Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach

During the last 30 years there have been a strong interest on stochastic processes on ultrametric spaces mainly due its connections with models of complex systems, such as glasses and proteins. These processes are very convenient for describing phenomena whose space of states display a hierarchical structure, see e.g. [9–13, 36, 61, 78, 80, 86, 94, 108, 111, 118, 122], and references therein. Avetisov et al. constructed a wide variety of models of ultrametric diffusion constrained by hierarchical energy landscapes, see [9–13]. From a mathematical point view, in these models the time-evolution of a complex system is described by a p-adic master equation (a parabolic-type pseudodifferential equation) which controls the time-evolution of a transition density function of a Markov process on an ultrametric space. This process describes the dynamics of the system in the space of configurational states which is approximated by an ultrametric space (\(\mathbb{Q}_{p}\)). This is the main motivation for developing a general theory of parabolic-type pseudodifferential equations.

Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space

We call a \emph{comb} a map $f:I\to [0,\infty)$, where $I$ is a compact interval, such that $\{f\ge \varepsilon\}$ is finite for any $\varepsilon$. A comb induces a (pseudo)-distance $\dtf$ on $\{f=0\}$ defined by $\dtf(s,t) = \max_{(s\wedge t, s\vee t)} f$. We describe the completion $\bar I$ of $\{f=0\}$ for this metric, which is a compact ultrametric space called \emph{comb metric space}. Conversely, we prove that any compact, ultrametric space $(U,d)$ without isolated points is isometric to a comb metric space. We show various examples of the comb representation of well-known ultrametric spaces: the Kingman coalescent, infinite sequences of a finite alphabet, the $p$-adic field and spheres of locally compact real trees. In particular, for a rooted, locally compact real tree defined from its contour process $h$, the comb isometric to the sphere of radius $T$ centered at the root can be extracted from $h$ as the depths of its excursions away from $T$.

We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the <em>c</em>-random walks. Such walks are used, e.g., in mathematical physics and population biology. The number variance problem consists in investigating if the variance of the number of “particles” $N_n(L)$ lying in the ball of radius $L$<em></em> at a given step $n$<em></em> remains bounded, or even better, converges to a finite limit, as $L\\to\\infty$. We give a necessary and sufficient condition and discuss its relationship to transience/recurrence property of the walk. Next we consider normalized fluctuations of $N_n(L)$ around the mean as $n\\to\\infty$ and $L$<em></em> is increased in an appropriate way. We prove convergence of finite dimensional distributions to a Gaussian process whose properties are discussed. As the $c$<em></em>-random walks mimic symmetric stable processes on $\\mathbb{R}$, we compare our results with those obtained by Hambly and Jones (2007, 2009), who studied the number variance problem for an infinite system of independent symmetric stable processes on $\\mathbb{R}$. Since the hierarchical group is an ultrametric space, corresponding results for symmetric stable processes and hierarchical random walks may be analogous or quite different, as has been observed in other contexts. An example of a difference in the present context is that for the stable processes a fluctuation limit process is a Gaussian process which is not Markovian and has long range dependent stationary increments, but the counterpart for hierarchical random walks is Markovian, and in a special case it has independent increments.

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the space D′0(X) of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem (an analog of the Kovalevskaya theorem).

We investigate the interrelations between labeled trees and ultrametric spaces generated by these trees. The labeled trees, which generate complete ultrametrics, totally bounded ultrametrics, and discrete ones, are characterized up to isomorphism. As corollary, we obtain a characterization of labeled trees generating compact ultrametrics and discrete totally bounded ultrametrics. It is also shown that every ultrametric space generated by labeled tree contains a dense discrete subspace.

This paper is divided into two parts. Part I deals with flows on arbitrary metric spaces and answers completely the question when they are parallelizable. We give an elementary proof for arbitrary locally compact separable metric spaces which, incidentally, also clarifies the role of Niemytskii's notion of an improper saddle point. Although our main result can be established very simply in special cases, by direct appeal to Feldbau's theorem [8], e.g., when the underlying space is an ANR, it does not appear to be so easily accessible in the general case considered here. In Part II we relate the geometry of parallelizable flows to questions concerning the existence of Lyapunov functions. Although we consider only the important particular case of flows on En x E determined by the trajectories of differential equations (and so do not need the full generality of Part I), our results extend verbatim to the case of flows discussed in I. We find, with the aid of suitable sections, that even when there is no stability at all there always exist functions with two of the three properties required of Lyapunov functions, and that the regularity properties of these functions depend essentially upon those of the sections. We carry out all constructions on the rectified flow. This affords conceptual simplification and, more important, results in a unified approach which brings out clearly the geometric picture that has been obscured, if not totally lost, in previous constructions (cf., e.g., [3], [4]). As illustration of our method we give simple proofs of the converses of the classical Lyapunov theorems.

