Markov Integrated Semigroups and their Applications to Continuous-Time Markov Chains

A Markov integrated semigroup G(t) is by definition a weaklystar differentiable and increasing contraction integrated semigroup on l _∞. We obtain a generation theorem for such semigroups and find that they are not integrated C ₀-semigroups unless the generators are bounded. To link up with the continuous-time Markov chains (CTMCs), we show that there exists a one-to-one relationship between Markov integrated semigroups and transition functions. This gives a clear probability explanation of G(t): it is just the mean transition time, and allows us to define and to investigate its q-matrix. For a given q-matrix Q, we give a criterion for the minimal Q-function to be a Feller-Reuter-Riley (FRR) transition function, this criterion gives an answer to a long-time question raised by Reuter and Riley (1972). This research was supported by the China Postdoctoral Science Foundation (No.2005038326).     

Eigentime identity for transient Markov chains

An eigentime identity is proved for transient symmetrizable Markov chains. For general Markov chains, if the trace of Green matrix is finite, then the expectation of first leap time is uniformly bounded, both of which are proved to be equivalent for single birth processes. For birth-death processes, the explicit formulas are presented. As an application, we give the bounds of exponential convergence rates of (sub-) Markov semigroup Pt from l_∞ to l_∞.     

Dual and Feller-Reuter-Riley transition functions

In this paper, we investigate duality and Feller-Reuter-Riley (FRR) property of continuous-time Markov chains (CTMCs). A criterion of dual q-functions is given in terms of their q-matrices. For a dual q-matrix Q, a necessary and sufficient conditions for the minimal Q-function to be a FRR transition function are also given. Finally, by using dual technique, we give a criterion of FRR Q-functions when Q is monotone.     

Contraction Integrated Semigroups and Their Application to Continuous-time Markov Chains

We introduce the notion of the contraction integrated semigroups and give the Lumer-Phillips characterization of the generator, and also the charaterazied generators of isometric integrated semigroups. For their application, a necessary and sufficient condition for q-matrices Q generating a contraction integrated semigroup is given, and a necessary and sufficient condition for a transition function to be a Feller-Reuter-Riley transition function is also given in terms of its q-matrix.     

On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process

In this paper, sufficient conditions are given for the existence of limiting distribution of a nonhomogeneous countable Markov chain with time-dependent transition intensity matrix. The method of proof exploits the fact that if the distribution of random process Q=(Q _t ) _t≥0 is absolutely continuous with respect to the distribution of ergodic random process Q°=(Q° _t ) _t≥0, then $Q_t \xrightarrow[{t \to \infty }]{{law}}\pi $ where π is the invariant measure of Q°. We apply this result for asymptotic analysis, as t→∞, of a nonhomogeneous countable Markov chain which shares limiting distribution with an ergodic birth-and-death process.     

Existence and Regularity of a Nonhomogeneous Transition Matrix under Measurability Conditions

This paper is about the existence and regularity of the transition probability matrix of a nonhomogeneous continuous-time Markov process with a countable state space. A standard approach to prove the existence of such a transition matrix is to begin with a continuous (in t≥0) and conservative matrix Q(t)=[q _ij (t)] of nonhomogeneous transition rates q _ij (t) and use it to construct the transition probability matrix. Here we obtain the same result except that the q _ij (t) are only required to satisfy a mild measurability condition, and Q(t) may not be conservative. Moreover, the resulting transition matrix is shown to be the minimum transition matrix, and, in addition, a necessary and sufficient condition for it to be regular is obtained. These results are crucial in some applications of nonhomogeneous continuous-time Markov processes, such as stochastic optimal control problems and stochastic games, and this was the main motivation for this work. Supported by NSFC and RFDP. The research of O. Hernández-Lerma was partially supported by CONACYT grant 45693-F.     

On a Class of Elliptic-Parabolic Equations on Unbounded Intervals

We study a class of degenerate elliptic second order differential operators acting on some polynomial weighted function spaces on [0,+[. We show that these operators are the generators of C ₀-semigroups of positive operators which, in turn, are the transition semigroups associated with right-continuous normal Markov processes with state space [0,+]. Approximation and qualitative properties of both the semigroups and the Markov processes are investigated as well. Most of the results of the paper depend on a representation of the semigroups we give in terms of powers of particular positive operators of discrete type we introduced and studied in a previous paper.     

Decay parameter and related properties of n-type branching processes

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions for n-type Markov branching processes on the basis of the 1-type Markov branching processes and 2-type Markov branching processes. Investigating such behavior is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for n-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Zn+ \ 0. It is shown that this λC can be directly obtained from the generating functions of the corresponding q-matrix. Moreover, the λC -invariant measures/vectors and quasi-distributions of such processes are deeply considered. λC -invariant measures and quasi-stationary distributions for the process on C are presented.     

Quasi-stationarity and quasi-ergodicity of general Markov processes

In this paper, we study the quasi-stationarity and quasi-ergodicity of general Markov processes. We show, among other things, that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t, x, y) (with respect to m) which is bounded in (x, y) for every t > 0, then X has a unique quasi-stationary distribution and a unique quasi-ergodic distribution. We also present several classes of Markov processes satisfying the above conditions.     

