Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

Lévy-driven Volterra Equations in Space and Time

We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.     

Branching Random Walks in Space–Time Random Environment: Survival Probability,Global and Local Growth Rates

We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the d-dimensional integer lattice, while at each time unit, they split into independent copies according to time–space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rate is given by the directional free energy.     

Numerical Solution of the Two-Sided Space–Time Fractional Telegraph Equation Via Chebyshev Tau Approximation

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg–Landau Equations

This paper considers the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by additive noise with α∈(0, 1). First, we give some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Second, we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equation in H. At last, we prove the finiteness of fractal dimension of random attractors.     

Degenerate Linear Evolution Equations with the Riemann–Liouville Fractional Derivative

We study the unique solvability of the Cauchy and Schowalter–Sidorov type problems in a Banach space for an evolution equation with a degenerate operator at the fractional derivative under the assumption that the operator acting on the unknown function in the equation is p-bounded with respect to the operator at the fractional derivative. The conditions are found ensuring existence of a unique solution representable by means of the Mittag-Leffler type functions. Some abstract results are illustrated by an example of a finite-dimensional degenerate system of equations of a fractional order and employed in the study of unique solvability of an initial-boundary value problem for the linearized Scott-Blair system of dynamics of a medium.     

On the Dirac–Klein–Gordon Equations in Three Space Dimensions

ABSTRACT

We establish a local existence result for Dirac–Klein–Gordon equations in three space dimensions, employing a null form estimate and a fixed point argument.     

Fuzzy Fractional Monodromy and the Section–Hyperboloid

A theorem about the matrix of fractional monodromy will be formulated. The monodromy corresponds to going around a fiber with a singular point of oscillator type with arbitrary resonance. The reason of fractional monodromy and fuzziness of such a monodromy is explained. Some ideas for the proof of the theorem are given. A few remarks about the semi-global structure of singular lagrangian fibration are made. Lecture held in the Seminario Matematico e Fisico di Milano on November 8, 2004 Received: June 2007     

Unconditionally Stable Algorithm for Solving the Three-Dimensional Nonstationary Navier–Stokes Equations

We propose an unconditionally stable method for solving the three-dimensional nonstationary Navier–Stokes equations in the velocity–pressure variables. The method is based on a conservative finite-difference scheme and the simultaneous solution of the momentum and continuity equations at each time layer. The velocity and pressure fields are calculated by using a parallel algorithm for solving systems of linear equations by the Gauss method.     

The Fundamental Solutions of the Space, Space-Time Riesz Fractional Partial Differential Equations with Periodic Conditions

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (SRFPDE) and the space-time Riesz fractional partial differential equation (STRFPDE) are discussed, respectively. Using methods of Fourier series expansion and Laplace transform, we derive the explicit expressions of the fundamental solutions for the SRFPDE and the STRFPDE, respectively.     

Stochastic Relations of Random Variables and Processes

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The main contributions of the paper are: a functional characterization of stochastic relations, necessary and sufficient conditions for the preservation of stochastic relations, and an algorithm for finding subrelations preserved by probability kernels. The theory is illustrated with applications to hidden Markov processes, population processes, and queueing systems.     

Congruence Convergence in Pp-Wave Space–Time

We argue that the well-known geodesic completeness property of pp-waves can be disregarded once the geodesics are extracted as being extended along sets of Brinkmann coordinates. We investigate this issue in the more general context of congruence convergence and show that the problem leads to various issues for nongeodesic congruences. Our consideration is mostly based on the null congruence expansion, and we also provide a generalized Raychaudhuri equation.     

An Approach to Time Integration of the Navier–Stokes Equations

Computational Mathematics and Mathematical Physics - An approach to the time integration of the Navier–Stokes equations for a compressible heat-conducting gas is developed. According to this...     

Maxwell–Dirac Equations in Four-Dimensional Minkowski Space

Abstract

I derive the global existence and asymptotic behavior of small amplitude solutions to the system of massive coupled classical Maxwell–Dirac equations in the four-dimensional Minkowski space. Because the physically defined energy of the system is not positive definite, I transform it into an equivalent system of Maxwell–Klein–Gordon equations, which I study with a method based on gauge invariant energy estimates and geometric properties of the equations.     

Gravitational Field Equations on Fefferman Space–Times

The total space \({\mathfrak M} \approx {\mathbb H}_1 \times S^1\) of the canonical circle bundle over the 3-dimensional Heisenberg group \({\mathbb H}_1\) is a space–time with the Lorentzian metric \(F_{\theta _0}\) (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form \(\theta _0\) on \({\mathbb H}_1\). The matter and energy content of \(\mathfrak M\) is described by the energy-momentum tensor \({T}_{\mu \nu }\) (the trace-less Ricci tensor of \(F_{\theta _0}\)) as an effect of the non flat nature of Feferman’s metric \(F_{\theta _0}\). We study the gravitational field equations \(R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }\) on \({\mathfrak M}\). We consider the first order perturbation \(g = F_{\theta _0} + \epsilon \, h\), \(\epsilon<< 1\), and linearize the field equations about \(F_{\theta _0}\). We determine a Lorentzian metric g on \({\mathfrak M}\) which solves the linearized field equations corresponding to a diagonal perturbation h.     

Reflected Symmetric α-Stable Processes and Regional Fractional Laplacian

In this paper we investigate the reflected symmetric α-stable processes and their generators. We show that the generators are regional fractional Laplacians on the closed region. In the case of 1≤α<2 their existence requires that on the boundary. Among other things we obtain the integration by parts formula of the regional fractional Laplacian and the semi-martingale decomposition of the reflected symmetric α-stable processes.     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

Ulam–Hyers Stability for the Darboux Problem for Partial Fractional Differential and Integro-differential Equations via Picard Operators

In the present paper we investigate some uniqueness and Ulam’s type stability concepts of fixed point equations due to Rus, for the Darboux problem of partial differential and integro-differential equations involving the Caputo fractional derivative. Our results are obtained by using weakly Picard operators theory.     

The Invariant and Quasi-invariant Transformations of the Stable Lévy Processes

Let ξ(t),t∈[0,1] be a strictly stable Lévy process with the index of stability α∈(0,2). By ℘ _ξ we denote the law of ξ in the Skorokhod space . For arbitrary ξ we construct ℘ _ξ -quasi-invariant semigroup of transformations of . Under some nondegeneracy condition on the spectral measure of the stable process we construct ℘ _ξ -quasi-invariant group of transformations of . In symmetric case this group is a group of the invariant transformations.      

Fractional Differential Equations with Nonlocal Integral and Integer–Fractional-Order Neumann Type Boundary Conditions

We introduce a new concept of the coupling of nonlocal integral and integer–fractional-order Neumann type boundary conditions, and discuss the existence and uniqueness of solutions for a coupled system of fractional differential equations supplemented with these conditions. The existence of solutions is derived from Leray–Schauder’s alternative and Schauder’s fixed point theorem, while the uniqueness of solutions is established by means of Banach’s contraction mapping principle. The results obtained in this paper are well illustrated with the aid of examples.