Markov Semigroups Research Articles

Quantum Markov semigroups for continuous-time open quantum random walk

Open quantum random walks in Attal et al. (Phys Lett A 376:1545–1548, 2012) are quantum generalizations of classical Markov chains. In this paper, we firstly consider a problem to construct a quantum Markov semigroup (QMS) $${\mathcal {T}}_{t}$$ directly from a continuous-time open quantum random walk (CTOQRW) developed in Pellegrini (J Stat Phys 154(3):838–865, 2014) and characterize its generator as well as its predual semigroup $${\mathcal {T}}_{*t}$$ . We show that this type of CTOQRWs brings about a large field of exploration for the behavior of open quantum system. Finally, we discuss some properties of this type of QMS constructed from CTOQRWs.

A new approach to the existence of invariant measures for Markovian semigroups

On etablit une approche en deux etapes pour demontrer l’existence des mesure invariantes finies pour un semigroupe de Markov donne. En fixant d’abord une mesure auxiliaire convenable, on demontre ensuite des conditions equivalentes a l’existence d’une mesure invariante finie qui est absolument continue par rapport a elle. Comme applications, on obtient une generalisation unificatrice des diverses versions du theoreme ergodique de Harris et on fournit une reponse a une question ouverte de Tweedie. On montre aussi que pour une EDP stochastique sur un triplet de Gelfand, la condition de coercivite stricte est suffisante pour garantir l’existence d’une seule mesure de probabilite pour le semigroupe associe, si une inegalite de type Harnack avec puissance est satisfaite. Un corollaire du resultat central montre que tout semigroupe uniformement borne sur $L^{p}$ possede une mesure invariante ; on donne des applications aux perturbations sectorielles des formes de Dirichlet.

The stationary distribution in a stochastic SIS epidemic model with general nonlinear incidence

In this paper, a stochastic SIS type epidemic model with general nonlinear incidence is investigated. The diffusion process in the model is indicated to be degenerate. A new technique is proposed to investigate the existence of stationary distribution. By using the theory of integral Markov semigroup, the threshold criterion is established to ensure the existence of unique stationary distribution for the model. The numerical examples are carried out to illustrate our theoretical results.

Functional inequalities for non-symmetric stochastic differential equations

A non-symmetric Markov semigroup usually has better properties than the corresponding symmetric one.For example, Wang (2017) provides a class of non-symmetric Markov semigroups which are hypercontractive (and thus converge exponentially in both $L^2$ and entropy),but the symmetric ones are even not ergodic. In this paper, we consider the inverse problem: search for reasonable conditions to ensure that a non-symmetric Markov semigroupand its symmetrization share the properties of exponential convergence, uniform integrability, hypercontractivity, and super boundedness.Since in the symmetric case these properties are precisely characterized by functional inequalities of the Dirichlet form,the key point of the study is to prove these inequalities for non-symmetric Markov processes. Stochastic differential equations driven by Brown motion or Levy jump process are investigated.

Threshold dynamics of a stochastic SIS epidemic model with nonlinear incidence rate

In this paper, we study a stochastic SIS epidemic model with nonlinear incidence rate. By employing the Markov semigroups theory, we verify that the reproduction number R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α) can be used to govern the threshold dynamics of the studied system. If R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α)>1, we show that there is a unique stable stationary distribution and the densities of the distributions of the solutions can converge in L1 to an invariant density. If R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α)<1, under mild extra conditions, we establish sufficient conditions for extinction of the epidemic. Our results show that larger white noise can lead to the extinction of the epidemic while smaller white noise can ensure the existence of a stable stationary distribution which leads to the stochastic persistence of the epidemic.

Contiguity of States and Super Wave Operators

This paper characterises contiguity for families of states defined on a von Neumann algebra. It extends earlier research of my own published in [1], motivated by scattering theory and the celebrated work of Lucien Le Cam in classical probability. It is well-known that Le Cam did a major contribution to develop asymptotic methods in statistical decision theory, defining various new concepts, among them, contiguity of probability measures. In particular, contiguity of quantum states allows to explore different aspects of the large time behaviour of noncommutative Markov semigroups, among them, the super wave operator characterisation. I illustrate these results via quantum and classical examples of evolutions.

