A Limit Theorem for Quantum Markov Chains Associated to Beam Splittings

We study special quantum Markov chains on a Fock space related to iterated beam splittings as introduced in [23]. Besides a characterization of the position distributions of the chain, we show some kind of weak convergence of such discrete time quantum Markov chains to a kind of continuous time quantum Markov process. Furthermore, we provide existence and uniqueness for the solution of a quantum stochastic differential equation related to this quantum Markov process both on an exponential domain and, on a larger domain, in a pointwise sense.

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Quantum stochastic differential equations on *-bialgebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100070006 ◽

1991 ◽

Vol 109 (3) ◽

pp. 571-595 ◽

Cited By ~ 8

Author(s):

Peter Glockner

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Existence And Uniqueness ◽

Independent Increment ◽

Quantum Stochastic Differential Equation ◽

Independent Increment Process ◽

Stationary Independent Increment ◽

Uniqueness Of A Solution

Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.

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Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

Advances in Applied Probability ◽

10.1017/apr.2020.40 ◽

2020 ◽

Vol 52 (4) ◽

pp. 1249-1283

Author(s):

Masatoshi Kimura ◽

Tetsuya Takine

Keyword(s):

Markov Chains ◽

Stationary Distribution ◽

Continuous Time ◽

State Transition ◽

Convex Polytope ◽

Linear Inequalities ◽

Systems Of Linear Inequalities ◽

Free Sets ◽

Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

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Strong ergodicity for continuous-time, non-homogeneous Markov chains

Journal of Applied Probability ◽

10.2307/3213529 ◽

1982 ◽

Vol 19 (3) ◽

pp. 692-694 ◽

Cited By ~ 2

Author(s):

Mark Scott ◽

Barry C. Arnold ◽

Dean L. Isaacson

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Continuous Time ◽

Homogeneous Markov Chain ◽

Strong Ergodicity

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.

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On a Class of Quantum Markov Chains on the Fock Space

Quantum Probability and Related Topics - QP-PQ: Quantum Probability and White Noise Analysis ◽

10.1142/9789814350891_0012 ◽

1994 ◽

pp. 215-237 ◽

Cited By ~ 4

Author(s):

W. Freudenberg

Keyword(s):

Markov Chains ◽

Fock Space ◽

Quantum Markov Chains

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Quasi-ergodicity for non-homogeneous continuous-time Markov chains

Journal of Applied Probability ◽

10.2307/3214422 ◽

1989 ◽

Vol 26 (3) ◽

pp. 643-648 ◽

Cited By ~ 7

Author(s):

A. I. Zeifman

Keyword(s):

Differential Equation ◽

Markov Chain ◽

Markov Chains ◽

State Space ◽

Continuous Time ◽

Sufficient Conditions ◽

Continuous Time Markov Chain ◽

Countable State Space ◽

Continuous Time Markov Chains ◽

Countable State

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

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Quantum Stochastic Differential Equation is Unitarily Equivalent to a Symmetric Boundary Value Problem in Fock Space

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025798000120 ◽

1998 ◽

Vol 01 (02) ◽

pp. 175-199 ◽

Cited By ~ 12

Author(s):

Alexander M. Chebotarev

Keyword(s):

Differential Equation ◽

Schrödinger Equation ◽

Boundary Value Problem ◽

Stochastic Differential Equation ◽

Schrodinger Equation ◽

Fock Space ◽

Boundary Value ◽

Weak Limit ◽

Quantum Stochastic Differential Equation ◽

Markov Evolution

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.

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Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

Journal of Applied Probability ◽

10.1017/s0021900200019331 ◽

2003 ◽

Vol 40 (02) ◽

pp. 327-345 ◽

Cited By ~ 8

Author(s):

Xianping Guo ◽

Onésimo Hernández-Lerma

Keyword(s):

Markov Chains ◽

Continuous Time ◽

Transition Rates ◽

Zero Sum Games ◽

Stationary Strategies ◽

Continuous Time Markov Chains ◽

Average Payoff ◽

Optimal Stationary Strategies ◽

Zero Sum

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

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QUANTUM STOCHASTIC ANALYSIS VIA WHITE NOISE OPERATORS IN WEIGHTED FOCK SPACE

Reviews in Mathematical Physics ◽

10.1142/s0129055x0200117x ◽

2002 ◽

Vol 14 (03) ◽

pp. 241-272 ◽

Cited By ~ 21

Author(s):

DONG MYUNG CHUNG ◽

UN CIG JI ◽

NOBUAKI OBATA

Keyword(s):

Differential Equation ◽

Differential Equations ◽

White Noise ◽

Fock Space ◽

Existence Of A Solution ◽

Quantum Stochastic Differential Equation ◽

Unique Existence ◽

Regularity Properties ◽

Weighted Fock Space ◽

White Noises

White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.

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On Multivariate Wide-sense Markov Processes

Nagoya Mathematical Journal ◽

10.1017/s0027763000012824 ◽

1968 ◽

Vol 33 ◽

pp. 7-19 ◽

Cited By ~ 31

Author(s):

V. Mandrekar

Keyword(s):

Differential Equation ◽

Markov Process ◽

Stochastic Differential Equation ◽

Markov Processes ◽

Wide Sense ◽

Linear Vector ◽

Gaussian Case

The idea of multivariate wide-sense Markov processes has been recently used by F.J. Beutler [1], In his paper, he shows that the solution of a linear vector stochastic differential equation in a wide-sense Markov process. We obtain here a characterization of such processes and as its consequence obtain the conditions under which it satisfies Beutler’s equation. Furthermore, in stationary Gaussian case we show that these are precisely stationary Gaussian Markov processes studied by J. Doob [5].

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