Caricature of hydrodynamics for lattice dynamics

We consider the lattice dynamics in the half-space, with zero boundary condition. The initial data are supposed to be random function. We introduce the family of initial measures {?? ₀ ^? , ? > 0} depending on a small scaling parameter ?. We assume that the measures ?? ₀ ^? are locally homogeneous for space translations of order much less than ? ^?1 and nonhomogeneous for translations of order ? ^?1. Moreover, the covariance of ?? ₀ ^? decreases with distance uniformly in ?. Given ?? ?? ? / 0, r ?? ? ₊ ^d , and ?? > 0, we consider the distributions of random solution in the time moments t = ??/? _?? and at lattice points close to [r/?] ?? ? ₊ ^d . Themain goal is to study the asymptotic behavior of these distributions as ? ?? 0 and to derive the limit hydrodynamic equations of the Euler or Navier-Stokes type.     

A geometric interpretation of the transition density of a symmetric Lévy process

We study for a class of symmetric Lévy processes with state space R n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t>0 and （δt）t>0. The first family of metrics describes the diagonal term pt（0）; it is induced by the characteristic exponent ψ of the Lévy process by dt（x, y） = ^1/2tψ（x-y）. The second and new family of metrics δt relates to ^1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt（x） = pt（0）e- δ2t （x,0） where pt（0） corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt（x）.     

Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings

The dual of an infinitely divisible distribution on ? ^d without Gaussian part defined in Sato (ALEA Lat. Am. J. Probab. Math. Statist. 3:67–110, 2007) is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping μ=Φ _f ? ρ of ρ to μ in the class of infinitely divisible distributions on ? ^d , where μ is the distribution of an improper stochastic integral of a nonrandom function f with respect to a Lévy process on ? ^d with distribution ρ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.     

Stochastic Comparison for Lévy-Type Processes

The main goal of this paper is to establish necessary and sufficient conditions for stochastic comparison of two general Lévy-type processes on ? ^d . By refining the test functions in Wang (Acta Math. Sin. Engl. Ser. 25:741–758, 2009), mainly the test functions of diffusion coefficients, we get the necessary conditions. The sufficiency of the conditions is obtained by constructing a new sequence of finite Lévy measures {ν ⁿ } _n≥1 different from the one in Wang (Acta Math. Sin. Engl. Ser. 25:741–758, 2009) to approach the Lévy measure ν.     

A Subclass of Type G Selfdecomposable Distributions on ℝ d

A new class of type G selfdecomposable distributions on ℝ ^d is introduced and characterized in terms of stochastic integrals with respect to Lévy processes. This class is a strict subclass of the class of type G and selfdecomposable distributions, and in dimension one, it is strictly bigger than the class of variance mixtures of normal distributions by selfdecomposable distributions. The relation to several other known classes of infinitely divisible distributions is established. Research of J. Rosiński supported, in part, by a grant from the National Science Foundation.     

Selfdecomposable Fields

In the present paper, we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel function) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context, we also study the so-called Urbanik classes of random fields. We follow this with the study of existence and selfdecomposability of integrated Volterra fields. Finally, we introduce infinitely divisible field-valued Lévy processes, give the Lévy-Itô representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of Lévy semistationary processes with a Gamma kernel and Ornstein–Uhlenbeck processes.     

The Hausdorff Dimension of Operator Semistable Lévy Processes

Let X={X(t)} _t≥0 be an operator semistable Lévy process in ? ^d with exponent E, where E is an invertible linear operator on ? ^d and X is semi-selfsimilar with respect to E. By refining arguments given in Meerschaert and Xiao (Stoch. Process. Appl. 115, 55–75, 2005) for the special case of an operator stable (selfsimilar) Lévy process, for an arbitrary Borel set B??₊ we determine the Hausdorff dimension of the partial range X(B) in terms of the real parts of the eigenvalues of E and the Hausdorff dimension of B.     

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index {{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}

Let B?=?(B ₁(t), . . . ,B _d (t)) be a d-dimensional fractional Brownian motion with Hurst index ???<?1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low H?lder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using ??standard?? tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates and call for an extension of Gaussian tools such as, for instance, the Malliavin calculus. After a first introductory paper (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics, 2011), this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as Lévy area. A summary in French may be found in Unterberger (Mode d??emploi de la théorie constructive des champs bosoniques, avec une application aux chemins rugueux, 2011).     

A lower bound in Nehari’s theorem on the polydisc

By theorems of Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2), Nehari??s theorem (i.e., if H _?? is a bounded Hankel form on H ²(D ^d ) with analytic symbol ??, then there is a function ?? in L ^??(T ^d ) such that ?? is the Riesz projection of g4) is known to hold on the polydisc D ^d for d > 1. A method proposed in Helson??s last paper is used to show that the constant C _d in the estimate ??????_?? ?? C _d ??H _?? ?? grows at least exponentially with d; it follows that there is no analogue of Nehari??s theorem on the infinite-dimensional polydisc.     

Uniform approximation of Poisson integrals of functions from the class H ω by de la Vallée Poussin sums

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C _?? ^q H _?? of Poisson integrals of functions from the class H _?? generated by convex upwards moduli of continuity ??(t) which satisfy the condition ??(t)/t ?? ?? as t ?? 0. As an implication, a solution of the Kolmogorov-Nikol??skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H ^??, 0 < ?? < 1, is obtained.     

Inhomogeneous Lévy Processes in Lie Groups and Homogeneous Spaces

We obtain a representation of an inhomogeneous Lévy process in a Lie group or a homogeneous space in terms of a drift, a matrix function and a measure function. Since the stochastic continuity is not assumed, our result generalizes the well-known Lévy–Itô representation for stochastic continuous processes with independent increments in ? ^d and its extension to Lie groups.     

