estimates for complex operators.

## Key findings

- Established hypocoercive estimates for Ornstein-Uhlenbeck semigroups.
- Derived explicit hypocoercive constants for non-local Jacobi semigroups.
- Provided a functional calculus for non-self-adjoint operators using intertwining relations.

## Abstract

The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup $\tilde{P} = (e^{-t\tilde{\mathbf A}})_{t \geq 0}$ to a target semigroup $P$ which is the object of study. This allows us to obtain conditions under which $P$ satisfies a hypocoercive estimate with exponential decay rate given by the spectral gap of $\tilde{\mathbf A}$. Along the way we also develop a functional calculus involving the non-self-adjoint resolution of identity induced by the intertwining relations. We apply these results in a general Hilbert space setting to two cases: degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on $\mathbb R^d$, and non-local Jacobi semigroups on $[0,1]^d$, which have been recently introduced and studied for $d=1$. In both cases we obtain hypocoercive estimates and are able to explicitly identify the hypocoercive constants