Markov Operators Research Articles

Equicontinuous families of Markov operators in view of asymptotic stability

The relation between the equicontinuity – the so-called e-property – and the stability of Markov operators is studied. In particular, it is shown that any asymptotically stable Markov operator with an invariant measure such that the interior of its support is non-empty satisfies the e-property.

About the $L^2$ analyticity of Markov operators on graphs

Let $\Gamma$ be a graph and $P$ be a reversible random walk on $\Gamma$. From the $L^2$ analyticity of the Markov operator $P$, we deduce that an iterate of odd exponent of $P$ is `lazy', that is there exists an integer $k$ such that the transition probability (for the random walk $P^{2k+1}$) from a vertex $x$ to itself is uniformly bounded from below. The proof does not require the doubling property on $\Gamma$ but only a polynomial control of the volume.

On non-linear Markov operators: surjectivity vs orthogonal preserving property

In the present paper, we consider non-linear Markov operators, namely polynomial stochastic operators. We introduce a notion of orthogonal preserving polynomial stochastic operators. The purpose of this study is to show that surjectivity of non-linear Markov operators is equivalent to their orthogonal preserving property.

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part II Irreversibility, norms and entropies

In this second part, we analyze the dissipation properties of generalized Poisson–Kac (GPK) processes, considering the decay of suitable L2-norms and the definition of entropy functions. In both cases, consistent energy dissipation and entropy functions depend on the whole system of primitive statistical variables, the partial probability density functions , while the corresponding energy dissipation and entropy functions based on the overall probability density do not satisfy monotonicity requirements as a function of time. These results provide new insights on the theory of Markov operators associated with irreversible stochastic dynamics. Examples from chaotic advection (standard map coupled to stochastic GPK processes) illustrate this phenomenon. Some complementary physical issues are also addressed: the ergodicity breaking in the presence of attractive potentials, and the use of GPK perturbations to mollify stochastic field equations.

A generalization of Kantorovich operators for convex compact subsets

In this article, we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators defined in the 1-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity.

Lower bounds and the asymptotic behaviour of positive operator semigroups

If $(T_{t})$ is a semigroup of Markov operators on an $L^{1}$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t\rightarrow \infty$. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius–Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.

Kazhdan projections, random walks and ergodic theorems

Abstract In this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. This construction exhibits useful properties and flexibility, and allows to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address several open questions. We give a direct comparison of properties (TE) and FE with Lafforgue’s reinforced Banach property (T); we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we construct non-compact ghost projections for warped cones. In this last case we conjecture that such warped cones provide counterexamples to the coarse Baum–Connes conjecture.

Large Deviations for Stationary Measures of Stochastic Nonlinear Wave Equations\\with Smooth White Noise

The paper is devoted to the derivation of the large deviations principle for the family of stationary measures of the Markov process generated by the flow of the damped nonlinear wave equations with vanishing white noise. One of the main novelties here is that we do not assume that the deterministic equation possesses a unique equilibrium and we do not impose roughness on the noise. We introduce a new mathematical tool called the generalized stationary measure, which, informally speaking, is a stationary measure that is not necessarily σ‐additive. We show that any Markov operator admits such a measure and use this to develop the Freidlin‐Wentzell and Khasminskii approaches to the infinite‐dimensional setting. We also extend Sowers' method when establishing exponential tightness. Some ingredients of the proof rely on rather nonstandard techniques.© 2017 Wiley Periodicals, Inc.

Uniform ergodicities and perturbation bounds of Markov chains on ordered Banach spaces

In this paper, we consider uniformly mean ergodic and uniformly asymptotical stable Markov operators on ordered Banach spaces. In terms of the ergodicity coefficient, we show the equivalence of uniform and weak mean ergodicities of Markov operators. This result allowed us to establish a category theorem for uniformly mean ergodic Markov operators. Furthermore, using properties of the ergodicity coefficient, we develop the perturbation theory for uniformly asymptotical stable Markov chains in the abstract scheme.

