Further results on the relationship between μ-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains

This note considers continuous-time Markov chains whose state space consists of an irreducible class, C, and an absorbing state which is accessible from C. The purpose is to provide results on μ-invariant and μ-subinvariant measures where absorption occurs with probability less than one. In particular, the well-known premise that the μ-invariant measure, m, for the transition rates be finite is replaced by the more natural premise that m be finite with respect to the absorption probabilities. The relationship between μ-invariant measures and quasi-stationary distributions is discussed.     

Semi-Markov processes and α-invariant distributions

A semi-Markov process is easily made Markov by adding some auxiliary random variables. This paper discusses the I-type quasi-stationary distributions of such “extended” processes, and the α-invariant distributions for the corresponding Markov transition probabilities; and we show that there is an intimate relation between the two. The results have relevance in the study of the time to “absorption” or “death” of semi-Markov processes. The particular case of a terminating renewal process is studied as an example.     

Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Let X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:

• equivalence classes of α-invariant K-connections on X
• α-invariant gauge classes of K-connections on X, and
• α-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic K ^?-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

     

Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains

The stationary conditional, doubly limiting conditional and limiting conditional mean ratio quasi-stationary distributions are given for continuous-time Markov chains with denumerable state space both in terms of the transition matrixP(t) and the infinitesimal, generatorQ.     

An existence theorem for probability measures invariant under a Markov kernel

LetP be a Markov kernel defined on a measurable space (X,A). A probability measure μ onA is said to beP-invariant if μ(A=∫P(x,A)dμ(x) for allA ∈A∈A. In this note we prove a criterion for the existence ofP-invariant probabilities which is, in particular, a substantial generalization of a classical theorem due to Oxtoby and Ulam ([5]). As another consequence of our main result, it is shown that every pseudocompact topological space admits aP-invariant Baire probability measure for any Feller kernelP.     

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.     

Iwasawa invariants for the False-Tate extension and congruences between modular forms

For an ordinary prime p?3, we consider the Hida family associated to modular forms of a fixed tame level, and their Selmer groups defined over certain Galois extensions of Q(μp) whose Galois group is G≅Zp?Zp. For Selmer groups defined over the cyclotomic Zp-extension of Q(μp), we show that if the μ-invariant of one member of the Hida family is zero, then so are the μ-invariants of the other members, while the λ-invariants remain the same only in a branch of the Hida family. We use these results to study the behavior of some invariants from non-commutative Iwasawa theory in the Hida family.     

Decay parameter and related properties of 2-type branching processes

We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for 2-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 = Z+2 \ 0. It is shown that this λC can be directly ...     

u-Invariants for forms of higher degree

Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. We give a survey of both old and new results on these u-invariants.     

Criteria for Feller transition functions

This paper presents some conditions for the minimal Q-function to be a Feller transition function, for a given q-matrix Q. We derive a sufficient condition that is stated explicitly in terms of the transition rates. Furthermore, some necessary and sufficient conditions are derived of a more implicit nature, namely in terms of properties of a system of equations (or inequalities) and in terms of the operator induced by the q-matrix. The criteria lead to some perturbation results. These results are applied to birth-death processes with killing, yielding some sufficient and some necessary conditions for the Feller property directly in terms of the rates. An essential step in the analysis is the idea of associating the Feller property with individual states.     

On Freudenburg's counterexample to the Fourteenth Problem of Hilbert

In this paper we study Freudenburg's counterexample to the fourteenth problem of Hilbert and counterexamples derived from it. We shall construct a generating set of a nonfinitely generated G_a-invariant ring given in Freudenburg's counterexample by making use of an integral sequence which was constructed inductively by Freudenburg. This generating set shall be used in describing a generating set of a nonfinitely generated G_a-invariant ring given in Daigle and Freudenburg's counterexample. Using these generating sets, we shall determine the Hilbert series of the above Freudenburg's and Daigle and Freudenburg's nonfinitely generated G_a-invariant rings, and find that these Hilbert series are rational functions. Then we also show that the Hilbert series of nonfinitely generated invariant rings appearing in the author's linear counterexamples are rational functions.     

Infinitesimal Torelli for elliptic surfaces revisited

In this article we give a new proof for the infinitesimal Torelli theorem for minimal elliptic surfaces without multiple fibers with Euler number at least 24 for nonconstant j-invariant. In the case of constant j-invariant we find a new proof in the case of Euler number at least 72. We also discuss several new counterexamples.     

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.     

Real <Emphasis Type="Italic">K</Emphasis>3 surfaces without real points,equivariant determinant of the Laplacian,and the Borcherds Φ-function

We consider an equivariant analogue of a conjecture of Borcherds. Let (Y, σ) be a real K3 surface without real points. We shall prove that the equivariant determinant of the Laplacian of (Y, σ) with respect to a σ-invariant Ricci-flat Kähler metric is expressed as the norm of the Borcherds Φ-function at the “period point”. Here the period of (Y, σ) is not the one in algebraic geometry.     

Homogenous projective factors for actions of semi-simple Lie groups

We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ _K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q. Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999     

Almost sure limit theorem for the maximum of a classof quasi-stationary sequences简

This paper investigates the problem of almost sure limit theorem for the maximum of quasi-stationary sequence based on the result of Turkman and Walker. We prove an almost sure limit theorem for the maximum of a class of quasi-stationary sequence under weak dependence conditions of D(uk,un) and αtn,ln = O(log log n).(1+ε).     

Separation Theorems for Group Invariant Polynomials

We study the existence of separation theorems by polynomials that are invariant under a group action. We show that if G is a finite subgroup of \(\textit{GL}(n,{\mathbb {C}})\), K is a set in \({\mathbb {C}}^{n}\) that is invariant under the action of G and z is a point in \({\mathbb {C}}^{n}\setminus K\) that can be separated from K by a polynomial Q, then z can be separated from K by a G-invariant polynomial P. Furthermore, if Q is homogeneous then P can be chosen to be homogeneous. As a particular case, if K is a symmetric polynomially convex compact set in \({\mathbb {C}}^{n}\) and \(z\notin K\) then there exists a symmetric polynomial that separates z and K.     

Flows, fixed points and rigidity for Kleinian groups

The main application of the techniques developed in this paper is to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group G, by a G-invariant pattern we mean a G-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of G. We prove that (in dimension at least three) for G ₁, G ₂ cocompact Kleinian groups, any quasiconformal map pairing a G ₁-invariant pattern to a G ₂-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas and Mj (Pattern rigidity in hyperbolic spaces: duality and pd subgroups, arxiv:math.GT/08094449, 2008) who proved rigidity for Poincare Duality subgroups. Pattern rigidity is a consequence of the study conducted in this paper of the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G ₁ and a quasiconformal conjugate h ^?1 G ₂ h of a cocompact group G ₂. We show that if the conjugacy h is not conformal then this group contains a flow, i.e. a non-trivial one parameter subgroup. Mostow rigidity is an immediate consequence.     

Surgery and equivariant Yamabe invariant

We consider the equivariant Yamabe problem, i.e., the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their G-invariant conformal subclasses. We prove a formula about how the G-Yamabe invariant changes under the surgery of codimension 3 or more, and compute some G-Yamabe invariants.     

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.