Two-term trace estimates for relativistic stable processes

We prove trace estimates for the relativistic α-stable process extending the result of Bañuelos and Kulczycki (2008) in the stable case.     

Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and GD(x,y) be the Green functions of rotationally invariant symmetric α-stable process in Rd and in an open set D, respectively, where 0<α<2. The inequality GD(x,y)GD(y,z)/GD(x,z)?c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c=c(D)∈(0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form GD(x,y)GD(z,w)/GD(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form GD(x,y)GD(z,w)/GD(x,w) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.     

Harnack inequalities for Ornstein-Uhlenbeck processes driven by Lévy processes

By using the existing sharp estimates of the density function for rotationally invariant symmetric α-stable Lévy processes and rotationally invariant symmetric truncated α-stable Lévy processes, we obtain that the Harnack inequalities hold for rotationally invariant symmetric α-stable Lévy processes with α∈(0,2) and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric α-stable Lévy process, while the logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated α-stable Lévy processes.     

Large Deviations for Additive Functionals of Symmetric Stable Processes

We establish the large deviation principle for additive functionals of symmetric α-stable processes employing the Gärtner-Ellis theorem.     

Boundary Behavior of α-Harmonic Functions on the Complement of the Sphere and Hyperplane

We study α-harmonic functions on the complement of the sphere and on the complement of the hyperplane in Euclidean spaces of dimension bigger than one, for α?∈?(1,2). We describe the corresponding Hardy spaces and prove the Fatou theorem for α-harmonic functions. We also give explicit formulas for the Martin kernel of the complement of the sphere and for the harmonic measure, Green function and Martin kernel of the complement of the hyperplane for the symmetric α-stable Lévy processes. Some extensions for the relativistic α-stable processes are discussed.     

Limit theorems of continuous-time random walks with tails

We study the functional limits of continuous-time random walks (CTRWs) with tails under certain conditions. We find that the scaled CTRWs with tails converge weakly to an α-stable Lévy process in D([0, 1]) with M ₁-topology but the corresponding scaled CTRWs converge weakly to the same limit in D([0, 1]) with J ₁-topology.     

On the Besov regularity of periodic Lévy noises

In this paper, we study the Besov regularity of Lévy white noises on the d-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general Lévy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-α-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical σ-field of the space of generalized functions. These results pave the way to the characterization of the n-term wavelet approximation properties of stochastic processes.     

Two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups

Two families of functions on (0,∞) are related to the theory of fractional powers of generators of strongly continuous semigroups—namely the family (σ_α(·,t))_t>0 of density functions of the one-sided stable semigroup of order α∈(0,1), and a family (e_α(·,μ))_μ>0 of Mittag-Leffler type functions. For the latter family we make this relation visible. We collect some important properties of these functions, and we improve a known result on the Laplace transform of the functions e_α(·,μ) and their derivatives in the sense that we enlarge the domains of the Laplace variable. We furthermore find an expression for the Laplace transform of the functions t?σ_α(x,t), x>0, in terms of the derivative e′_α(·,μ).     

Simulation of geometric stable and other limiting multivariate distributions arising in random summation scheme

A method for simulation of ν_α-stable random vectors based on their representation as location and scale mixtures of α-stable random vectors is discussed. The method has applications in modeling multivariate financial portfolios and Monte Carlo estimation procedures.     

A class of semigroups regularized in space and time

We consider α-times integrated C-regularized semigroups, which are a hybrid between semigroups regularized in space (C-semigroups) and integrated semigroups regularized in time. We study the basic properties of these objects, also in absence of exponential boundedness. We discuss their generators and establish an equivalence theorem between existence of integrated regularized semigroups and well-posedness of certain Cauchy problems. We investigate the effect of smoothing regularized semigroups by fractional integration.     

Small, nm-stable compact G-groups

We prove that if (H, G) is a small, nm-stable compact G-group, then H is nilpotent-by-finite, and if additionally NM(H) < ω or NM(H) = ω ^α for some ordinal α, then H is abelian-by-finite. Both results are significant steps towards the proof of the conjecture that each small, nm-stable compact G-group is abelian-by-finite. We provide counter-examples to the NM-gap conjecture, that is we give examples of small, nm-stable compact G-groups of infinite ordinal NM-rank.     

A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions

We study the problem of parameter estimation for Ornstein–Uhlenbeck processes driven by symmetric α-stable motions, based on discrete observations. A least squares estimator is obtained by minimizing a contrast function based on the integral form of the process. Let h be the length of time interval between two consecutive observations. For both the case of fixed h and that of h → 0, consistencies and asymptotic distributions of the estimator are derived. Moreover, for both of the cases of h, the estimator has a higher order of convergence for the Ornstein–Uhlenbeck process driven by non-Gaussian α-stable motions (0 < α < 2) than for the process driven by the classical Gaussian case (α = 2).     

Exponential ergodicity and regularity for equations with Lévy noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8]. Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28], with a different method.     

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C^1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.     

Optimal algorithms for the α-neighbor p-center problem

Assigning multiple service facilities to demand points is important when demand points are required to withstand service facility failures. Such failures may result from a multitude of causes, ranging from technical difficulties to natural disasters. The α-neighbor p-center problem deals with locating p service facilities. Each demand point is assigned to its nearest α service facilities, thus it is able to withstand up to α − 1 service facility failures. The objective is to minimize the maximum distance between a demand point and its αth nearest service facility. We present two optimal algorithms for both the continuous and discrete α-neighbor p-center problem. We present experimental results comparing the performance of the two optimal algorithms for α = 2. We also present experimental results showing the performance of the relaxation algorithm for α = 1, 2, 3.     

A note on the weak invariance principle for local times

We prove uniform L²-convergence of local times of aperiodic recurrent random walks to local times of strictly α-stable processes with α > 1.     

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.     

Random attractor for stochastic lattice dynamical systems with α-stable Lévy noises

The present paper is devoted to the existence of a random attractor for stochastic lattice dynamical systems with α-stable Lévy noises.