On the convolution and subordination of convex functions

In this work we present some new results on convolution and subordination in geometric function theory. We prove that the class of convex functions of order α is closed under convolution with a prestarlike function of the same order. Using this, we prove that subordination under the convex function order α is preserved under convolution with a prestarlike function of the same order. Moreover, we find a subordinating factor sequence for the class of convex functions. The work deals with several ideas and techniques used in geometric function theory, contained in the book Convolutions in Geometric Function Theory by Ruscheweyh (1982).     

Decomposability for stable processes

We characterize all possible independent symmetric α-stable (SαS) components of an SαS process, 0<α<2. In particular, we focus on stationary SαS processes and their independent stationary SαS components. We also develop a parallel characterization theory for max-stable processes.     

Convex and alpha-prestarlike subordination chains

In this paper we consider convex subordination chains (c.s.c.) and alpha-prestarlike subordination chains (α-p.s.c.) over (0,1] on the unit disc in the complex plane. We obtain sufficient conditions for a mapping f(z,t) to be an α-p.s.c. over (0,1], which generalize a well-known result of Ruscheweyh [St. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l'Université de Montreal, 1982]. We also obtain sufficient conditions for injectivity for nonanalytic mappings on the unit disc, and give certain examples of α-c.s.c. over (0,1] on the unit disc.     

Starlikeness and subordination properties of a linear operator

In this paper, we introduce a new class of p-valent analytic functions defined by using a linear operator L_k^α. For functions in this class H_k^α(p,λh) we estimate the coefficients. Furthermore, some subordination properties related to the operator L_k^α are also derived.     

Abrupt Lévy processes

Among Lévy processes with unbounded variation, we distinguish the abrupt ones, which are characterised by infinitely sharp extrema. Stable processes with parameter α>1 and creeping Lévy processes are abrupt. We give a characterisation of abrupt processes and study their Dini derivatives at all points of their trajectories.     

Mutual subordination of multivariate stationary processes over any locally compact abelian group

Summary Our purpose is to extend Kolmogorov's theorem [5, Th. 10] on mutual subordination for univariate weakly stationary stochastic processes over the (discrete) group of integers to multivariate processes over any (Hausdorff) locally compact abelian (lca) group. This extension is given in Theorems (1.12) and (3.4) below. We shall lean heavily on the joint paper [10] on the decomposition of matricial measures, to which the present paper may be regarded as a sequel.In Section 1 of the paper we shall define and prove theorems on the concept of E-subordination, where E is a projection-valued measure. In Section 2 we shall examine the structure of stationary processes over an lca group. In Section 3 we shall consider the concept of subordination of stationary processes. Finally in Section 4, we shall apply our subordination theorems to deduce that matrixvalued functions in L ₂ on the unit circle having no negative frequencies have a constant rank a.e. (Lebesgue) (4.2), a theorem of F. and M. Riesz (4.3), and a theorem on wandering subspaces due to Robertson [9] (4.4).The author is grateful to Prof. P. R. Masani for his generous help in formulating this paper.     

Anomalous diffusion approach to dielectric spectroscopy data with independent low- and high-frequency exponents

We advance a perspective outcome of tempered α-stable processes used in modeling of anomalous diffusion, a physical mechanism underlying the non-Debye relaxations. The tempered processes are characterized by a heavy tail truncation in time and have finite moments, but they also save some useful features of a purely skewed α-stable process. Due to these features, the relaxation phenomena get a transient character being shown in their asymptotic behavior. From the stochastic subordination scenario of the tempered anomalous diffusion we derive relaxation functions with independent low- and high-frequency exponents falling in the range (0, 1]. Those functions can be used to model all types of experimentally observed two-power-law relaxation patterns.     

