Search

Bertoin, Roynette and Yor (2004) described new connections between the class LE of Lévy‐Laplace exponents Ψ (also called the class of (sub)critical branching mechanism) and the class of Bernstein functions (BF) which are internal, that is, those Bernstein functions ϕ s.t. Ψ∘ϕ remains a Bernstein function for every Ψ. We complete their work and illustrate how the class of internal function is rich from the stochastic point of view. It is well known that every ϕ∈BF corresponds to (i) a subordinator (Xt)t ≥ 0 (or equivalently to transition semigroups ℙ(Xt∈dx)t≥0 and (ii) a Lévy measure μ (which controls the jumps of the subordinator). It is also known that, on (0,∞), the measure ℙ(Xt∈dx)/t converges vaguely to dδ0(dx)+μ(dx) as t→0, where d is the drift term, but rare are the situations where we can compare the transition semigroups with the Lévy measure. Our extensive investigations on the composition of Lévy‐Laplace exponents Ψ with Bernstein functions show, for instance, these remarkable facts: ϕ is internal is equivalent to (a) ϕ2∈BF or to (b) tμ(dx)−ℙ(Xt∈dx) is a positive measure on (0,∞). We also provide conditions on μ insuring internality for ϕ and illustrate how Lévy‐Laplace exponents are closely connected to the class of Thorin Bernstein function. [ABSTRACT FROM AUTHOR]

Using the Blossom method in the quasi extended Chebyshev space, we first construct a class of generalized cubic Bernstein functions in the generalized cubic polynomial space { 1 , 3 t 2 - 2 t 3 , (1 - t) 2 λ (t) , t 2 μ (t) } . Our work includes many previous works as special cases by selecting specific shape functions λ (t) and μ (t) . The resulting generalized cubic Bézier curves and their properties are discussed. The corner cutting method is also proposed to compute curves efficiently and stably. A generalized Bernstein operator is given and spectral analysis on a specific case is carried out. The results show that the obtained generalized Bézier curves can better fit the control polygon compared to the Bézier curves. We further deduce the closed-form solution of generalized cubic B-splines that possess local shape functions. Some important properties are presented, including local support property, nonnegativity, partition of unity, total positivity property, C 2 continuity, linear independence and so on. Some numerical examples shows that the resulting generalized cubic B-spline curves can approximate the control polygon more flexibly. [ABSTRACT FROM AUTHOR]

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, W 0 (t) , in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert W function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions. [ABSTRACT FROM AUTHOR]

We consider here convolution operators, in the Caputo sense, with nonsingular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel's parameters and, consequently, of the jumps' density function. [ABSTRACT FROM AUTHOR]

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, W0(t), in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert W function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions.

In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result. [ABSTRACT FROM AUTHOR]

The generalization of fractional Brownian motion in the white and grey noise spaces has been recently carried over, following the Mandelbrot–Van Ness representation, through Riemann–Liouville type fractional operators. Our aim is to extend this construction by means of more general fractional derivatives and integrals, which we define as Fourier-multiplier operators and then specialize by means of Bernstein functions. More precisely, we introduce a general class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein–Uhlenbeck process. The greater generality of our model, with respect to the previous ones, allows a higher flexibility and a wider applicability. We derive here some properties of this class of processes (such as continuity, occupation density, variance asymptotics and persistence) according to the conditions satisfied by the Fourier symbol of the operator or the Bernstein function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation. [ABSTRACT FROM AUTHOR]

