Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

Smoothness,asymptotic smoothness and the Blum-Hanson property

We isolate various sufficient conditions for a Banach space X to have the so-called Blum-Hanson property. In particular, we show that X has the Blum-Hanson property if either the modulus of asymptotic smoothness of X has an extremal behaviour at infinity, or if X is uniformly Gâteaux smooth and embeds isometrically into a Banach space with a 1-unconditional finite-dimensional decomposition.     

Quasisimilarity of power bounded operators and Blum-Hanson property

We construct a power bounded operator on a Hilbert space which is not quasisimilar to a contraction. To this aim, we solve an open problem from operator ergodic theory showing that there are power bounded Hilbert space operators without the Blum-Hanson property. We also find an example of a power bounded operator quasisimilar to a unitary operator which is not similar to a contraction, thus answering negatively open questions raised by Kérchy and Cassier. On the positive side, we prove that contractions on ?p spaces (1?p<∞) possess the Blum-Hanson property.     

On surjective second order non-linear Markov operators and associated nonlinear integral equations

It was known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simpleces, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.     

Piecewise linear least squares approximations of Frobenius-Perron operators

We propose a piecewise linear numerical method based on least squares approximations for computing stationary density functions of Frobenius-Perron operators associated with piecewise C² and stretching mappings of the unit interval. We prove the weak convergence of the method for a class of Frobenius-Perron operators, and the numerical results show that it is also norm convergent and has a better convergence rate than the piecewise linear Markov approximation method.     

Super and weak Poincaré inequalities for hypoelliptic operators

Sufficient conditions are presented for super/weak Poincare inequalities to hold for a class of hypoelliptic operators on noncompact manifolds. As applications, the essential spectrum and the convergence rate of the associated Markov semigroup are described for Gruschin type operators on R2 and Kohn-Laplacian type operators on the Heisenberg group.     

Convolution products of probability measures on completely simple semigroups

Convolution products of probability measures are considered in the context of completely simple semigroups. Given a sequence of measures (μn)⊂Prob(S), where S is a finite completely simple semigroup, results are proven which (1) relate the supports of the measures in the sequence to the supports of the tail limit measures, and (2) determine necessary and sufficient conditions for convergence of the convolution products in the case of rectangular groups. An example showing how the theory can be used to analyze the convergence behavior of non-homogeneous Markov chains is included.     

Convergence to diffusions with regular boundaries

Earlier results on weak convergence to diffusion processes [8] are generalized to cases where the limiting diffusions may have regular boundaries. The boundaries may be adhesive or reflecting, and in each case we give two different sets of conditions for convergence. It is shown that these conditions are necessary and sufficient for convergence in the same sense as the conditions in [8]. We also extend our results to cases where the coefficients of the diffusions have simple discontinuities, in particular we thereby answer an open question by Keilson and Wellner [9]. Finally we formulate alternative sets of conditions for convergence, with these new sets being more convenient for instance when the sequence under investigation consists of pure jump Markov processes in continuous time.     

On the convergence of difference schemes for a heat conduction equation

Sufficient conditions are obtained for the convergence of difference schemes for the numerical solution of the Cauchy problem for a heat conduction equation in two space variables. The sufficient conditions are derived in a form similar to those for the convergence of a sequence of linear positive operators in the Korovkin theorem. As an application it is shown that difference schemes that are widely used in practice can easily be checked for convergence by these conditions.     

On asymptotics for Vaserstein coupling of Markov chains

We prove that strong ergodicity of a Markov process is linked with a spectral radius of a certain “associated” semigroup operator, although, not a “natural” one. We also give sufficient conditions for weak ergodicity and provide explicit estimates of the convergence rate. To establish these results we construct a modification of the Vaserstein coupling. Some applications including mixing properties are also discussed.     

Equivalence of conditions for convergence of iterative methods for singular equations

Equivalence is shown between different conditions for convergence of iterative methods for consistent singular systems of linear equations on Banach spaces. These systems appear in many applications, such as Markov chains and Markov processes. The conditions considered relate the range and null spaces of different operators.     