Inverse problems for ergodicity of Markov chains

For continuous-time Markov chains, we provide criteria for non-ergodicity, non-algebraic ergodicity, non-exponential ergodicity, and non-strong ergodicity. For discrete-time Markov chains, criteria for non-ergodicity, non-algebraic ergodicity, and non-strong ergodicity are given. Our criteria are in terms of the existence of solutions to inequalities involving the Q-matrix (or transition matrix P in time-discrete case) of the chain. Meanwhile, these practical criteria are applied to some examples, including a special class of single birth processes and several multi-dimensional models.     

An extension of a Phillips's theorem to Banach algebras and application to the uniform continuity of strongly continuous semigroups

In this work we present an extension to arbitrary unital Banach algebras of a result due to Phillips [R.S. Phillips, Spectral theory of semigroups of linear operators, Trans. Amer. Math. Soc. 71 (1951) 393-415] (Theorem 1.1) which provides sufficient conditions assuring the uniform continuity of strongly continuous semigroups of linear operators. It implies that, when dealing with the algebra of bounded operators on a Banach space, the conditions of Phillips's theorem are also necessary. Moreover, it enables us to derive necessary and sufficient conditions in terms of essential spectra which guarantee the uniform continuity of strongly continuous semigroups. We close the paper by discussing the uniform continuity of strongly continuous groups (T(t))_t∈R acting on Banach spaces with separable duals such that, for each t∈R, the essential spectrum of T(t) is a finite set.     

Applications of Feller–Reuter–Riley Transition Functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.     

Decay Properties of <Emphasis Type="Italic">n</Emphasis>-type Markov Branching Processes with Disasters

In this paper, we consider the decay properties of n-type Markov branching processes with disasters, including the decay parameter and invariant measures. It is first proved that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest. Then, the exact value of the decay parameter \(\lambda _C\) is obtained. We prove that the decay parameter can be easily expressed explicitly. Furthermore, we prove that the Markov branching process with disasters is always \(\lambda _C\)-positive. The invariant vectors and the invariant measures are given explicitly.     

Sulľapplicazione della funzione ellitticap u alla teoria dei poligoni di poncelet

Abstract  In this paper we study strongly continuous positive semigroups on particular classes of weighted continuous function space on a locally compact Hausdorff space X having a countable base. In particular we characterize those positive semigroups which are the transition semigroups of suitable Markov processes. Some applications are also discussed. Keywords Positive semigroup, Markov transition function, Markov process, Weighted continuous function space, Degenerate second order differential operator Mathematics Subject Classification (2000) 47D06, 47D07, 60J60     

The complexity of the fixed point property

It is NP-complete to determine whether a given ordered set has a fixed point free order-preserving self-map. On the way to this result, we establish the NP-completeness of a related problem: Given ordered sets P and Q with t-tuples (p ₁, ... , p _t) and (q ₁, ... , q _t) from P and Q respectively, is there an order-preserving map f: P→Q satisfying f(p _i)≥q _i for each i=1, ... , t?     

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at −∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(⋅)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I−S)/2 is Fredholm on the variable Lebesgue space , then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces and with some exponents ql and qr lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively.     

On the convergence of sequences of stationary jump Markov processes

This paper presents two main results: first, a Liapunov type criterion for the existence of a stationary probability distribution for a jump Markov process; second, a Liapunov type criterion for existence and tightness of stationary probability distributions for a sequence of jump Markov processes. If the corresponding semigroups T_N(t) converge, under suitable hypotheses on the limit semigroup, this last result yields the weak convergence of the sequence of stationary processes (T_N(t), π_N) to the stationary limit one.     

Random evolutions in discrete and continuous time

This paper points out a connection between random evolutions and products of random matrices. This connection is useful in predicting the long-run growth rate of a single-type, continuously changing population in randomly varying environments using only observations at discrete points in time. A scalar Markov random evolution is specified by the n×n irreducible intensity matrix or infinitesimal generator Q = (q_ij) of a time-homogeneous Markov chain and by n finite real growth rates (scalars) s_i. The scalar Markov random evolution is the quantity M^C(t) = exp(Σⁿ_j=1s_jg^C_j (t)), where g^C_j(t) is the occupancy times in state j up to time t. The scalar Markov product of random matrices induced by this scalar Markov random evolution is the quantity M^D(t) = exp(Σⁿ_j=1s_jg^D_j (t)), where g^D_j(t) is the occupancy time in state j up to and including t of the discrete-time Markov chain with stochastic one-step transition matrix P = e^Q. We show that lim_t→∞(1/t)E(logM^D(t))=lim_t→∞(1/t)E(logM^C(t)) but that in general lim_t→∞(1/t)logE(M^C(t)) ≠ lim_t→∞(1/t)logE(M^D(t)). Thus the mean Malthusian parameter of population biologists is invariant with respect to the choice of continuous or discrete time, but the rate of growth of average population size is not. By contrast with a single-type population, in multitype populations whose growth is governed by non-commuting operators, the mean Malthusian parameter may be destined for a less prominent role as a measure of long-run growth.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.