Compactness of symmetric Markov semigroups and boundedness of eigenfunctions

Let X X be an irreducible m m -symmetric Markov process on E E with strong Feller property. In addition, suppose X X has a tightness property. We then show that the semigroup of X X is a compact operator on L 2 ( E ; m ) L^2(E;m) and every eigenfunction has a bounded continuous version.

Long-time behaviour of a stochastic chemostat model with distributed delay

ABSTRACTIn this paper, we analyse a stochastic chemostat model with distributed delay and degenerate diffusion. We transform the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the positive solutions can converge in to an invariant density. The existence of a stable stationary distribution implies stochastic persistence of the microorganism.

Intrinsic Ultracontractivity and Ground State Estimates of Non-local Dirichlet forms on Unbounded Open Sets

In this paper we consider a large class of symmetric Markov processes $${X=(X_t)_{t\ge0}}$$ on $${\mathbb{R}^d}$$ generated by non-local Dirichlet forms, which include jump processes with small jumps of $${\alpha}$$ -stable-like type and with large jumps of super-exponential decay. Let $${D\subset \mathbb{R}^d}$$ be an open (not necessarily bounded and connected) set, and $${X^D=(X_t^D)_{t \ge 0}}$$ be the killed process of X on exiting D. We obtain explicit criterion for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup $${(P^{D}_t)_{t\ge0}}$$ of XD. When D is a horn-shaped region, we further obtain two-sided estimates of ground state in terms of jumping kernel of X and the reference function of the horn-shaped region D.

Constructing Sampling Schemes via Coupling: Markov Semigroups and Optimal Transport

In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.

Generators of Quantum One-Dimensional Diffusions

\bad Quantum dynamical semigroups represent a noncommutative analogue of (sub)Markov semigroups in classical probability: while the latter are semigroups of maps in functional spaces, the former are semigroups of maps in operator algebras having certain properties of positivity and normalization. In this paper we describe quantum dynamical semigroups, which are the noncommutative analogues of classical diffusions on $R$ and $R_{+}$, and demonstrate various properties of the semigroup and its generator depending on the boundary condition. We also give a proof of a result describing the domain of the generator of “noncommutative diffusion on $R_{+}$ with extinction at 0” and give an explicit example of the trace-class operator in this domain, which does not belong to the domain of closure of the initial operator.

A model of HBV infection with intervention strategies: dynamics analysis and numerical simulations.

In this paper, we analyze the effect of environment noise on the transmission dynamics of a stochastic hepatitis B virus (HBV) infection model with intervention strategies. By using the Markov semigroups theory, we define the stochastic basic reproduction number and find it can be used to govern disease extinction or persistence. When it is less than one, under a mild extra condition, the stochastic system has a disease-free equilibrium and the disease is predicted to die out with probability one. When it is greater than one, under mild extra conditions, the model admits a stationary distribution which means the persistence of the disease. Thus, we observe that larger intensity of noise (resulting in a smaller stochastic basic reproduction number) can suppress the emergence of hepatitis B outbreak. Numerical simulations are also carried out to investigate the influence of information intervention strategies that may change individual behavior and protect the susceptible from infection. Our analysis shows that the environmental noise can greatly a ect the long-term behavior of the system, highlighting the importance of the role of intervention strategies in the control of hepatitis B.

Viscosity solutions to Hamilton–Jacobi–Bellman equations associated with sublinear Lévy(-type) processes

Using probabilistic methods we study the existence of viscosity solutions to non-linear integro-differential equations $$\partial_t u(t,x) - \sup_{\alpha \in I} \bigg( b_{\alpha}(x) \cdot \nabla_x u(t,x) + \frac{1}{2} \text{tr}\left(Q_{\alpha}(x) \cdot \nabla^2_x u(t,x)\right) +\int_{y \neq 0} \big(u(t,x+y)-u(t,x)-\nabla_x u(t,x) \cdot h(y) \big) \, \nu_{\alpha}(x,dy) \bigg) = 0$$ with initial condition $u(0,x)= \varphi(x)$; here $(b_{\alpha}(x),Q_{\alpha}(x),\nu_{\alpha}(x,dy))$, $\alpha \in I$, $x \in \mathbb{R}^d$, is a family of L\'evy triplets and $h$ is some truncation function. The solutions, which we construct, are of the form $u(t,x) = T_t \varphi(x)$ for a sublinear Markov semigroup $(T_t)_{t \geq 0}$ with representation $$T_t \varphi(x) = \mathcal{E}^x \varphi(X_t):= \sup_{\mathbb{P} \in \mathfrak{P}_x} \int_{\Omega} \varphi(X_t) \, d\mathbb{P}$$ where $(X_t)_{t \geq 0}$ is a stochastic process and $\mathfrak{P}_x$, $x \in \mathbb{R}^d$, are families of probability measures. The key idea is to exploit the connection between sublinear Markov semigroups and the associated Kolmogorov backward equation. In particular, we obtain new existence and uniqueness results for viscosity solutions to Kolmogorov backward equations associated with L\'evy(-type) processes for sublinear expectations and Feller processes on classical probability spaces.