Simple random walk on long range percolation clusters I: heat kernel bounds

In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any d ?? 1 and for any exponent ${s \in (d, (d+2) \wedge 2d)}$ giving the rate of decay of the percolation process, we show that the return probability decays like ${t^{-{d}/_{s-d}}}$ up to logarithmic corrections, where t denotes the time the walk is run. Our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. The bounds and accompanying understanding of the geometry of the cluster play a crucial role in the companion paper (Crawford and Sly in Simple randomwalk on long range percolation clusters II: scaling limit, 2010) where we establish the scaling limit of the random walk to be ??-stable Lévy motion.     

Numerical Techniques in Lévy Fluctuation Theory

This paper presents a framework for numerical computations in fluctuation theory for Lévy processes. More specifically, with $\bar X_t:= \sup_{0\le s\le t} X_s$ denoting the running maximum of the Lévy process X _t , the aim is to evaluate ${\mathbb P}(\bar X_t \le x)$ for t,x?>?0. We do so by approximating the Lévy process under consideration by another Lévy process for which the double transform ${\mathbb E} e^{-\alpha \bar X_{\tau(q)}}$ is known, with τ(q) an exponentially distributed random variable with mean 1/q; then we use a fast and highly accurate Laplace inversion technique (of almost machine precision) to obtain the distribution of $\bar X_t$ . A broad range of examples illustrates the attractive features of our approach.     

Local Subexponentiality and Self-decomposability

The class of exponential tilts of convolution equivalent distributions is determined. As a corollary, the local subexponentiality of one-sided infinitely divisible distributions is characterized. It is applied to the subexponentiality of the densities of a self-decomposable distribution and its Lévy measure. Bondesson’s conjecture on the density of the Lévy measure of a lognormal distribution is solved as an example. Results of Denisov et al. on the distributions of random sums are extended to the two-sided case. Finally, the local subexponentiality of the distribution of the supremum of a random walk is characterized.     

M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps

We study parametric inference for multidimensional stochastic differential equations with jumps from some discrete observations. We consider a case where the structure of jumps is mainly controlled by a random measure which is generated by a Lévy process with a Lévy measure f_θ(z)dz, and we admit the case ∫ f_θ(z)dz = ∞ in which infinitely many small jumps occur even in any finite time intervals. We propose an estimating function under this complicated situation and show the consistency and the asymptotic normality. Although the estimators in this paper are not completely efficient, the method can be applied to comparatively wide class of stochastic differential equations, and it is easy to compute the estimating equations. Therefore, it may be useful in applications. We also present some simulation results for some simple models. Final version 25 December 2004     

Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation

In a previous paper (Benaych-Georges in Related Convolution 2006), we defined the rectangular free convolution ?_λ. Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ?_λ -infinitely divisible distributions, which preserves limit theorems. We give an interpretation of this correspondence in terms of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their ?_λ-infinitely divisible correspondents when the dimensions go from one to infinity in a ratio λ.     

Sub-Markovian C 0-Semigroups Generated by Fractional Laplacian with Gradient Perturbation

We present a sufficient condition for fractional Laplacian with gradient perturbation to generate a sub-Markovian C ₀-semigroup on ${L^1(\mathbb{R}^d, dx)}$ . The condition also yields the ultracontractivity of the semigroup and an upper on-diagonal estimate of the associated transition kernel. Based on the subordination technique, the extension to general pure jump Lévy process with gradient perturbation is studied. As a direct application, we obtain sufficient conditions for the strong Feller property of stochastic differential equations driven by additive Lévy process.     

The Z \to c\bar c \to \gamma \gamma *, Z \to b\bar b \to \gamma \gamma * triangle diagrams and the Z → γψ, Z → γγ decays

We expounded an approach for studying the Z ?? ??? and Z ?? ???? decay based on the sum rules for the $Z \to c\bar c \to \gamma \gamma *$ and $Z \to b\bar b \to \gamma \gamma *$ amplitudes and their derivatives. We calculate the branching ratios of the Z ?? ??? and Z ?? ???? decays under different suppositions about the saturation of the sum rules. We find the lower bounds of ?? _?? BR(Z ?? ???) = 1.95 · 10 ^?7 and ?? _?? BR(Z ?? ????) = 7.23 · 10^?7 and discuss deviations from the lower bounds including the possibility of BR[Z ?? ??J/??(1S)] ?? BR[Z ?? ????(1S)] ?? 10 ^?6 , which is probably measurable at the LHC. Moreover, we calculate the angle distributions in the Z ?? ??? and Z ?? ???? decays.     

Truncation and aliasing errors for Whittaker-Kotelnikov-Shannon sampling expansion

Let B _?? ^p , 1 ?? p < ??, be the space of all bounded functions from L _p (?) which can be extended to entire functions of exponential type ??. The uniform error bounds for truncated Whittaker-Kotelnikov-Shannon series based on local sampling are derived for functions f ?? B _?? ^p without decay assumption at infinity. Then the optimal bounds of the aliasing error and truncation error of Whittaker-Kotelnikov-Shannon expansion for non-bandlimited functions from Sobolev classes U(W _p ^r (?)) are determined up to a logarithmic factor.     

Harnack Inequality and Hölder Regularity Estimates for a Lévy Process with Small Jumps of High Intensity

We consider a Lévy process in ? ^d (d≥3) with the characteristic exponent $$\varPhi(\xi)=\frac{|\xi|^2}{\ln(1+|\xi|^2)}-1.$$ The scale invariant Harnack inequality and a priori estimates of harmonic functions in Hölder spaces are proved.