Let M n denote the algebra of all n × n complex matrices. Let A, B be positive definite matrices in M n and | | | · | | | be a unitarily invariant norm on M n . We establish some estimates of the right hand side of some Hermite–Hadamard-type inequalities in which Bernstein functions are involved. Furthermore, if f is a Bernstein function on (0 , ∞) , we obtain a bound for | | | f (A) - f (B) | | | in terms of | | | A - B | | | , as follows f (A) - f (B) ≤ max { ‖ f ′ (A) ‖ , ‖ f ′ (B) ‖ } A - B . We also show that if X ∈ M n and f is a Bernstein function on (0 , ∞) , then f (A) X - X f (B) ≤ max ‖ f ′ (A) ‖ , ‖ f ′ (B) ‖ A X - X B . These results extend and unify some known results for operator monotone functions; a subclass of Bernstein functions. [ABSTRACT FROM AUTHOR]

In this work, we refine the results of Sendov and Shan [New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures, J. Theor. Probab. 28 (2015), 1689–1725] on subordinators obtained by the class of Bernstein functions stable by the Mellin-Euler differential operator I−xddxI-x\tfrac{\text{d}}{\text{d}x} by giving a stochastic interpretation, proving monotonicity properties of the related distributions and providing additional extensions.

The present work is devoted to developing two numerical techniques based on fractional Bernstein polynomials, namely fractional Bernstein operational matrix method, to numerically solving a class of fractional integro-differential equations (FIDEs). The first scheme is introduced based on the idea of operational matrices generated using integration, whereas the second one is based on operational matrices of differentiation using the collocation technique. We apply the Riemann–Liouville and fractional derivative in Caputo’s sense on Bernstein polynomials, to obtain the approximate solutions of the proposed FIDEs. We also provide the residual correction procedure for both methods to estimate the absolute errors. Some results of the perturbation and stability analysis of the methods are theoretically and practically presented. We demonstrate the applicability and accuracy of the proposed schemes by a series of numerical examples. The numerical simulation exactly meets the exact solution and reaches almost zero absolute error whenever the exact solution is a polynomial.We compare the algorithms with some known analytic and semi-analytic methods. As a result, our algorithm based on the Bernstein series solution methods yield better results and show outstanding and optimal performance with high accuracy orders compared with those obtained from the optimal homotopy asymptotic method, standard and perturbed least squares method, CAS and Legendre wavelets method, and fractional Euler wavelet method.

We investigate the behavior near zero of the integrated density of states for random Schrödinger operators Φ (− Δ) + V ω in L 2 (ℝ d) , d ≥ 1 , where Φ is a complete Bernstein function such that for some α ∈ (0 , 2 ] , one has Φ (λ) ≍ λ α / 2 , λ ↘ 0 , and V ω (x) = ∑ i ∈ ℤ d q i (ω) W (x − i) is a random nonnegative alloy-type potential with compactly supported single site potential W. We prove that there are constants C , C ̃ , D , D ̃ > 0 such that − C ≤ liminf λ ↘ 0 λ d / α | log F q (D λ) | log N (λ) and limsup λ ↘ 0 λ d / α | log F q (D ̃ λ) | log N (λ) ≤ − C ̃ , where F q is the common cumulative distribution function of the lattice random variables q i . For typical examples of F q , the constants D and D ̃ can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, the large class of operator monotone functions of the Laplacian. This class includes both local and nonlocal kinetic terms such as the Laplace operator, its fractional powers, the quasi-relativistic Hamiltonians and many others. [ABSTRACT FROM AUTHOR]

In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence. [ABSTRACT FROM AUTHOR]

By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. This work could be seen as an-in depth analysis of a specific class, the one with the self-similarity property, of the general inverse of increasing Markov processes introduced in [15]. [ABSTRACT FROM AUTHOR]

In an important class of linear viscoelastic media the stress is the superposition of a Newtonian term and a stress relaxation term. It is assumed that the creep compliance is a Bernstein class function, which entails that the relaxation function is LICM. In this paper, the effect of Newtonian viscosity term on wave propagation is examined. It is shown that Newtonian viscosity dominates over the features resulting from stress relaxation. For comparison the effect of unbounded relaxation function is also examined. In both cases, the wave propagation speed is infinite, but the high-frequency asymptotic behavior of attenuation is different. Various combinations of Newtonian viscosity and relaxation functions, and the corresponding creep compliances are summarized. [ABSTRACT FROM AUTHOR]