Non-atomic bivariate copulas and implicitly dependent random variables

Two (continuous) random variables X and Y are implicitly dependent if there exist Borel functions α and β such that α∘X=β∘Y almost surely. The copulas of such random variables are exactly the copulas that are factorizable as the ∗-product of a left invertible copula and a right invertible copula. Consequently, every implicit dependence copula assigns full mass to the graph of f(x)=g(y) for some measure-preserving functions f and g but the converse is not true in general.We obtain characterizations of a copula C assigning full mass to the graph of f(x)=g(y) in terms of a partial factorizability of its Markov operator TC and in terms of the non-atomicity of two newly defined associated σ-algebras σC and σC∗, in which case C is called non-atomic. As an application, we give a broad sufficient condition under which a copula with fractal support has an implicit dependence support. Under certain extra conditions, we explicitly compute the left invertible and right invertible factors of the copula with fractal support.

Variational principles and Lagrangian functions for stochastic processes and their dissipative statistical descriptions

A variational Lagrangian formulation for stochastic processes and for the evolution equations of the associated probability density functions is developed. Particular attention is dedicated to Poisson–Kac processes possessing finite propagation velocity. The variational formulation in terms of Lagrangian and Hamiltonian densities permits to address different forms of “reversibility” characterizing these processes with respect to processes driven by nowhere differentiable Wiener fluctuations, and associated with the concepts of dynamic and statistical reversibility. The latter property, i.e. the statistical reversibility, implies that the extended Markov operator associated with Poisson–Kac and Generalized Poisson–Kac processes forms a group, continuously parametrized with respect to time.

Value of Persistent Information

We develop a theory of how the value of an agent's information advantage depends on the persistence of information. We focus on strategic situations with strict conflict of interest, formalized as stochastic zero‐sum games where only one of the players observes the state that evolves according to a Markov operator. Operator Q is said to be better for the informed player than operator P if the value of the game under Q is higher than under P regardless of the stage game. We show that this defines a convex partial order on the space of ergodic Markov operators. Our main result is a full characterization of this partial order, intepretable as an ordinal notion of persistence relevant for games. The analysis relies on a novel characterization of the value of a stochastic game with incomplete information.

Tốc độ hội tụ trong định lý giới hạn trung tâm cho bước đi ngẫu nhiên trong một chiều

In this paper, we study the model of random walk with state space . We use the method of moments as in Depauw et al.’s paper (2009) and Lam Hoang Chuong’s paper (2014) to prove that this random walk conveges in distribution to a normal law (Theorem 1.3) and give its rate also (Theorem 3.1). More precisely, with be the corresponding Markov operator of the previous random walk and a given function f, we solve the Poisson equation and then treat the limits of its solutions, the rate of the convergence is instantly given by the convergence of the moment of random walk.

On the reflected random walk on R+

Let ρ be a borelian probability measure on R having a moment of order 1 and a drift λ = ∫Rydρ(y) < 0. Consider the random walk on R+ starting at x ∈ R+ and defined for any n ∈N by \begin{eqnarray*} \left\{\begin{array}{rl} X_0&=x \\ X_{n+1} & = |X_n+Y_{n+1}| \end{array}\right. \end{eqnarray*} where (Yn) is an iid sequence of law ρ. We denote P the Markov operator associated to this random walk and, for any borelian bounded function f on R+, we call Poisson’s equation the equation f = g − Pg with unknown function g. In this paper, we prove that under a regularity condition on ρ and f, there is a solution to Poisson’s equation converging to 0 at infinity. Then, we use this result to prove the functional central limit theorem and it’s almost-sure version.

Analysis of Polya-Gamma Gibbs sampler for Bayesian logistic analysis of variance

We consider the intractable posterior density that results when the one-way logistic analysis of variance model is combined with a flat prior. We analyze Polson, Scott and Windle’s (2013) data augmentation (DA) algorithm for exploring the posterior. The Markov operator associated with the DA algorithm is shown to be trace-class.

On a Characterization of Gaussian Transition Operators

In this paper we give a detailed proof of the characterization of the Gaussian Markov operators mapping a set of Gaussian probability measures into itself. We also characterize the integral operators with Gaussian kernels that map the set of unnormalized Gaussian measures into itself.

