Exponential Ergodicity of Classical and Quantum Markov Birth and Death Semigroups

Raffaella Carbone () and Franco Fagnola ()
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Raffaella Carbone: Università di Pavia, Dipartimento di Matematica
Franco Fagnola: Università di Genova, Dipartimento di Matematica

A chapter in Proceedings of the International Conference on Stochastic Analysis and Applications, 2004, pp 169-183 from Springer

Abstract: Abstract With a quantum Markov semigroup (T t )t≥0 on B(h) with a faithful normal invariant state ρ we associate the semigroup (T t )t≥0 on Hilbert-Schmidt operators on h (the L 2(ρ) space) defined by T t (ρ s/2 xρ (1−s)/2) = ρ s/2 T t (x)ρ (1−s)/2. This allows us to study the spectrum of the infinitesimal generator of (T t )t≥0 and deduce informations on the speed of convergence to equilibrium of the given semigroup. We apply this idea to show that some quantum Markov semigroups related to birthand-death processes converge to equilibrium exponentially rapidly in L 2(ρ). Moreover, through unitary transformations, we extend these results to other semigroups as, for instance, the quantum Ornstein-Uhlenbeck semigroup.

Keywords: Invariant State; Invariant Vector; Quantum Open System; Markov Semigroup; Identity Preserve (search for similar items in EconPapers)
Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4020-2468-9_11

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DOI: 10.1007/978-1-4020-2468-9_11

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