Asymptotic Inequality Research Articles

Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations

We introduce a new framework of Markovian lifts of stochastic Volterra integral equations (SVIEs for short) with completely monotone kernels. We define the state space of the Markovian lift as a separable Hilbert space which incorporates the singularity or regularity of the kernel into the definition. We show that the solution of an SVIE is represented by the solution of a lifted stochastic evolution equation (SEE for short) defined on the Hilbert space and prove the existence, uniqueness and Markov property of the solution of the lifted SEE. Furthermore, we establish an asymptotic log-Harnack inequality and some consequent properties for the Markov semigroup associated with the Markovian lift via the asymptotic coupling method.

Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.

A Universal Power Law Governing the Accuracy of Wave Function-Based Electronic Structure Calculations

A universal power law governing the accuracy of wave function-based electronic structure calculations is derived from first principles. The resulting expression ΔE(N,N)N≳19π2gNN, where g is a system-specific factor assuming values between zero and one and ≳ stands for asymptotic inequality at the limit of N→∞, allows facile estimation of the error ΔE(N,N) in the electronic energy of a singlet state of an N-electron system computed with a basis set of N one-electron functions. Several approaches to the estimation of the factor g, which depends on the on-top two-electron density, are presented.

Stochastic functional differential equations with infinite delay under non-Lipschitz coefficients: Existence and uniqueness, Markov property, ergodicity, and asymptotic log-Harnack inequality

This paper focuses on a class of stochastic functional differential equations with infinite delay and non-Lipschitz coefficients. Under one-sided super-linear growth and non-Lipschitz conditions, this paper establishes the existence and uniqueness of strong solutions and strong Markov properties of the segment processes. Under additional assumption on non-degeneracy of the diffusion coefficient, exponential ergodicity for the segment process is derived by using asymptotic coupling method. In addition, the asymptotic log-Harnack inequality is established for the associated Markovian semigroup by using coupling and change of measures, which implies the asymptotically strong Feller property. Finally, an example is given to demonstrate these results.

Asymptotic Log-Harnack Inequality for Stochastic Fractional MHD Equations with Degenerate Noise

<p style='text-indent:20px;'>In this paper we consider the stochastic <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional (<inline-formula><tex-math id="M2">\begin{document}$ d = 2, 3 $\end{document}</tex-math></inline-formula>) magneto-hydrodynamic (MHD) equations with fractional dissipations <inline-formula><tex-math id="M3">\begin{document}$ ((-\Delta)^\alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^\beta) $\end{document}</tex-math></inline-formula> driven by degenerate noise. Here <inline-formula><tex-math id="M5">\begin{document}$ \alpha, \beta $\end{document}</tex-math></inline-formula> represent the parameters of the fractional dissipations corresponding to the velocity field and that to the magnetic field respectively. Using the asymptotic coupling method, the asymptotic log-Harnack inequality is established when <inline-formula><tex-math id="M6">\begin{document}$ \alpha\wedge\beta\geq\frac{1}{2}+\frac{d}{4} $\end{document}</tex-math></inline-formula>. As applications, we derive the gradient estimate, asymptotic irreducibility, asymptotic heat kernel estimate and ergodicity for the model.</p>

Additive regression splines with total variation and non negative garrote penalties

This study examines a penalized additive regression spline estimator with total variation and non negative garrote-type penalties. The proposed estimator is obtained based on a two-stage procedure. In the first stage, an initial estimator is obtained via total variation penalization. The total variation penalty enables data-adaptive knot selection and regularizes the overall smoothness of the estimator. The second stage imposes the non negative garrote penalty on the estimated functional components to attain variable selectivity. Regarding the theoretical aspect, a non asymptotic oracle inequality for the total variation penalized estimator is established under some regularity conditions. Based on the oracle inequality, we prove that the estimator attains the optimal rate of convergence up to a logarithmic factor, which in turn leads to the selection and estimation consistency of the second-stage garrote estimator. Numerical studies are presented to illustrate the usefulness of a combination of these two penalties. The results show that the proposed method outperforms existing methods.

On the unbalanced cut problem and the generalized Sherrington–Kirkpatrick model

We establish a strict asymptotic inequality between a class of graph partition problems on the sparse Erdős–Rényi and random regular graph ensembles with the same average degree. Along the way, we establish a variational representation for the ground state energy for generalized mixed p -spin glasses and derive strict comparison inequalities for such models as the alphabet changes.

Asymptotic Log-Harnack Inequality and Ergodicity for 3D Leray-α Model with Degenerate Type Noise

The asymptotic log-Harnack inequality is proved for Leray-α model with degenerate type noise using the asymptotic coupling method. In particular, we don’t impose any lower bound assumption for the viscosity constant. As applications, we also derive ergodicity and further asymptotic properties for stochastic 3D Leray-α model.

Asymptotic Lech's inequality

We explore the classical Lech's inequality relating the Hilbert–Samuel multiplicity and colength of an m-primary ideal in a Noetherian local ring (R,m). We prove optimal versions of Lech's inequality for sufficiently deep ideals in characteristic p>0, and we conjecture that they hold in all characteristics.Our main technical result shows that if (R,m) has characteristic p>0 and Rˆ is reduced, equidimensional, and has an isolated singularity, then for any sufficiently deep m-primary ideal I, the colength and Hilbert–Kunz multiplicity of I are sufficiently close to each other. More precisely, for all ε>0, there exists N≫0 such that for any I⊆R with ℓ(R/I)>N, we have (1−ε)ℓ(R/I)≤eHK(I)≤(1+ε)ℓ(R/I).

Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises

We study the asymptotic properties of the stochastic Cahn-Hilliard equation with the logarithmic free energy by establishing different dimension-free Harnack inequalities according to various kinds of noises. The main characteristics of this equation are the singularities of the logarithmic free energy at 1 and −1 and the conservation of the mass of the solution in its spatial variable. Both the space-time colored noise and the space-time white noise are considered. For the highly degenerate space-time colored noise, the asymptotic log-Harnack inequality is established under the so-called essentially elliptic conditions. And the Harnack inequality with power is established for non-degenerate space-time white noise.

Asymptotic log-Harnack inequality and applications for stochastic 2D hydrodynamical-type systems with degenerate noise

In this paper, an asymptotic log-Harnack inequality and some consequent properties are established via the asymptotic coupling method for a class of stochastic 2D hydrodynamical-type systems driven by degenerate noise. The main results are applicable to the stochastic 2D Navier–Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 2D Boussinesq equations, stochastic 2D magnetic Benard problem, stochastic 3D Leray- $$\alpha $$ model and also stochastic shell models of turbulence in the degenerate noise case.

Asymptotic log-Harnack inequality and applications for SPDE with degenerate multiplicative noise

The asymptotic log-Harnack inequality is established for SPDE with degenerate multiplicative noise by the coupling method. As applications, the gradient estimate, asymptotic strong Feller property, irreducibility, heat kernel estimate and ergodicity are derived.

Asymptotic lower bound of class numbers along a Galois representation

Let T be a free Zp-module of finite rank equipped with a continuous Zp-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a lower bound of the additive p-adic valuation of the class number of Kn, which is the Galois extension field of K fixed by the stabilizer of T/pnT. By applying this result, we prove an asymptotic inequality which describes an explicit lower bound of the class numbers along a tower K(A[p∞])/K for a given abelian variety A with certain conditions in terms of the Mordell–Weil group. We also prove another asymptotic inequality for the cases when A is a Hilbert–Blumenthal or CM abelian variety.

Ergodicity of 3D Leray-α model with fractional dissipation and degenerate stochastic forcing

By using the asymptotic coupling method, the asymptotic log-Harnack inequality is established for the transition semigroup associated to the 3D Leray-[Formula: see text] model with fractional dissipation driven by highly degenerate noise. As applications, we derive the asymptotic strong Feller property and ergodicity for the stochastic 3D Leray-[Formula: see text] model with fractional dissipation, which is the stochastic 3D Navier–Stokes equation regularized through a smoothing kernel of order [Formula: see text] in the nonlinear term and a [Formula: see text]-fractional Laplacian. The main results can be applied to the classical stochastic 3D Leray-[Formula: see text] model ([Formula: see text]), stochastic 3D hyperviscous Navier–Stokes equation ([Formula: see text]) and stochastic 3D critical Leray-[Formula: see text] model ([Formula: see text]).

Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory

The asymptotic log-Harnack inequality is established for several kinds of models on stochastic differential systems with infinite memory: non-degenerate SDEs, neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate, uniqueness of the invariant probability measure, asymptotic gradient estimate (hence, asymptotically strong Feller property), as well as asymptotic irreducibility.

Asymptotic Markov inequality on Jordan arcs

Markov's inequality for the derivative of algebraic polynomials is considered on -smooth Jordan arcs. The asymptotically best estimate is given for the th derivative for all . The best constant is related to the behaviour around the endpoints of the arc of the normal derivative of the Green's function of the complementary domain. The result is deduced from the asymptotically sharp Bernstein inequality for the th derivative at inner points of a Jordan arc, which is derived from a recent result of Kalmykov and Nagy on the Bernstein inequality on analytic arcs. In the course of the proof we shall also need to reduce the analyticity condition in this last result to -smoothness. Bibliography: 21 titles.

Асимптотическое неравенство Маркова на жордановых дугах

Изучаются неравенства Маркова для производной алгебраического многочлена на $C^2$-гладкой жордановой дуге. Получена асимптотически точная оценка для производной порядка $k$ при всех $k=1,2,…$ . Наилучшая константа в этой оценке связана с поведением в окрестности концов дуги нормальной производной функции Грина дополнения к дуге. Данный результат выводится из асимптотически точного неравенства Бернштейна для $k$-й производной во внутренних точках жордановой дуги. Последний результат в свою очередь получен с использованием недавнего результата С. И. Калмыкова и Б. Надя о неравенстве Бернштейна на аналитических дугах. В ходе доказательства будет ослаблено условие аналитичности дуг из результата Калмыкова и Надя до условия $C^2$-гладкости. Библиография: 21 название.

Cox process functional learning

This article addresses the problem of functional supervised classification of Cox process trajectories, whose random intensity is driven by some exogenous random covariable. The classification task is achieved through a regularized convex empirical risk minimization procedure, and a non asymptotic oracle inequality is derived. We show that the algorithm provides a Bayes-risk consistent classifier. Furthermore, it is proved that the classifier converges at a rate which adapts to the unknown regularity of the intensity process. Our results are obtained by taking advantage of martingale and stochastic calculus arguments, which are natural in this context and fully exploit the functional nature of the problem.

Moving fronts in integro-parabolic reaction-advection-diffusion equations

We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reactionadvection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method.

An asymptotic variance inequality for instrumental variable estimators signaling proportional bias increases

An asymptotic variance inequality for instrumental variable (IV) estimators is proposed, which suggests a critical variance that signals proportional increases in the bias of IV estimators through the augmentation of a set of instruments.