A universal envelope for Gaussian processes and their kernels

The central theme in our paper deals with mathematical issues involved in the answer to the following question: How can we generate stochastic processes from their correlation data? Since Gaussian processes are determined by moment information up to order two, we focus on the Gaussian case. Two functional analytic tools are used here, in more than one variant. They are: operator factorization; and direct integral decompositions in the form of Karhunen-Loève expansions. We define and study a new interplay between the theory of positive definition functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. The three classes of processes we study are as follows: Processes associated with: (a) arbitrarily given sigma finite regular measures on a fixed Borel measure space; (b) with Hilbert spaces of sigma-functions; and (c) with systems of self-similar measures arising in the theory of iterated function systems. Starting with a non-degenerate positive definite function K on some fixed set S, we show that there is a choice of a universal sample space Ω, which can be realized as a “boundary” of (S,K). Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.     

A class of Gaussian processes with fractional spectral measures

We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures σ (generalized spectral measures), and our focus here is on the case when the measure σ is a singular measure. We characterize the processes arising from σ when σ is in one of the classes of affine selfsimilar measures. Our analysis makes use of Kondratiev white noise spaces. With the use of a priori estimates and the Wick calculus, we extend and sharpen (see Theorem 7.1) earlier computations of Ito stochastic integration developed for the special case of stationary increment processes having absolutely continuous measures. We further obtain an associated Ito formula (see Theorem 8.1).     

Large deviations for conditional Volterra processes

In this article we investigate a problem of large deviations for continuous Gaussian Volterra processes, conditioned to follow a fixed trajectory up to a fixed time T > 0, in order to establish the behavior of the process in the near future after T and to give an asymptotic estimate of the exit probability of its bridge. Some examples are considered.     

An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self‐delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program‐size complexity H (s) of a given finite binary string s. In the standard way, H (s) is defined as the length of the shortest input string for U to output s. In the other way, the so‐called universal probability m is introduced first, and then H (s) is defined as –log₂ m (s) without reference to the concept of program‐size. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator‐valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour‐El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi‐POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi‐POVM. The validity of this definition is discussed. In what follows, we introduce an operator version (s) of H (s) in a Hilbert space of infinite dimension using a universal semi‐POVM, and study its properties. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The contiguity of probability measures and asymptotic inference in continuous time stationary diffusions and Gaussian processes with known covariance

We establish contiguity of families of probability measures indexed by T, as T → ∞, for classes of continuous time stochastic processes which are either stationary diffusions or Gaussian processes with known covariance. In most cases, and in all the examples we consider in Section 4, the covariance is completely determined by observing the process continuously over any finite interval of time. Many important consequences pertaining to properties of tests and estimators, outlined in Section 5, will then apply.     

On some nonself mappings

Let X be a Banach space, let K be a non–empty closed subset of X and let T : K → X be a non–self mapping. The main result of this paper is that if T satisfies the contractive–type condition (1.1) below and maps ?K (?K the boundary of K) into K then T has a unique fixed point in K.     

Stochastic processes and their spectral representations over non-archimedean fields

In this paper, we study stochastic processes with values in finite- and infinite-dimensional vector spaces over infinite fields K of zero characteristic with nontrivial non-Archimedean norms. For different types of stochastic processes controlled by measures with values in K and in complete topological vector spaces over K, we study stochastic integrals, vector-valued measures, and integrals in spaces over K. We also prove theorems on spectral decompositions of non-Archimedean stochastic processes.     

Fixed point properties of semigroups of non-expansive mappings

In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.     

Stochastic control with exit time and constraints,application to small time attainability of sets

We study the existence of a solution of controlled stochastic differential equations remaining in a given set of constraints at any time smaller than the exit time of a given open set. We also investigate the small time attainability of a given closed set K, i.e., the property that, for all arbitrary small time horizon T and for all initial condition in a sufficiently small neighborhood of K, there exists a solution to the controlled stochastic differential equation which reaches K before T.     

Kolmogorov numbers of Riemann–Liouville operators over small sets and applications to Gaussian processes

We investigate compactness properties of the Riemann–Liouville operator R_α of fractional integration when regarded as operator from L₂[0,1] into C(K), the space of continuous functions over a compact subset K in [0,1]. Of special interest are small sets K, i.e. those possessing Lebesgue measure zero (e.g. fractal sets). We prove upper estimates for the Kolmogorov numbers of R_α against certain entropy numbers of K. Under some regularity assumption about the entropy of K these estimates turn out to be two-sided. By standard methods the results are also valid for the (dyadic) entropy numbers of R_α. Finally, we apply these estimates for the investigation of the small ball behavior of certain Gaussian stochastic processes, as e.g. fractional Brownian motion or Riemann–Liouville processes, indexed by small (fractal) sets.     