A remark on the simultaneous <Emphasis Type="Italic">α</Emphasis>-stabilization of linear plants

In the present paper, we consider the problem of simultaneous α-stabilization (i.e., stabilization with degree of stability exceeding a given number α > 0) for linear scalar stationary plants of arbitrary order, for which we obtain numerically verifiable necessary conditions for simultaneous α-stabilization and a sufficient condition together with a constructive algorithm for the stabilizing controller design. To solve these problems, we use an approach based on a one-to-one transformation of the α-stability domains in the spaces of coefficients of polynomials.     

Convergence of trajectories in fractal interpolation of stochastic processes

The notion of fractal interpolation functions (FIFs) can be applied to stochastic processes. Such construction is especially useful for the class of α-self-similar processes with stationary increments and for the class of α-fractional Brownian motions. For these classes, convergence of the Minkowski dimension of the graphs in fractal interpolation of the Hausdorff dimension of the graph of original process was studied in [Herburt I, Małysz R. On convergence of box dimensions of fractal interpolation stochastic processes. Demonstratio Math 2000;4:873–88. [11]], [Małysz R. A generalization of fractal interpolation stochastic processes to higher dimension. Fractals 2001;9:415–28. [15]], and [Herburt I. Box dimension of interpolations of self-similar processes with stationary increments. Probab Math Statist 2001;21:171–8. [10]].We prove that trajectories of fractal interpolation stochastic processes converge to the trajectory of the original process. We also show that convergence of the trajectories in fractal interpolation of stochastic processes is equivalent to the convergence of trajectories in linear interpolation.     

Stationary points of O’Hara’s knot energies

In this article we study the regularity of stationary points of the knot energies E ^(α) introduced by O’Hara (Topology 30(2):241–247, 1991; Topol Appl 48(2):147–161, 1992; Topol Appl 56(1):45–61, 1994) in the range ${\alpha\in(2,3)}$ . In a first step we prove that E ^(α) is C ¹ on the set of all regular embedded curves belonging to ${{H^{(\alpha+1)/2,2}(\mathbb {R}{/}\mathbb {Z}, \mathbb {R}^n)}}$ and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E ^(α) plus a positive multiple of the length. We show that stationary points of finite energy are of class C ^∞—so especially all local minimizers of E ^(α) among curves with fixed length are smooth.     

Continuous-Time Markov Decision Processes with State-Dependent Discount Factors

We consider continuous-time Markov decision processes in Polish spaces. The performance of a control policy is measured by the expected discounted reward criterion associated with state-dependent discount factors. All underlying Markov processes are determined by the given transition rates which are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. By using the dynamic programming approach, we establish the discounted reward optimality equation (DROE) and the existence and uniqueness of its solutions. Under suitable conditions, we also obtain a discounted optimal stationary policy which is optimal in the class of all randomized stationary policies. Moreover, when the transition rates are uniformly bounded, we provide an algorithm to compute (or?at least to approximate) the discounted reward optimal value function as well as a discounted optimal stationary policy. Finally, we use an example to illustrate our results. Specially, we first derive an explicit and exact solution to the DROE and an explicit expression of a discounted optimal stationary policy for such an example.     

On stationary Markov processes with polynomial conditional moments

We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say m is a polynomial of degree at most m. We show that then under some additional, natural technical assumption there exists a family of orthogonal polynomial martingales. More precisely we show that such a family of processes is completely characterized by the sequence {(α_n, p_n)}_{n ? 0} where α′_ns are some positive reals while p′_ns are some monic orthogonal polynomials. Bakry and Mazet (Séminaire de Probabilit?s, vol. 37, 2003) showed that under some additional mild technical conditions each such sequence generates some stationary Markov process with polynomial regression.

We single out two important subclasses of the considered class of Markov processes. The class of harnesses that we characterize completely. The second one constitutes of the processes that have independent regression property and are stationary. Processes with independent regression property so to say generalize ordinary Ornstein–Uhlenbeck (OU) processes or can also be understood as time scale transformations of Lévy processes. We list several properties of these processes. In particular we show that if these process are time scale transforms of Lévy processes then they are not stationary unless we deal with classical OU process. Conversely, time scale transformations of stationary processes with independent regression property are not Lévy unless we deal with classical OU process.     

