# On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

## TL;DR

This paper introduces non-local Jacobi operators as generators of ergodic Markov semigroups, analyzing their spectral properties, convergence rates, and contractivity, with explicit formulas and bounds, extending classical Jacobi polynomial theory.

## Contribution

It develops a spectral and convergence theory for non-local Jacobi operators, including explicit semigroup expansions, spectrum characterization, and hypocoercivity results, generalizing classical Jacobi operator properties.

## Key findings

- Semigroup admits a series expansion with generalized Jacobi polynomials
- Spectrum of the non-self-adjoint generator is fully characterized
- Variance decay is hypocoercive with explicit constants

## Abstract

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability measures and study their spectral and convergence properties. In particular, we derive a series expansion of the semigroup in terms of explicitly defined polynomials, which generalize the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators and semigroups, with the local objects serving as reference points for transferring properties from the local to the non-local case.