Markovian Coupling Research Articles

Optimal Markovian coupling for finite activity Lévy processes

Optimal Markovian coupling for finite activity Lévy processes

On Convergence Rate Bounds for a Class of Nonlinear Markov Chains

A new approach is developed for evaluating the convergence rate for nonlinear Markov chains (MC) based on the recently developed spectral radius technique of Markovian coupling for linear MC and the idea of small nonlinear perturbations of linear MC. The method further enhances recent advances in the problem of convergence for such models.

Measuring High-Order Phonon Correlations in an Optomechanical Resonator.

We use single photon detectors to probe the motional state of a superfluid ^{4}He resonator of mass ∼1 ng. The arrival times of Stokes and anti-Stokes photons (scattered by the resonator's acoustic mode) are used to measure the resonator's phonon coherences up to the fourth order. By postselecting on photon detection events, we also measure coherences in the resonator when ≤3 phonons have been added or subtracted. These measurements are found to be consistent with predictions that assume the acoustic mode to be in thermal equilibrium with a bath through a Markovian coupling.

Crossing time in the dissipative Landau\u2013Zener quantum dynamics

We study the dynamics of a quantum two-state system driven through an avoided crossing under the influence of a super-Ohmic environment. We determine the Landau–Zener probability employing the numerical exact quasi-adiabatic path integral and a Markovian weak coupling approach. Increasing the driving time in the numerical protocol, we find converged results which shows that super-Ohmic environments only influence the Landau Zener probability within a finite crossing time window. This crossing time is qualitatively determined by the environmental cut-off energy. At weak coupling, we show that the Markovian weak coupling approach provides an accurate description. Since pure dephasing of a super-Ohmic bath is non-Markovian, this highlights that pure dephasing hardly influences the Landau–Zener probability. The finite crossing time window, thus, results from the suppression of relaxation once the energy splitting exceeds the environmental cut-off energy.Graphical abstract

The Strassen invariance principle for certain non-stationary Markov–Feller chains

We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov–Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression.

Input-Output Theory with Quantum Pulses.

We present a formalism that accounts for the interaction of a local quantum system, such as an atom or a cavity, with traveling pulses of quantized radiation. We assume Markovian coupling of the stationary system to the input and output fields and nondispersive asymptotic propagation of the pulses before and after the interaction. This permits derivation of a master equation where the input and output pulses are treated as single oscillator modes that both couple to the local system in a cascaded manner. As examples of our theory, we analyze reflection by an empty cavity with phase noise, stimulated atomic emission by a quantum light pulse, and formation of a Schrödinger-cat state by the dispersive interaction of a coherent pulse and a single atom in a cavity.

Synchronization of networked harmonic oscillators subject to Markovian jumping coupling strengths

To explore the effect the switching coupling strengths on synchronization of complex dynamical networks, both an impulsive coupling protocol and a piecewise constant coupling protocol are designed by using periodic sampling velocity data for a group of harmonic oscillators. Moreover, the convergence of these protocols with switching coupling strengths is discussed and some sufficient conditions are established under which the networked harmonic oscillators could achieve collective oscillatory behavior in almost sure sense. It is shown that the almost sure exponential synchronization can be reached even when the coupling strength is switched between enabling and not enabling to achieve synchronization. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results.

Coupling the Kolmogorov diffusion: maximality and efficiency considerations

AbstractThis is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral or, more generally, together with a finite number of iterated time integrals). We show that in this case there can be no Markovian maximal coupling. Indeed, there can be no efficient Markovian coupling strategy (efficient for all pairs of distinct starting values), where the notion of efficiency extends the terminology of Burdzy and Kendall (2000). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling strategy.

Distributed H ∞ filtering over sensor networks with heterogeneous Markovian coupling intercommunication delays

This study is concerned with distributed H∞ filtering for continuous-time linear systems over sensor networks with heterogeneous Markovian coupling intercommunication delays. The set of sensor nodes forms a sensing and communication network whose topology is modelled by a directed graph that describes the measurement exchanged among neighbouring sensor nodes. Heterogeneous random coupling delays modelled by a Markov process are considered in the intercommunication between interacting sensor nodes. A refined two-step decoupling technique is presented to deal with the complicated coupling of exchanged measurement in the presence of the delays. A sufficient condition on the existence of desired distributed H∞ filters is derived such that the resultant filtering error system is mean square exponentially stable with prescribed weighting average H∞ performance. Two illustrative examples are given to show the effectiveness of the proposed results.