On the class of Markov chains with finite convergence time

We study the necessary and sufficient conditions for a finite ergodic Markov chain to converge in a finite number of transitions to its stationary distribution. Using this result, we describe the class of Markov chains which attain the stationary distribution in a finite number of steps, independent of the initial distribution. We then exhibit a queueing model that has a Markov chain embedded at the points of regeneration that falls within this class. Finally, we examine the class of continuous time Markov processes whose embedded Markov chain possesses the property of rapid convergence, and find that, in the case where the distribution of sojourn times is independent of the state, we can compute the distribution of the system at time t in the form of a simple closed expression.     

Nonzero fixed points of power-bounded linear operators

This paper provides a variety of sufficient conditions for the existence of a nonzero fixed point of a power-bounded linear operator defined on a real Banach space. In the case of power-bounded positive operators on a Banach lattice, among the conditions we provide are not being strongly stable along with commuting with a compact operator or being quasicompact. These results apply directly to Markov operators. In the case of an arbitrary power-bounded operator on a Hilbert space, being uniformly asymptotically regular and not strongly stable guarantees the existence of a nonzero fixed point.

     

Recurrency and Transiency of Markov Processes Governed by Coupled Systems of Second–Order Elliptic Operators

Markov processes corresponding to coupled systems of the form k=1,2,...,m, are considered. Here L_k are second–order linear elliptic operators and d_k,j are non–negative functions. We prove sufficient conditions for the recurrency and transiency of the Markov processes corresponding to such systems and alos provide some examples which show that it is possible to have a recurrent system even if some or all its components are transient; where by components, we mean the Markov processes engendered by the uncoupled equations. We also provide all example which shows that a system composed of recurrent processes may itself be transient. Finally, we consider necessary and sufficient conditions for the exterior Dirichlet problem of the coupled system to have a bounded solution as.     

Asymptotic Optimality of Isoperimetric Constants

In this paper, we investigate the existence of L ²(π)-spectral gaps for π-irreducible, positive recurrent Markov chains with a general state space Ω. We obtain necessary and sufficient conditions for the existence of L ²(π)-spectral gaps in terms of a sequence of isoperimetric constants. For reversible Markov chains, it turns out that the spectral gap can be understood in terms of convergence of an induced probability flow to the uniform flow. These results are used to recover classical results concerning uniform ergodicity and the spectral gap property as well as other new results. As an application of our result, we present a rather short proof for the fact that geometric ergodicity implies the spectral gap property. Moreover, the main result of this paper suggests that sharp upper bounds for the spectral gap should be expected when evaluating the isoperimetric flow for certain sets. We provide several examples where the obtained upper bounds are exact.     

Contraction mappings underlying undiscounted Markov decision problems

This paper is concerned with the properties of the value-iteration operator0 which arises in undiscounted Markov decision problems. We give both necessary and sufficient conditions for this operator to reduce to a contraction operator, in which case it is easy to show that the value-iteration method exhibits a uniform geometric convergence rate. As necessary conditions we obtain a number of important characterizations of the chain and periodicity structures of the problem, and as sufficient conditions, we give a general “scrambling-type” recurrency condition, which encompasses a number of important special cases. Next, we show that a data transformation turns every unichained undiscounted Markov Renewal Program into an equivalent undiscounted Markov decision problem, in which the value-iteration operator is contracting, because it satisfies this “scrambling-type” condition. We exploit this contraction property in order to obtain lower and upper bounds as well as variational characterizations for the fixed point of the optimality equation and a test for eliminating suboptimal actions.     

Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

Subgeometric ergodicity for continuous-time Markov chains

In this paper, subgeometric ergodicity is investigated for continuous-time Markov chains. Several equivalent conditions, based on the first hitting time or the drift function, are derived as the main theorem. In its corollaries, practical drift criteria are given for ?-ergodicity and computable bounds on subgeometric convergence rates are obtained for stochastically monotone Markov chains. These results are illustrated by examples.     

On the convergence of moments in stationary Markov chains

Necessary and sufficient conditions are given for the convergence of the first moment of functionals of Markov chains.     

Second-order methods for accretive inclusions in a Banach space

We consider equations with set-valued accretive operators in a Banach space, whose solutions are understood in the sense of inclusion. By using the resolvent, we reduce these equations to equations with single-valued operators. For the constructed problems, we suggest a continuous and an iteration second-order method and obtain sufficient conditions for their strong convergence in some class of Banach spaces.