In this article we present technique how to produce completely monotone functions using linear functionals and already known families of completely monotone functions. After that, using mean value theorems, we construct means of Cauchy type that have monotonicity properties. [ABSTRACT FROM AUTHOR]

The Catalan number sequence is one of the most famous number sequences in combinatorics and is well studied in the literature. In this paper we further investigate its fundamental properties related to the moment problem and prove for the first time that it is an infinitely divisible Stieltjes moment sequence in the sense of S.-G. Tyan. Besides, any positive real power of the sequence is still a Stieltjes determinate sequence. Some more cases including (a) the central binomial coefficient sequence (related to the Catalan sequence), (b) a double factorial number sequence and (c) the generalized Catalan (or Fuss–Catalan) sequence are also investigated. Finally, we pose two conjectures including the determinacy equivalence between powers of nonnegative random variables and powers of their moment sequences, which is supported by some existing results. [ABSTRACT FROM AUTHOR]

In a recent study of certain merging-splitting models of animal-group size (Degond et al. in J Nonlinear Sci 27(2):379–424, 2017), it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense, corresponding to unbounded growth of group size. In the present paper we show that for any such initial distribution with a power-law tail, the solution approaches a self-similar spreading form. A one-parameter family of such self-similar solutions exists, with densities that are completely monotone, having power-law behavior in both small and large size regimes, with different exponents. [ABSTRACT FROM AUTHOR]

We start by providing an explicit characterization and analytical properties, including the persistence phenomena, of the distribution of the extinction time T of a class of non-Markovian self-similar stochastic processes with two-sided jumps that we introduce as a stochastic time-change of Markovian self-similar processes. For a suitably chosen time-change, we observe, for classes with two-sided jumps, the following surprising facts. On the one hand, all the T 's within a class have the same law which we identify in a simple form for all classes and reduces, in the spectrally positive case, to the Fréchet distribution. On the other hand, each of its distribution corresponds to the law of an extinction time of a single Markov process without positive jumps, leaving the interpretation that the time-change has annihilated the effect of positive jumps. The example of the non-Markovian processes associated to Lévy stable processes is detailed. [ABSTRACT FROM AUTHOR]

The space-fractional and the time-fractional Poisson processes are two well-known models of fractional evolution. They can be constructed as standard Poisson processes with the time variable replaced by a stable subordinator and its inverse, respectively. The aim of this paper is to study nonhomogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with nonstationary increments), denoted by. Firstly, we consider the Poisson process time-changed by H and we obtain its explicit distribution and governing equation. Then, by using the right-continuous inverse of H, we define an inhomogeneous analog of the time-fractional Poisson process. [ABSTRACT FROM AUTHOR]

Abstract We give several new characterizations of completely monotone functions and Bernstein functions via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on N 0. We give a complete answer to the following question: Can we affirm that a function f is completely monotone (resp. a Bernstein function) if we know that the sequence f (k) k is completely monotone (resp. alternating)? This approach constitutes a kind of converse to Hausdorff's moment characterization theorem in the context of completely monotone sequences. [ABSTRACT FROM AUTHOR]

