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

Abstract

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operator. We show that these operators extend to the generator of an ergodic Markov semigroup with a unique invariant probability measure and study its spectral and convergence properties. In particular, we give a series expansion of the semigroup in terms of explicitly defined polynomials, which are counterparts of 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/semigroups, with the local Jacobi operator/semigroup serving as a reference object for transferring properties to the non-local ones.

On interweaving relations

“…In particular, for β ≥ 1/2 and up to waiting a warming-up time ln(1+ 1 σ ), before which Corollary (16) provides no information and is less good than (25), we get after this period an exponential rate of convergence equal to 1 (the best possible asymptotical one would be 2, i.e. twice the spectral gap of L β,σ ).…”

“…where λ 1 ≥ 2β > 2 and refer here and below to [16,Section 5] for a thorough review of the Jacobi semigroup.…”

“…We remark that, when λ 1 2 = n ∈ N, there exists a homeomorphism between J β and the radial part of the Laplace-Beltrami operator on the n-sphere, which leads to the curvature-dimension condition CD(λ 1 − 1, λ 1 ), see [9] for the definition. We deduce from [16,Proposition 3.6], choosing in the notation thereout µ = λ 1 2 and ≡ 0, the following interweaving relation between the symmetric and other Jacobi semigroups.…”

“…We close this example by mentioning that in [16] interweaving relations are established between the symmetric Jacobi semigroup and a class of non-local and non-self-adjoint Markov semigroups on the unit interval [0, 1].…”