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

Patrick Cheridito,Pierre Patie,Anna Srapionyan,Aditya Vaidyanathan

doi:10.5802/jep.148

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.

Highlights

Abstract. — In this paper, we introduce and study non-local Jacobi operators, which generalize the classical Jacobi operators
An important ingredient of our approach is the notion of an intertwining relation, which is a type of commutation relationship for linear operators
As we show in Lemma 3.2 below, φ is a Bernstein function; that is, φ : [0, ∞) → [0, ∞)

Summary

Main results on non-local Jacobi operators and semigroups

We state our main results concerning the non-local operator J defined in (1.1). Note that for h ≡ 0, one has β = βμ and, by moment identification and determinacy, it is checked that (2.10) boils down, up to a multiplicative constant, to the Rodrigues representation of the classical Jacobi polynomials P(nμ) given in (A.6) In this sense, (Pφn)n 0 and (Vφn)n 0 generalize (P(nμ))n 0 in different ways, related to different representations of these orthogonal polynomials. Note that the optimal entropy decay rate is obtained only for μ = λ1/2 > 1, in which case, λ(lμog)S = 2λ1, while otherwise λ(lμog)S < 2λ1; see, e.g. Fontenas [26] We review these notions for the classical Jacobi semigroup in the appendix. Note that Qτ is an ergodic Markov semigroup on L2(β) with β as an invariant measure, and its generator is given by −φ(τ)(−J) = log Qτ ; see Sato [48, Chap. Assumption A allows us to obtain the existence and uniqueness of an invariant probability measure

Proofs

Examples

Invariant measure and L2-properties