Linear and Nonlinear Dissipative Dynamics

Zegarliński, Bogusław

doi:10.1016/s0034-4877(16)30035-0

Cited by 3 publications

(2 citation statements)

References 46 publications

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“…Similar analysis for Dunkl-type operators in the infinite dimensional setting but in a simplified context corresponding to the very particular case of a root system A 1 , was studied in [16]. In this paper we extend these results to the case of general root systems in all dimensions.…”

Section: Introductionmentioning

confidence: 52%

Infinite dimensional systems of particles with interactions given by Dunkl operators

Velicu¹

2019

Preprint

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Firstly we consider a finite dimensional Markov semigroup generated by Dunkl laplacian with drift terms. Using gradient bounds we show that for small coefficients this semigroup has an invariant measure. We then extend this analysis to an infinite dimensional semigroup on pR N q Z d which we construct using gradient bounds, and finally we study the existence of invariant measures and ergodicity properties.

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Section: Introductionmentioning

confidence: 52%

Infinite dimensional systems of particles with interactions given by Dunkl operators

Velicu¹

2019

Preprint

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“…In the late 20th century, Olkiewicz and Zegarlinski completed a crucial step by generalising Gross' log-Sobolev inequalities [14] in non-commutative L p space, and proved their equivalence with hypercontractivity under some regular conditions [25]. New results and progress has thus emerged in this line of research [26,27,28,29,30] (see also [31] and the references therein). Specifically, noncommutative log-Sobolev inequalities enable the analysis of the dynamics of von Neumann algebras, including their most interesting case -the time evolution of a single quantum state through a Markov process [32].…”

Section: Introductionmentioning

confidence: 99%

Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles

Cheng

Hsieh

2019

Journal of Mathematical Physics

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Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative Lp spaces, and have proven useful in bounding the time evolution of classical/quantum Markov processes, among many other applications. In this paper, we consider yet another fundamental setting -matrix-valued functions -and prove new Sobolev-type inequalities for them. Our technical contributions are two-fold: (i) we establish a series of matrix Poincaré inequalities for separably convex functions and general functions with Gaussian unitary ensembles inputs; and (ii) we derive Φ-Sobolev inequalities for matrix-valued functions defined on Boolean hypercubes and for those with Gaussian distributions. Our results recover the corresponding classical inequalities (i.e. real-valued functions) when the matrix has one dimension. Finally, as an application of our technical outcomes, we derive the upper bounds for a fundamental entropic quantity -the Holevo quantity -in quantum information science since classical-quantum channels are a special instance of matrix-valued functions. This is obtained through the equivalence between the constants in the strong data processing inequality and the Φ-Sobolev inequality.16

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Infinite Dimensional Systems of Particles with Interactions Given by Dunkl Operators

Velicu

2020

Potential Anal

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Linear and Nonlinear Dissipative Dynamics

Cited by 3 publications

References 46 publications

Infinite dimensional systems of particles with interactions given by Dunkl operators

Infinite dimensional systems of particles with interactions given by Dunkl operators

Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles

Infinite Dimensional Systems of Particles with Interactions Given by Dunkl Operators

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