LetKndenote the number of types of a sample of sizentaken from an exchangeable coalescent process (Ξ-coalescent) with mutation. A distributional recursion for the sequence (Kn)n∈ℕis derived. If the coalescent does not have proper frequencies, i.e. if the characterizing measure Ξ on the infinite simplex Δ does not have mass at 0 and satisfies ∫Δ∣x∣Ξ(dx)/(x,x)<∞, where ∣x∣:=∑i=1∞xiand (x,x)≔∑i=1∞xi2forx=(x1,x2,…)∈Δ, thenKn/nconverges weakly asn→∞ to a limiting variableKthat is characterized by an exponential integral of the subordinator associated with the coalescent process. For so-called simple measures Ξ satisfying ∫ΔΞ(dx)/(x,x)<∞, we characterize the distribution ofKvia a fixed-point equation.

Coalescents with multiple collisions (also called Λ-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Consider an infinite population with immigration labelled at each generation byN:= {1, 2, …}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a model for genealogy with immigration, focusing on simple distinguished coalescents, i.e. such that when a coagulation occurs all the blocks involved merge as a single block. These processes are characterized by two finite measures on [0, 1] denoted byM= (Λ0, Λ1). We call themM-coalescents. We show by martingale arguments that the condition of coming down from infinity for theM-coalescent coincides with that obtained by Schweinsberg for the Λ-coalescent. In the same vein as Bertoin and Le Gall,M-coalescents are associated with some stochastic flows. The superprocess embedded can be viewed as a generalized Fleming-Viot process with immigration. The measures Λ0and Λ1respectively specify the reproduction and the immigration. The coming down from infinity of theM-coalescent will be interpreted as the initial types extinction: after a certain time all individuals are immigrant children.

The ability to price risks and devise optimal investment strategies in the presence of an uncertain random market is the cornerstone of modern finance theory. We first consider the simplest such problem of a so-called European call initially solved by Black and Scholes using Ito stochastic calculus for markets modelled by a log-Brownien stochastic process. A simple and powerful formalism is presented which allows us to generalize the analysis to a large class of stochastic processes, such as ARCH, jump or Levy processes. We also address the case of correlated Gaussian processes, which is shown to be a good description of three different market indices (MATIF, CAC40, FTSE100). Our main result is the introduction of the concept of an optimal strategy in the sense of (functional) minimization of the risk with respect to the portfolio. If the risk may be made to vanish for particular continuous uncorrelated 'quasiGaussian' stochastic processes (including Black and Scholes model), this is no longer the case for more general stochastic processes. The value of the residual risk is obtained and suggests the concept of risk-corrected option prices. In the presence of very large deviations such as in Levy processes, new criteria for rational fixing of the option prices are discussed. We also apply our method to other types of options, `Asian', `American', and discuss new possibilities (`doubledecker'...). The inclusion of transaction costs leads to the appearance of a natural characteristic trading time scale.

One of the standard goals of in nite-dimensional topology is the topological classi cation of linear metric spaces and their convex subsets. The origins of this problem go back to the questions of Fr echet [Fr] and Banach [Ba] whether all in nite-dimensional separable Fr echet (Banach) spaces are homeomorphic (we explain our terminology at the end of the introduction). These questions have been a rmatively answered by the combined results of Anderson and Kadec (see [BP]) which may be stated as follows: every separable Fr echet space is homeomorphic to a Hilbert space. This result was generalized by Toru nczyk [To] for the nonseparable case. Since then the main interest in the eld has turned to topological classi cation of incomplete or nonlocally convex linear metric spaces. Quite recently, Cauty [Ca2] has constructed an example of a separable linear complete metric space X which is not an AR. In particular, X is not homeomorphic to any convex subset of a locally convex linear metric space. The topological classi cation of incomplete locally convex metric linear spaces, or even incomplete pre-Hilbert spaces, is much more complex than for the complete ones. First, there exist pre-Hilbert spaces of an arbitrary Borel class. Moreover, the Borel type invariant does not classify such spaces, and in the class of -compact spaces there exist uncountably many pairwise nonhomeomorphic such spaces [BP], [Ca1]. Until recently, there was hope that, in spite of these di culties, the classi cation problem could be restricted to subspaces of Hilbert spaces. The result of Bessaga and Dobrowolski [BD] stating that every locally convex linear -compact metric space is homeomorphic to a pre-Hilbert space looked like a promising rst step. The main objective of this paper is to show that, in general, it is not possible to restrict ones attention to subspaces of Hilbert spaces only. We show that the classes of pre-Hilbert,