Convergence theorem of common fixed points for a family of nonspreading mappings in Hilbert space

We introduce the concept of K-mapping of a finite family of nonspreading mappings {T_i}_i=1^N{\{T_i\}_{i=1}^N} and we show that the fixed point set of the K-mapping is the set of common fixed points of {T_i}_i=1^N{\{T_i\}_{i=1}^N}. Moreover, we prove strong convergence theorem of the Ishikawa iterative process to a common fixed point of a finite family of nonspreading mappings in Hilbert space under certain control conditions.     

On the number of singular points of weak solutions to the Navier‐Stokes equations

We consider a suitable weak solution to the three‐dimensional Navier‐Stokes equations in the space‐time cylinder Ω × ]0, T[. Let Σ be the set of singular points for this solution and Σ (t) ≡ {(x, t) ∈ Σ}. For a given open subset ω ? Ω and for a given moment of time t ∈]0, T[, we obtain an upper bound for the number of points of the set Σ(t) ? ω. © 2001 John Wiley & Sons, Inc.     

Random number of units for K-out-of-n systems

This paper takes up the reliability and preventive replacement problems for a K-out-of-n system, where K is a stochastic parameter provided. Firstly, we consider the case when K is predefined as constant numbers as is done with the traditional method, and obtain the system reliability R(t), mean time to failure (MTTF), and replacement policies, in which the number n of units for replacement and replacement time T of operation are, respectively, optimized. Secondly, we focus on the above discussions again when K cannot be predefined constantly, but it is a random variable with an estimated probability function. Furthermore, we give approximate computations in an easier way for MTTF, optimal number n* and replacement time T*, respectively.     

Strong approximation of semimartingales and statistical processes

Summary As an application of general convergence results for semimartingales, exposed in their book Limit Theorems for Stochastic Processes, Jacod and Shiryaev obtained a fundamental result on the convergence of likelihood ratio processes to a Gaussian limit. We strengthen this result in a quantitative sense and show that versions of the likelihood ratio processes can be defined on the space of the limiting experiment such that we get pathwise almost sure approximations with respect to the uniform metric. The approximations are considered under both sequences of measures, the hypothesisP ⁿ and the alternative . A consequence is e.g. an estimate for the speed of convergence in the Prohorov metric. New approximation techniques for stochastic processes are developed.This article was processed by the author using the L^AT_EX style filepljourIm from Springer-Verlag.     

Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations

We study a class of integrable and discontinuous measure-valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution belonging to the domain of attraction of a stable law. These processes, computed on a test function f, are semimartingales whose martingale terms are identified with integrals of f with respect to a martingale measure. According to a representation theorem of continuous (respectively purely discontinuous) martingale measures as stochastic integrals with respect to a white noise (resp. to a POISSON process), we prove that the measure-valued processes that we consider are solutions of stochastic differential equations in the space of L² (Ω)-valued vector measures.     

Local empirical processes near boundaries of convex bodies

We investigate the behaviour of Poisson point processes in the neighbourhood of the boundary ∂K of a convex body K in ,d ≥ 2. Making use of the geometry of K, we show various limit results as the intensity of the Poisson process increases and the neighbourhood shrinks to ∂K. As we shall see, the limit processes live on a cylinder generated by the normal bundle of K and have intensity measures expressed in terms of the support measures of K. We apply our limit results to a spatial version of the classical change-point problem, in which random point patterns are considered which have different distributions inside and outside a fixed, but unknown convex body K.     

Holomorphic liftings and Bergman kernel estimates for 𝒟ℱ𝒩‐domains

Let E be a 𝒟ℱ𝒩‐space and let U ⊂ E be open. By applying the nuclearity of the Fréchet space ℋ︁(U) of holomorphic functions on U we show that there are finite measures μ on U leading to Bergman spaces of μ ‐square integrable holomorphic functions. We give an explicit construction for μ by using infinite dimensional Gaussian measures. Moreover, we prove boundary estimates for the corresponding Bergman kernels K_μ on the diagonal and we give an application of our results to liftings of μ ‐square integrable Banach space valued holomorphic functions over U. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Central limit theorems in D[0, 1]

Summary Let X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {X _n} of independent copies of X, let S _n=X₁++X_n. Conditions which are either necessary or sufficient for the weak convergence of n ^–1/2(S _n–ES_n) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.     

Cantorian space–time,Fantappie’s final group,accelerated universe and other consequences

In this paper we introduce Mohamed El Naschie’s ϵ^(∞) Cantorian space–time in connection with stochastic self-similar processes to give a possible explanation of the segregation of the Universe at fixed scale; then by considering the Fanntappie’s transformation group we show how the universe could appear accelerated on Cantorian space–time.