Sharp large deviations for sums of bounded from above random variables

We show large deviation expansions for sums of independent and bounded from above random variables. Our moderate deviation expansions are similar to those of Cram′er(1938), Bahadur and Ranga Rao(1960), and Sakhanenko(1991). In particular, our results extend Talagrand's inequality from bounded random variables to random variables having finite(2 + δ)-th moments, where δ∈(0, 1]. As a consequence,we obtain an improvement of Hoeffding's inequality. Applications to linear regression, self-normalized large deviations and t-statistic are also discussed.     

Some limit properties for the coefficients of the q-Catalan numbers

In the paper, some limit properties, such as, moderate deviation principle and law of large numbers, for the coefficients of the q-Catalan numbers and the generalized q-Catalan numbers are obtained.     

The genuinely nonlinear Graetz—Nusselt ultraparabolic equation

We study a second-order quasilinear ultraparabolic equation whose matrix of the coefficients of the second derivatives is nonnegative, depends on the time and spatial variables, and can change rank in the case when it is diagonal and the coefficients of the first derivatives can be discontinuous. We prove that if the equation is a priori known to enjoy the maximum principle and satisfies the additional “genuine nonlinearity” condition then the Cauchy problem with arbitrary bounded initial data has at least one entropy solution and every uniformly bounded set of entropy solutions is relatively compact in L _loc ¹ . The proofs are based on introduction and systematic study of the kinetic formulation of the equation in question and application of the modification of the Tartar H-measures proposed by E. Yu. Panov.     

Large deviations for invariant measures of general stochastic reaction–diffusion systems

In this paper we prove a large deviations principle for the invariant measures of a class of reaction–diffusion systems in bounded domains of $\text{R}{}^{\mathrm{d}}\text{,}\text{d?1}$, perturbed by a noise of multiplicative type. We consider reaction terms which are not Lipschitz-continuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate. To cite this article: S. Cerrai, M. Röckner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Absolute convergence of double series of Fourier-Haar coefficients for functions of bounded p-variation

We consider functions of two variables of bounded p-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of series of Fourier-Haar coefficients of one-variable functions which have a bounded Wiener p-variation or belong to the class Lip ??. We show that the obtained results are unimprovable. We also formulate N-dimensional analogs of the main result and its corollaries.     

Asymptotic synchronization in n-dimensional second order dissipative lattices of coupled oscillators

By introducing a new norm which is equivalent to the usual norm in the phase space, we prove that for n-dimensional second order dissipative lattices of coupled oscillators with external periodic forces under Dirichlet, Neumann and periodic boundary conditions, if the system is bounded dissipative and the coupled coefficients are both large enough, the asymptotic synchronization will occur. And we give a concrete bounded dissipative second order lattices system. Our results show that the bounds of the difference between the components of any solution are directly proportional to mn/2 and inversely proportional to the coupled coefficients, where m is the mesh size and n is the space dimension of lattice points.     

Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications

In this paper Euler-Maruyama approximation for SDE with non-Lipschitz coefficients is proved to converge uniformly to the solution in Lp-space with respect to the time and starting points. As an application, we also study the existence of solution and large deviation principle for anticipative SDE with random initial condition.     

Large deviation results for the nonparametric regression function estimator on functional data

This paper is devoted to the study of large deviation behavior in the setting of the estimation of the regression function on functional data. A large deviation principle is stated for a process Z _n , defined below, allowing to derive a pointwise large deviation principle for the Nadaraya- Watson-type l-indexed regression function estimator as a by-product. Moreover, a uniform over VC-classes Chernoff type large deviation result is stated for the deviation of the l-indexed regression estimator.     

Gradient estimates for diffusion semigroups with singular coefficients

Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the Rd-case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of Rd with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the Rd-case and that the coefficients can be Hölder continuous. Our results also imply a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying diffusion semigroup.     

Cramér-type moderate deviations for stationary sequences of bounded random variables

We derive Cramér-type moderate deviations for stationary sequences of bounded random variables. Our results imply the moderate deviation principles and a Berry–Esseen bound. Applications to quantile coupling inequalities, functions of ?-mixing sequences, and contracting Markov chains are discussed.     

Stability of weak solutions for the compressible Navier-Stokes-Poisson equations

In this paper, we consider the isentropic compressible Navier-Stokes-Poisson equations arising from transport of charged particles or motion of gaseous stars in astrophysics. We are interested in the case that the viscosity coefficients depend on the density and shall degenerate in the appearance of (density) vacuum, and show the L 1 -stability of weak solutions for arbitrarily large data on spatial multi-dimensional bounded or periodic domain or whole space.     

Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

We prove comparison theorems for the H ^∞-calculus that allow to transfer the property of having a bounded H ^∞-calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ^∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ^∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L _p -scale. We apply the results to give new proofs on the H ^∞-calculus for elliptic differential operators, including Schrödinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ^∞-calculus in certain Sobolev spaces and that the Stokes operator on bounded domains Ω with ?Ω ∈ C ^1,1 has a bounded H ^∞-calculus in the Helmholtz scale L _p,σ (Ω), p ∈ (1,∞).     

A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization

In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by (1 + log(H/h))², where H and h are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.     

Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay

The principle of competitive exclusion is extended to n-species nonautonomous Lotka-Volterra competition systems of differential equations with infinite delay. It is shown that if the coefficients are bounded, continuous and satisfy certain inequalities, then any solution with initial function in an appropriate space will have n−1 of its components tend to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation.     

Gradient estimates for Dirichlet parabolic problems in unbounded domains

We consider autonomous parabolic Dirichlet problems in a regular unbounded open set Ω⊂R^N involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution.     

A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids

We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows us to solve elliptic differential equations with variable coefficients for dimensions d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator-dependent stencil for multilinear finite elements. Numerical simulation results are presented for a three-dimensional problem on a curvilinear bounded domain and for a six-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditioner.     

Large deviations and support theorem for diffusion processes via rough paths

We use the continuity theorem of Lyons for rough paths in the p-variation topology to produce an elementary approach to the large deviation principle and the support theorem for diffusion processes. The proofs reduce to establish the corresponding results for Brownian motion itself as a rough path in the p-variation topology, 2<p<3, and the technical step is to handle the Lévy area in this respect. Some extensions and applications are discussed.     

Application of homogenization and large deviations to a parabolic semilinear equation

We study the behavior of the solution of a partial differential equation with a linear parabolic operator with non-constant coefficients varying over length scale δ and nonlinear reaction term of scale 1/?. The behavior is required as ? tends to 0 with δ small compared to ?. We use the theory of backward stochastic differential equations corresponding to the parabolic equation. Since δ decreases faster than ?, we may apply the large deviations principle with homogenized coefficients.     

On the properties of generalized frames

In this paper, we introduce the notion of generalized frames and study their properties. Discrete and integral frames are special cases of generalized frames. We give criteria for generalized frames to be integral (discrete). We prove that any bounded operator A with a bounded inverse acting from a separable space H to L ₂(Ω) (where Ω is a space with countably additive measure) can be regarded as an operator assigning to each element x ∈ H its coefficients in some generalized frame.     

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.