Maximal Coupling of Euclidean Brownian Motions

We prove that the mirror coupling is the unique maximal Markovian coupling of two Euclidean Brownian motions starting from single points and discuss the connection between the uniqueness of maximal Markovian coupling of Brownian motions and certain mass transportation problems.

Avoidance Coupling

We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide. In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling keeping apart Omega(n/log n) random walks, taking turns to move in discrete time.

Characterization of maximal Markovian couplings for diffusion processes

Necessary conditions for the existence of a maximal Markovian coupling of diffusion processes are studied. A sufficient condition described as a global symmetry of the processes is revealed to be necessary for the Brownian motion on a Riemannian homogeneous space. As a result, we find many examples of a diffusion process which admits no maximal Markovian coupling. As an application, we find a Markov chain which admits no maximal Markovian coupling for specified starting points.

Numerically Optimized Markovian Coupling and Mixing in One‐Dimensional Maps

Numerically Optimized Markovian Coupling and Mixing in One‐Dimensional Maps

Optimally Coupling the Kolmogorov Diffusion, and Related Optimal Control Problems

AbstractWe discuss the optimal Markovian coupling before an exponential time of the Kolmogorov diffusion, and a class of related stochastic control problems in which the aim is to hit the origin before an exponential time. We provide a scaling argument for the optimal control in the near field and use rational WKB approximation to obtain the optimal control in the far field, and compare these analytical results with numerical experiments. In some of these optimal control problems, in which the advection velocity field is bounded, we show that the probability of success agrees exactly with its leading-order asymptotic approximation in some areas of the plane, up to an undetermined multiplicative constant. We conjecture a necessary and sufficient condition for this behaviour, which is strongly supported by numerical experiments.

Local conditions for the stochastic comparison of particle systems

We study the stochastic comparison of interacting particle systems where the state space of each particle is a finite set endowed with a partial order, and several particles may change their value at a time. For these processes we give local conditions, on the rates of change, that assure their comparability. We also analyze the case where one of the processes does not have any changes that involve several particles, and obtain necessary and sufficient conditions for their comparability. The proofs are based on the explicit construction of an order-preserving Markovian coupling. We show the applicability of our results by studying the stochastic comparison of infinite-station Jackson networks and batch-arrival, batch-service, and assemble-transfer queueing networks.

Local conditions for the stochastic comparison of particle systems

We study the stochastic comparison of interacting particle systems where the state space of each particle is a finite set endowed with a partial order, and several particles may change their value at a time. For these processes we give local conditions, on the rates of change, that assure their comparability. We also analyze the case where one of the processes does not have any changes that involve several particles, and obtain necessary and sufficient conditions for their comparability. The proofs are based on the explicit construction of an order-preserving Markovian coupling. We show the applicability of our results by studying the stochastic comparison of infinite-station Jackson networks and batch-arrival, batch-service, and assemble-transfer queueing networks.

NECESSARY AND SUFFICIENT CONDITIONS FOR THE STOCHASTIC COMPARISON OF JACKSON NETWORKS

External and internal monotonicity properties for Jackson networks have been established in the literature with the use of coupling constructions. Recently, Lopez et al. derived necessary and sufficient conditions for the (strong) stochastic comparison of two-station Jackson networks with increasing service rates, by constructing a certain Markovian coupling. In this article, we state necessary and sufficient conditions for the stochastic comparison of L-station Jackson networks in the general case. The proof is based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.

Stochastic domination and Markovian couplings

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P ≤ stP', we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.

Existence and Application of Optimal Markovian Coupling with Respect to Non-Negative Lower Semi-Continuous Functions

In this paper, for two given transition probabilities, the existence of the optimal Markovian coupling with respect to a non-negative lower semi-continuous function is proved. As an application of this result, the well-known Strassen's theorem is generalized. Moreover, it is proved that the existence of an order-preserving Markovian coupling of two given jump processes is equivalent to their stochastical comparability.

Existence and Application of Optimal Markovian Coupling with Respect to Non-Negative Lower Semi-Continuous Functions

Existence and Application of Optimal Markovian Coupling with Respect to Non-Negative Lower Semi-Continuous Functions