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

Patie, Pierre; Vaidyanathan, Aditya

doi:10.48550/arxiv.1905.07042

Cited by 2 publications

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“…Similar results have been obtained by Patie and Savov in [40] and Achleitner et al in [1]. Our hypocoercive estimate is obtained via intertwining, which suggests that hypocoercivity may be studied purely from this viewpoint, an idea that is further investigated in the recent work by the second and fourth co-authors [42].…”

Section: 3supporting

confidence: 90%

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

Cheridito¹,

Patie²,

Srapionyan³

et al. 2019

Preprint

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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.

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Section: 3supporting

confidence: 90%

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

Cheridito¹,

Patie²,

Srapionyan³

et al. 2019

Preprint

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“…We now describe a refined version of a interweaving relation between degenerate and non-degenerate hypoelliptic Ornstein-Uhlenbeck semigroups on R d , d ≥ 1, that was identified in [40]. In that paper, the authors exploit the interweaving relations to obtain the hypocoercive estimate with explicit constants for the convergence to equilibrium in the weighted Hilbert space of the degenerate hypoelliptic Ornstein-Uhlenbeck semigroups which are non-normal.…”

Section: Degenerate Hypoelliptic Ornstein-uhlenbeck Processesmentioning

confidence: 99%

“…(where α(ε) is defined in ( 14)). The bounds (40) and ( 41) are not directly comparable, since they concern different distributions, namely m 0 P (ε) t+τ (βε) and m 0 P (ε) t and the former is not just a deterministic time translate through P of the latter. Nevertheless, to highlight the potential advantage of (40), let us make the following observation.…”

Section: It Follows Thatmentioning

confidence: 99%

“…The bounds (40) and ( 41) are not directly comparable, since they concern different distributions, namely m 0 P (ε) t+τ (βε) and m 0 P (ε) t and the former is not just a deterministic time translate through P of the latter. Nevertheless, to highlight the potential advantage of (40), let us make the following observation. Let X (ε) ≔ (X (ε) t ) t≥0 be a diffusion process associated to the Markov semigroup P (ε) , with small ε ∈ (0, 1/2), starting with X (ε) 0 uniformly distributed over [0, 1].…”

Section: It Follows Thatmentioning

confidence: 99%

“…Relying on (40), we consider the position X (ε) T 2 +T 3 , where T 2 is independent from X (ε) and has the same law than τ ε and T 3 ≥ 0 is such that…”

Section: It Follows Thatmentioning

confidence: 99%

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On interweaving relations

Miclo,

Patie

2019

Preprint

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Interweaving relations are introduced and studied here in a general Markovian setting as a strengthening of usual intertwining relations between semigroups, obtained by adding a randomized delay feature. They provide a new classification scheme of the set of Markovian semigroups which enables to transfer from a reference semigroup and up to an independent warm-up time, some ergodic, analytical and mixing properties including the ϕ-entropy convergence to equilibrium, the hyperboundedness and when the warm-up time is deterministic the cut-off phenomena. We also present several useful transformations that preserve interweaving relations. We provide a variety of examples of interweaving relations ranging from classical, discrete, and non-local Laguerre and Jacobi semigroups to degenerate hypoelliptic Ornstein-Uhlenbeck semigroups and some non-colliding particle systems.

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

Cited by 2 publications

References 0 publications

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

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

On interweaving relations

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