Optimal control of demand-paging systems

The Oldroyd 8-constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. In this paper, we focus on the normal stress difference responses to large-amplitude oscillatory shear (LAOS) flow. The nonlinearity of the polymeric liquids, triggered by LAOS, causes these responses at even multiples of the test frequency. We call responses at a frequency higher than twice the test frequency higher harmonics. The normal stress difference responses for the Oldroyd 8-constant framework has recently yielded to the exact analytical solution. However, in its closed form, Bessel functions appear 24 times, each within summations to infinity. In this paper, to simplify the exact solution, we expand it in a Taylor series. We truncate the series after its 17th power of the shear rate amplitude. Our main result reduces to the well-known expression for the special cases of the corotational Jeffreys and corotational Maxwell fluids. Whereas these special cases yielded to the Goddard integral expansion (GIE), the more general Oldroyd 8-constant framework does not. We use Ewoldt grids to show our main result to be highly accurate for the corotational Jeffreys and corotational Maxwell fluids. For these two special cases, our solutions agree closely with the exact solutions as long as Wi/De<3310. We compare our main result, for the special case of the Johnson–Segalman fluid, with measurements on dissolved polyisobutylene in the isobutylene oligomer. For this, we use the Spriggs relations to generalize our main result to multimode, which then agrees closely with the measurements.

Our aim in this paper is to develop a theory of purity and to prove in a unified conceptual way the existence of almost split morphisms, almost split sequences and almost split triangles in abstract homotopy categories, a rather omnipresent class of categories of interest in representation theory. Our main tool for doing this is the classical Brown representability theorem. Mathematics Subject Classifications (2000): 16Gxx, 18Gxx, 18Exx, 55U35. Abstract homotopy categories were introduced by E. Brown in the mid-sixties as the proper framework for the study of the homotopy theory of CW-complexes. In this setting, he proved in (18) his celebrated representability theorem, a variant of which has recently found important applications in the stable module category of a modular group algebra and more generally in compactly generated triangulated categories, mainly through the work of Rickard (42) and Neeman (40). Our main purpose in this paper is to develop a theory of purity and a theory of existence of almost split morphisms in an abstract homotopy category, using the Brown representability theorem as a main catalyzing tool. This aim is justified by the fact that abstract homotopy categories are om- nipresent in representation theory. They include module categories, locally finitely presented categories with products, compactly generated triangulated categories, categories of projective modules, and stable categories modulo projectives over left coherent and right perfect rings, categories of injective modules and stable categories modulo injectives over right Morita rings, (stable) categories of Cohen- Macaulay modules over Gorenstein rings, and many others. Abstract homotopy

Degradation of polymer coatings in service occurs due to the cumulative effect of a vast number of aggressive photons, molecules, or other assaults that occur at random locations and random intervals such that the Central Limit Theorem can be applied. This permits use of the Normal distribution to describe the extent and variation in the damage that occurs. In addition, initial network formation is also governed by stochastic processes as small reactive molecules react to form the polymer. Molecular dynamics simulations show that extensive variation is possible in the connectedness and local topology of a crosslinked polymer, so there are statistically occurring flaws and inconsistencies, even before exposure in service. The original defects, together with the damage caused by degradation, determine the performance of coatings. For deterioration in service, accumulation of damage is modelled as a stochastic process with the properties of the Normal distribution, using the mean to represent the most likely state of the material. Inserting this in very well-known theories of material properties such as gloss and toughness shows how the kinetics of failure of coatings may arise. Models include basic physical property values of materials and show how they affect the outcome. This approach also shows how nonsimultaneous degradation leads to an apparent induction period in the change of properties that may not originate with autocatalytic degradation chemistry. Equations generated in this approach demonstrate that different properties are sensitive in different ways to the degradation process and thus decay with a different rate, even though the basic mechanism of deterioration is identical.