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Lyapunov Densities For Markov Processes: An Application To Quantum Systems With Non-Demolition Measurements

Özkan Karabacak, Horia Cornean, Rafael Wisniewski

Abstract

Stochastic convergence of discrete time Markov processes has been analysed based on a dual Lyapunov approach. Using some existing results on ergodic theory of Markov processes, it has been shown that existence of a properly subinvariant function (counterpart of the Lyapunov density in deterministic systems) implies sweeping of a Markov process out of the sets where this function is integrable. Such a function can be used as a certificate of convergence in probability of a stochastic system. We apply this technique to Markov processes induced by a quantum system with non-demolition measurement and propose dual Lyapunov certificates to certify sweeping.

Lyapunov Densities For Markov Processes: An Application To Quantum Systems With Non-Demolition Measurements

Abstract

Stochastic convergence of discrete time Markov processes has been analysed based on a dual Lyapunov approach. Using some existing results on ergodic theory of Markov processes, it has been shown that existence of a properly subinvariant function (counterpart of the Lyapunov density in deterministic systems) implies sweeping of a Markov process out of the sets where this function is integrable. Such a function can be used as a certificate of convergence in probability of a stochastic system. We apply this technique to Markov processes induced by a quantum system with non-demolition measurement and propose dual Lyapunov certificates to certify sweeping.
Paper Structure (8 sections, 11 theorems, 46 equations)

This paper contains 8 sections, 11 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Sweeping Markov Operators
  3. Preliminaries: Hopf Decomposition, Subinvariant Functions and Sweeping
  4. Main Result: Dual Lyapunov Function for Sweeping of Markov Processes
  5. An Extension of Frobenius-Perron Operators
  6. Application To Quantum Systems With Non-Demolition Measurements
  7. Conclusion
  8. A Lemma for the Proof of $\mathcal{P}\rho_1<\rho_1$ in Example \ref{['ex:QND']}

Key Result

lemma 1

If $u\in L_1(X)$ and $u\geq 0$ then

Theorems & Definitions (22)

  • definition 1: Hopf decomposition Foguel1969TheProcesses
  • lemma 1: Foguel1969TheProcesses
  • lemma 2: Foguel1969TheProcesses
  • definition 2: Sweeping operator
  • definition 3: Admissible family
  • definition 4: Locally integrable function
  • lemma 3
  • proof
  • definition 5: Smoothing operator
  • lemma 4
  • ...and 12 more