Distributions with Slow Tails and Ergodicity of Markov Semigroups in Infinite Dimensions

2009 ◽  
pp. 13-79 ◽  
Author(s):  
Sergey Bobkov ◽  
Boguslaw Zegarlinski
Keyword(s):  
Markov Semigroups ◽  
Infinite Dimensions

scholarly journals Markov semigroups with hypocoercive-type generator in infinite dimensions: Ergodicity and smoothing

2016 ◽  
Vol 270 (9) ◽  
pp. 3173-3223 ◽  
Author(s):  
V. Kontis ◽  
M. Ottobre ◽  
B. Zegarlinski
Keyword(s):  
Markov Semigroups ◽  
Infinite Dimensions

scholarly journals Ergodicity of Markov Semigroups with Hörmander Type Generators in Infinite Dimensions

2011 ◽  
Vol 37 (3) ◽  
pp. 199-227 ◽  
Author(s):  
Federica Dragoni ◽  
Vasilis Kontis ◽  
Bogusław Zegarliński
Keyword(s):  
Markov Semigroups ◽  
Infinite Dimensions

Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups

2021 ◽  
Vol 67 (5) ◽  
pp. 2878-2909
Author(s):  
Ivan Bardet ◽  
Marius Junge ◽  
Nicholas Laracuente ◽  
Cambyse Rouze ◽  
Daniel Stilck Franca
Keyword(s):  
Markov Semigroups

scholarly journals When are maps preserving semi-inner products linear?

2021 ◽  
Author(s):  
Paweł Wójcik
Keyword(s):  
Hilbert Space ◽  
Infinite Dimensions ◽  
Normed Spaces ◽  
Linear Isometry ◽  
Inner Products ◽  
Finite Dimensional ◽  
Uniformly Smooth ◽  
Non Linear ◽  
Extra Assumption ◽  
Continuous Injection

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

scholarly journals Correction to: Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions

2021 ◽  
Author(s):  
Nawaf Bou-Rabee ◽  
Andreas Eberle
Keyword(s):  
Monte Carlo ◽  
Infinite Dimensions ◽  
Hamiltonian Monte Carlo

A Correction to this paper has been published: 10.1007/s40072-020-00175-6

scholarly journals Convergence Rates for Quantum Evolution and Entropic Continuity Bounds in Infinite Dimensions

2019 ◽  
Vol 374 (2) ◽  
pp. 823-871 ◽  
Author(s):  
Simon Becker ◽  
Nilanjana Datta
Keyword(s):  
Convergence Rates ◽  
Lipschitz Continuity ◽  
Open Quantum Systems ◽  
High Energy ◽  
Quantum Systems ◽  
Quantum Evolution ◽  
Infinite Dimensions ◽  
Quantum Brownian Motion ◽  
Infinite Dimensional ◽  
Energy Constrained

Abstract By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite dimensions, which are stronger than previously known bounds. We extensively discuss applications of our theory to quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann equation, and quantum Brownian motion. Next, we obtain explicit log-Lipschitz continuity bounds for entropies of infinite-dimensional quantum systems, and classical capacities of infinite-dimensional quantum channels under energy-constraints. These bounds are determined by the high energy spectrum of the underlying Hamiltonian and can be evaluated using Weyl’s law.

scholarly journals Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions

1986 ◽  
Vol 65 (3) ◽  
pp. 368-405 ◽  
Author(s):  
Michael G Crandall ◽  
Pierre-Louis Lions
Keyword(s):  
Viscosity Solutions ◽  
Infinite Dimensions ◽  
Jacobi Equations
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