Concentration of quantum states from quantum functional and transportation cost inequalities

Quantum functional inequalities (e.g., the logarithmic Sobolev and Poincaré inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a so-called transportation cost inequality of order 2 (TC2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (TC2) in turn implies a transportation cost inequality of order 1 (TC1). In this paper, we introduce quantum generalizations of the inequalities (TC1) and (TC2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us. We establish that these inequalities are related to each other, and to the quantum modified logarithmic Sobolev- and Poincaré inequalities, as in the classical case. We also show that these inequalities imply certain concentration-type results for the invariant state of the underlying semigroup. We consider the example of the depolarizing semigroup to derive concentration inequalities for any finite dimensional full-rank quantum state. These inequalities are then applied to derive upper bounds on the error probabilities occurring in the setting of finite blocklength quantum parameter estimation.

Large deviations for invariant measures of stochastic differential equations with jumps

ABSTRACTLarge deviation principle (LDP) for the invariant measures of stochastic differential equations with jumps is obtained. First, we prove given as the solutions of the stochastic differential equations have invariant measures . In order to ensure the uniqueness of the invariant measures, the strong Feller property and irreducibility for the Markov semigroup are proved. Moreover, the LDP for the invariant measures is established. The proof of the LDP for is based on the LDP for , which have been studied by Budhiraja et al. [5].

Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory

The asymptotic log-Harnack inequality is established for several kinds of models on stochastic differential systems with infinite memory: non-degenerate SDEs, neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate, uniqueness of the invariant probability measure, asymptotic gradient estimate (hence, asymptotically strong Feller property), as well as asymptotic irreducibility.

Adaptive invariant density estimation for ergodic diffusions over anisotropic classes

Consider some multivariate diffusion process $\mathbf{X}=(X_{t})_{t\geq0}$ with unique invariant probability measure and associated invariant density $\rho$, and assume that a continuous record of observations $X^{T}=(X_{t})_{0\leq t\leq T}$ of $\mathbf{X}$ is available. Recent results on functional inequalities for symmetric Markov semigroups are used in the statistical analysis of kernel estimators $\widehat{\rho}_{T}=\widehat{\rho}_{T}(X^{T})$ of $\rho$. For the basic problem of estimation with respect to $\mathrm{sup}$-norm risk under anisotropic Holder smoothness constraints, the proposed approach yields an adaptive estimator which converges at a substantially faster rate than in standard multivariate density estimation from i.i.d. observations.

Invariant States for a Quantum Markov Semigroup Constructed from Quantum Bernoulli Noises

Quantum Bernoulli noises are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal-time. In this paper, we consider a quantum Markov semigroup constructed from quantum Bernoulli noises. Among others, we show that the semigroup has infinitely many faithful invariant states that are diagonal, and satisfies the quantum detailed balance condition.

Characterization of Decoherence-Free Subsystems

In this paper we compare the notion of decoherence-free subsystems with the structure of the decoherence-free subalgebra for a uniformly continuous quantum Markov semigroup on ℬ ( h ) . We show that the existence of a faithful normal invariant state and the atomicity of the decoherence-free subalgebra induce a decomposition of h that allows us to identify every decoherence-free subsystem.

Irreducibility and strong Feller property for non-linear SPDEs

ABSTRACTUnder some non-degeneracy condition, the strong Feller property and irreducibility are studied for non-linear stochastic partial differential equations driven by multiplicative noise within the framework called ‘variational approach’. Our result for irreducibility can be applied to equations with locally monotone coefficients. In some special cases, we discuss the Hölder continuity of the associated Markov semigroups. The main results are applied to several examples such as stochastic Burgers equation, stochastic porous media equation and stochastic fast diffusion equation.