Two-sided heat kernel estimates for censored stable-like processes

In this paper, we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C ^1,1 open sets in ${\mathbb R^d}$ , where d ≥ 1 and ${\alpha\in (1, 2)}$ . We first show that the semigroup of the censored α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C ^1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C ^1,1 open sets.     

On (strong) α-favorability of the Wijsman hyperspace

The Banach-Mazur game as well as the strong Choquet game are investigated on the Wijsman hyperspace from the nonempty player's (i.e. α's) perspective. For the strong Choquet game we show that if X is a locally separable metrizable space, then α has a (stationary) winning strategy on X iff it has a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X. The analogous result for the Banach-Mazur game does not hold, not even if X is separable, as we show that α may have a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X, and not have one on X. We also show that there exists a separable 1st category metric space such that α has a (stationary) winning strategy on its Wijsman hyperspace. This answers a question of Cao and Junnila (2010) [6].     

Subcompact cardinals, squares, and stationary reflection

We generalise Jensen’s result on the incompatibility of subcompactness with □. We show that α ⁺-subcompactness of some cardinal less than or equal to α precludes ${\square _\alpha }$ , but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling’s hierarchy of weak squares, showing which cases are precluded by α ⁺-subcompactness, and again we demonstrate the optimality of our results by forcing the strongest possible squares under these restrictions to hold.     

Domain representability and the Choquet game in Moore and BCO-spaces

In this paper we investigate the role of domain representability and Scott-domain representability in the class of Moore spaces and the larger class of spaces with a base of countable order. We show, for example, that in a Moore space, the following are equivalent: domain representability; subcompactness; the existence of a winning strategy for player α (= the nonempty player) in the strong Choquet game Ch(X); the existence of a stationary winning strategy for player α in Ch(X); and Rudin completeness. We note that a metacompact ?ech-complete Moore space described by Tall is not Scott-domain representable and also give an example of ?ech-complete separable Moore space that is not co-compact and hence not Scott-domain representable. We conclude with a list of open questions.     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

Continuity of integrated density of states — independent randomness

In this paper we discuss the continuity properties of the integrated density of states for random models based on that of the single site distribution. Our results are valid for models with independent randomness with arbitrary free parts. In particular in the case of the Anderson type models (with stationary, growing, decaying randomness) on the ν dimensional lattice, with or without periodic and almost periodic backgrounds, we show that if the single site distribution is uniformly α-Hölder continuous, 0 < α ≤ 1, then the density of states is also uniformly α-Hölder continuous.     

The Mann–Whitney <Emphasis Type="Italic">U</Emphasis>-statistic for α-dependent sequences

We give the asymptotic behavior of the Mann–Whitney U-statistic for two independent stationary sequences. The result applies to a large class of short-range dependent sequences, including many nonmixing processes in the sense of Rosenblatt [17]. We also give some partial results in the long-range dependent case, and we investigate other related questions. Based on the theoretical results, we propose some simple corrections of the usual tests for stochastic domination; next we simulate different (nonmixing) stationary processes to see that the corrected tests perform well.     

Subordination in the sense of Bochner of L p -semigroups and associated Markov processes

We show that the subordination induced by a convolution semigroup（subordination in the sense of Bochner）of a C0-semigroup of sub-Markovian operators on an Lpspace is actually associated to the subordination of a right（Markov）process.As a consequence,we solve the martingale problem associate with the Lp-infinitesimal generator of the subordinate semigroup.We also prove quasi continuity properties for the elements of the domain of the Lp-generator of the subordinate semigroup.It turns out that an enlargement of the base space is necessary.A main step in the proof is the preservation under such a subordination of the property of a Markov process to be a Borel right process.We use several analytic and probabilistic potential theoretical tools.