P. L. Chebyshev posed and solved (1857, 1859) the problem of finding an improper rational fraction least deviating from zero in the uniform metric on a closed interval among rational fractions whose denominator is a fixed polynomial of a given degree m that is positive on the interval and whose numerator is a polynomial of a given degree n ≥ m with unit leading coefficient. In 1884, A.A.Markov solved a similar problem in the case when the denominator is the square root of a given positive polynomial. In the 20th century, this research direction was developed by S.N.Bernstein, N. I.Akhiezer, and other mathematicians. For example, in 1964, G. Szegő extended Chebyshev's result to the case of trigonometric fractions using the methods of complex analysis. In this paper, using the methods of real analysis and developing Bernstein's approach, we find the best uniform approximation on the period by trigonometric polynomials of a certain order for an infinite series of proper trigonometric fractions of a special form. It turned out that, in the periodic case, it is natural to formulate some results in terms of the generalized Poisson kernel Πρ,ξ(t) = (cosξ)Pρ(t) + (sinξ)Qρ(t), which is a linear combination of the Poisson kernel Pρ(t) = (1 − ρ2)/[2(1 + ρ2 − 2ρ cos t)] and the conjugate Poisson kernel Qρ(t) = (ρ sin t)/(1 + ρ2 − 2ρ cos t), where ρ ∈ (−1, 1) and ξ ∈ R. We find the best uniform approximation on the period by the subspace Tn of trigonometric polynomials of order at most n for the linear combination Πρ,ξ(t) + (−1)nΠρ,ξ(t + π) of the generalized Poisson kernel and its translation. For ξ = 0, this yields Bernstein's known results on the best uniform approximation on [−1, 1] of the fractions 1/(x2 − a2) and x/(x2 − a2), |a| > 1, by algebraic polynomials. For ξ = π/2, we obtain the weight analogs (with the weight 1−x2x2) of these results. In addition, we find the value of the best uniform approximation on the period by the subspace Tn of a special linear combination of the mentioned Poisson kernel Pρ and the Poisson kernel Kρ for the biharmonic equation in the unit disk. [ABSTRACT FROM AUTHOR]

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

In this work we develop an original and thorough analysis of the (non)-smoothness properties of the semigroups, and their heat kernels, associated to a large class of continuous state branching processes with immigration. Our approach is based on an in-depth analysis of the regularity of the absolutely continuous part of the invariant measure combined with a substantial refinement of Ogura's spectral expansion of the transition kernels. In particular, we find new representations for the eigenfunctions and eigenmeasures that allow us to derive delicate uniform bounds that are useful for establishing the uniform convergence of the spectral representation of the semigroup acting on linear spaces that we identify. We detail several examples which illustrate the variety of smoothness properties that CBI transition kernels may enjoy and also reveal that our results are sharp. Finally, our technique enables us to provide the (eventually) strong Feller property as well as the rate of convergence to equilibrium in the total variation norm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of infinitely smooth and “nearly” locally supported radial basis functions (RBFs), derived from the general solution of a heat equation arising from the American option pricing problem. These basis functions are expressed in terms of “the repeated integrals of the complementary error function” and provide highly efficient tools to solve the free boundary partial differential equation resulting from the related option pricing model. We introduce an integral operator with a function-dependent lower limit which is employed as a basic tool to prove the radial positive definiteness of the proposed basis functions and could be of independent interest in the RBF theory. We then show that using the introduced functions as expansion bases in the context of an RBF-based meshless collocation scheme, we could exactly impose the transparent boundary condition accompanying the heat equation. We prove that the condition numbers of the resulting collocation matrices are orders of magnitude less than those arising from other popular RBF families used in current literature. Some other properties of these bases such as their Fourier transforms as well as some useful representations in terms of positive Borel measures will also be discussed. [ABSTRACT FROM AUTHOR]

In this article, the first hitting times of generalized Poisson processesNf(t), related to Bernštein functionsfare studied. For the space-fractional Poisson processes,Nα(t),t> 0 (corresponding tof=xα), the hitting probabilitiesP{Tαk< ∞} are explicitly obtained and analyzed. The processesNf(t) are time-changed Poisson processesN(Hf(t)) with subordinatorsHf(t) and here we studyand obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the formwhereare generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process. [ABSTRACT FROM AUTHOR]

We study coagulation-fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior crossing over from exponent $$-\frac{2}{3}$$ for small size to $$-\frac{3}{2}$$ for large size, with an exponential cutoff. [ABSTRACT FROM AUTHOR]