A note on a Wiener-Wintner theorem for mean ergodic Markov amenable semigroups

We prove a Wiener-Wintner ergodic type theorem for a Markov representation S = { S g : g ∈ G } \mathcal {S} = \{ S_g : g\in G \} of a right amenable semitopological semigroup G G . We assume that S \mathcal {S} is mean ergodic as a semigroup of linear Markov operators acting on ( C ( K ) , ‖ ⋅ ‖ sup ) (C(K), \| \cdot \|_{\sup }) , where K K is a fixed Hausdorff, compact space. The main result of the paper is necessary and sufficient conditions for mean ergodicity of a distorted semigroup { χ ( g ) S g : g ∈ G } \{ \chi (g)S_g : g\in G \} , where χ \chi is a semigroup character. Such conditions were obtained before under the additional assumption that S \mathcal {S} is uniquely ergodic.

Trace Class Markov Chains for Bayesian Inference with Generalized Double Pareto Shrinkage Priors

AbstractBayesian shrinkage methods have generated a lot of interest in recent years, especially in the context of high‐dimensional linear regression. In recent work, a Bayesian shrinkage approach using generalized double Pareto priors has been proposed. Several useful properties of this approach, including the derivation of a tractable three‐block Gibbs sampler to sample from the resulting posterior density, have been established. We show that the Markov operator corresponding to this three‐block Gibbs sampler is not Hilbert–Schmidt. We propose a simpler two‐block Gibbs sampler and show that the corresponding Markov operator is trace class (and hence Hilbert–Schmidt). Establishing the trace class property for the proposed two‐block Gibbs sampler has several useful consequences. Firstly, it implies that the corresponding Markov chain is geometrically ergodic, thereby implying the existence of a Markov chain central limit theorem, which in turn enables computation of asymptotic standard errors for Markov chain‐based estimates of posterior quantities. Secondly, because the proposed Gibbs sampler uses two blocks, standard recipes in the literature can be used to construct a sandwich Markov chain (by inserting an appropriate extra step) to gain further efficiency and to achieve faster convergence. The trace class property for the two‐block sampler implies that the corresponding sandwich Markov chain is also trace class and thereby geometrically ergodic. Finally, it also guarantees that all eigenvalues of the sandwich chain are dominated by the corresponding eigenvalues of the Gibbs sampling chain (with at least one strict domination). Our results demonstrate that a minor change in the structure of a Markov chain can lead to fundamental changes in its theoretical properties. We illustrate the improvement in efficiency resulting from our proposed Markov chains using simulated and real examples.

Commute Maps: Separating Slowly Mixing Molecular Configurations for Kinetic Modeling.

Identification of the main reaction coordinates and building of kinetic models of macromolecular systems require a way to measure distances between molecular configurations that can distinguish slowly interconverting states. Here we define the commute distance that can be shown to be closely related to the expected commute time needed to go from one configuration to the other, and back. A practical merit of this quantity is that it can be easily approximated from molecular dynamics data sets when an approximation of the Markov operator eigenfunctions is available, which can be achieved by the variational approach to approximate eigenfunctions of Markov operators, also called variational approach of conformation dynamics (VAC) or the time-lagged independent component analysis (TICA). The VAC or TICA components can be scaled such that a so-called commute map is obtained in which Euclidean distance corresponds to the commute distance, and thus kinetic models such as Markov state models can be computed based on Euclidean operations, such as standard clustering. In addition, the distance metric gives rise to a quantity we call total kinetic content, which is an excellent score to rank input feature sets and kinetic model quality.

Three-dimensional Gaussian fluctuations of non-commutative random surfaces along time-like paths

We construct a continuous-time non-commutative Markov operator on U(glN) with corresponding morphisms U(glN)→End(L2(U(N)))⊗∞ at any fixed times 0≤t1<t2<…. This is an analog of a continuous-time non-commutative Markov operator on the group von Neumann algebra vN(U(N)) constructed in [16], and is a variant of discrete-time non-commutative random walks on U(glN)[2,9].It is also shown that when restricting to the Gelfand–Tsetlin subalgebra of U(glN), the non-commutative Markov operator matches a (2+1)-dimensional random surface model introduced in [7]. As an application, it is then proved that the moments converge to an explicit Gaussian field along time-like paths. Combining with [7] which showed convergence to the Gaussian free field along space-like paths, this computes the entire three-dimensional Gaussian field. In particular, it matches a Gaussian field from eigenvalues of random matrices [5].