In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Lévy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bernštein functions for each time t. By means of these processes, a generalization of subordinate semigroups in the sense of Bochner is proposed. Because of time-inhomogeneity, two-parameter semigroups (propagators) arise and we provide a Phillips formula which leads to time dependent generators. The inverse processes are also investigated and the corresponding governing equations obtained in the form of generalized variable order fractional equations. An application to a generalized subordinate Brownian motion is also examined. [ABSTRACT FROM AUTHOR]

The study of distributed order calculus usually concerns fractional derivatives of the form ∫ 0 1 ∂ α u m ( d α ) for some measure m , eventually a probability measure. In this paper an approach based on Lévy mixing is proposed. Non-decreasing Lévy processes associated with Lévy triplets of the form ( a ( y ) , b ( y ) , ν ( d s , y ) ) are considered and the parameter y is randomized by means of a probability measure. The related subordinators are studied from different points of view. Some distributional properties are obtained and the interplay with inverse local times of Markov processes is explored. Distributed order integro-differential operators are introduced and adopted in order to write explicitly the governing equations of such processes. An application to slow diffusions (delayed Brownian motion) is discussed. [ABSTRACT FROM AUTHOR]

We prove that for any Bernstein function ψ the operator − ψ ( A ) generates a bounded holomorphic C 0 -semigroup ( e − t ψ ( A ) ) t ≥ 0 on a Banach space, whenever − A does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that ( e − t ψ ( A ) ) t ≥ 0 is holomorphic in the holomorphy sector of ( e − t A ) t ≥ 0 , and if ( e − t A ) t ≥ 0 is sectorially bounded in this sector then ( e − t ψ ( A ) ) t ≥ 0 has the same property. We also obtain new sufficient conditions on ψ in order that, for every Banach space X , the semigroup ( e − t ψ ( A ) ) t ≥ 0 on X is holomorphic whenever ( e − t A ) t ≥ 0 is a bounded C 0 -semigroup on X . These conditions improve and generalize well-known results by Carasso–Kato and Fujita. [ABSTRACT FROM AUTHOR]

We study the product formula $$(fg)(A) = f(A)g(A)$$ in the framework of (unbounded) functional calculus of sectorial operators $$A$$ . We give an abstract result, and, as corollaries, we obtain new product formulas for the holomorphic functional calculus, an extended Stieltjes functional calculus and an extended Hille-Phillips functional calculus. Our results generalise previous work of Hirsch, Martinez and Sanz, and Schilling. [ABSTRACT FROM AUTHOR]

Abstract: We create a new, functional calculus, approach to approximation formulas for -semigroups on Banach spaces restricted to the domains of fractional powers of their generators. This approach allows us to equip the approximation formulas with rates which appear to be optimal in a natural sense. In the case of analytic semigroups, we improve our general results obtaining better convergence rates which are optimal in that case too. The setting of analytic semigroups includes also the case of convergence on the whole space. As an illustration of our approach, we deduce optimal convergence rates in classical approximation formulas for -semigroups restricted to the domains of fractional powers of their generators. [Copyright &y& Elsevier]

We completely determine the set of s, t > 0 for which the function Ls,t(x) := x- Γ(x+t)/Γ(x+s) xs-t+1 is a Bernstein function, that is Ls,t(x) is positive with completely monotonic derivative on (0,∞). The complete monotonicity of several closely related functions is also established. [ABSTRACT FROM AUTHOR]

Ramanujan's sequence θ( n), n=0,1,2,... , is defined by $\frac{e^{n}}{2}=\sum_{j=0}^{n-1}\frac{n^{j}}{j!}+\frac{n^{n}}{n!} \theta(n)$. It is possible to define, in a simple manner, the function θ( x) for all nonnegative real numbers x. We show that the function $\lambda(x):=x (\theta(x)-\frac{1}{3} )$ is a Bernstein function on [0,∞), that is, λ( x) is nonnegative with completely monotonic derivative on [0,∞). This implies some earlier results concerning complete monotonicity of the function θ( x) on [0,∞). [ABSTRACT FROM AUTHOR]

In 1958, Jeffreys (Geophys J R Astron Soc 1:92-95) proposed a power law of creep, generalizing the logarithmic law earlier introduced by Lomnitz, to broaden the geophysical applications to fluid-like materials including igneous rocks. This generalized law, however, can be applied also to solid-like viscoelastic materials. We revisit the Jeffreys-Lomnitz law of creep by allowing its power law exponent α, usually limited to the range 0 ≤ α ≤ 1 to all negative values. This is consistent with the linear theory of viscoelasticity because the creep function still remains a Bernstein function, that is positive with a completely monotone derivative, with a related spectrum of retardation times. The complete range α ≤ 1 yields a continuous transition from a Hooke elastic solid with no creep $\left(\alpha \,\to\, -\infty\right)$ to a Maxwell fluid with linear creep $\left(\alpha \,=\,1\right)$ passing through the Lomnitz viscoelastic body with logarithmic creep $\left(\alpha\, =0\right)$, which separates solid-like from fluid-like behaviors. Furthermore, we numerically compute the relaxation modulus and provide the analytical expression of the spectrum of retardation times corresponding to the Jeffreys-Lomnitz creep law extended to all α ≤ 1. [ABSTRACT FROM AUTHOR]

We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci and Sokal. In addition, we consider the relationship of functions of W with the class of completely monotonic functions and show that W is a complete Bernstein function. [ABSTRACT FROM AUTHOR]

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and visualized on artificial data sets by classical multidimensional scaling. A distance-based discriminant algorithm and a robust multidimensional centroid estimate illustrate the theory, closely connected to the Gaussian kernels of Machine Learning. [ABSTRACT FROM AUTHOR]

Abstract: The theory of quasi-arithmetic means represents a powerful tool in the study of covariance functions across space–time. In the present study we use quasi-arithmetic functionals to make inferences about the permissibility of averages of functions that are not, in general, permissible covariance functions. This is the case, e.g., of the geometric and harmonic averages, for which we obtain permissibility criteria. Also, some important inequalities involving covariance functions and preference relations as well as algebraic properties can be derived by means of the proposed approach. In particular, quasi-arithmetic covariances allow for ordering and preference relations, for a Jensen-type inequality and for a minimal and maximal element of their class. The general results shown in this paper are then applied to the study of spatial and spatio-temporal random fields. In particular, we discuss the representation and smoothness properties of a weakly stationary random field with a quasi-arithmetic covariance function. Also, we show that the generator of the quasi-arithmetic means can be used as a link function in order to build a space–time nonseparable structure starting from the spatial and temporal margins, a procedure that is technically sound for those working with copulas. Several examples of new families of stationary covariances obtainable with this procedure are shown. Finally, we use quasi-arithmetic functionals to generalise existing results concerning the construction of nonstationary spatial covariances, and discuss the applicability and limits of this generalisation. [Copyright &y& Elsevier]

Abstract: We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in -fat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded -fat open sets with respect to these processes with their Euclidean boundaries. [Copyright &y& Elsevier]

There is a great demand for statistical modelling of phenomena that evolve in both space and time, and thus, there is a growing literature on covariance function models for spatio-temporal processes. Although several nonseparable space–time covariance models are available in the literature, very few of them can be used for spatially anisotropic data. In this paper, we propose a new class of stationary nonseparable covariance functions that can be used for both geometrically and zonally anistropic data. In addition, we show some desirable mathematical features of this class. Another important aspect, only partially covered by the literature, is that of spatial nonstationarity. We show a very simple criteria allowing for the construction of space–time covariance functions that are nonseparable, nonstationary in space and stationary in time. Part of the theoretical results proposed in the paper will then be used for the analysis of Irish wind speed data as in HASLETT and RAFTERY ( Applied Statistics, 38, 1989, 1). [ABSTRACT FROM AUTHOR]