Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay

Potential Analysis - In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with...     

The Asymptotic Behavior of the Stochastic Nonlinear Schrdinger Equation With Multiplicative Noise

The nonlinear Schrdinger equation is one of the basic models for nonlinear waves.In some circumstances,randomness has to be taken into account and it often occurs through a random potential.Here,we consider the following equation     

Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach

We prove well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise, including the nonconservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower-order terms and treat the resulting equation by a fixed point argument based on generalizations of Strichartz estimates proved by Marzuola et al. (J Funct Anal 255(6):1479–1553, 2008). This approach makes it possible to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrödinger equation. In contrast to the latter, we obtain well-posedness in the full range \([1, 1 + 4/d)\) of admissible exponents in the nonlinear part (where \(d\) is the dimension of the underlying Euclidean space), i.e., in exactly the same range as in the deterministic case.     

Hölder Continuous Solutions to Complex Hessian Equations

We prove the Hölder continuity of the solution to complex Hessian equation with the right hand side in L ^p , \(p>\frac {n}{m}\) , 1 < m < n, in a m-strongly pseudoconvex domain in ? ⁿ under some additional conditions on the density near the boundary and on the boundary data.     

The Asymptotic Behavior of the Stochastic Nonlinear Schr(o)dinger Equation With Multiplicative Noise

The nonlinear Schr(o)dinger equation is one of the basic models for nonlinear waves. In some circumstances, randomness has to be taken into account and it often occurs through a random potential. Here, we consider the following equation     

Estimates for Correlation in Dynamical Systems: From Hölder Continuous Functions to General Observables

For many dynamical systems that are popular in applications, estimates are known for the decay of correlation in the case of Hölder continuous functions. In the present article, we suggest an approach that allows us to obtain estimates for correlation in dynamical systems in the case of arbitrary functions. This approach is based on approximation and estimates are obtained with the use of known estimates for Hölder continuous functions. We apply our approach to transitive Anosov diffeomorphisms and derive the central limit theorem for the characteristic functions of certain sets with boundary of zero measure.     

Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity

Let {X(t)} _t∈? be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability 1, the trajectories of X have the same global Hölder regularity over any compact interval, i.e. the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon occurs with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: Does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent? In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.     

Hölder categories

Hölder categories are invented to provide an axiomatic foundation for the study of categories of archimedean lattice-ordered algebraic structures. The basis of such a study is Hölder’s Theorem (1908), stating that the archimedean totally ordered groups are precisely the subgroups of the additive real numbers ? with the usual addition and ordering, which remains the single most consequential result in the studies of lattice-ordered algebraic systems since Birkhoff and Fuchs to the present. This study originated with interest in W*, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the ?-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on W*. In the course of this study, certain abstract notions jumped to the forefront. Two of these, in particular, seem to have been mostly overlooked; some notion of simplicity appears to be essential to any kind of categorical study of W*, as are the quasi-initial objects in a category. Once these two notions have been brought into the conversation, a Hölder category may then be defined as one which is complete, well powered, and in which (a) the initial object I is simple, and (b) there is a simple quasi-initial coseparator R. In this framework it is shown that the epireflective hull of R is the least monoreflective class. And, when I = R — that is, the initial element is simple and a coseparator — a theorem of Bezhanishvili, Morandi, and Olberding, for bounded archimedean f-algebras with identity, can be be generalized, as follows: for any Hölder category subject to the stipulation that the initial object is a simple coseparator, every uniformly nontrivial reflection — meaning that the reflection of each non-terminal object is non-terminal — is a monoreflection. Also shown here is the fact that the atoms in the class of epireflective classes are the epireflective hulls of the simple quasi-initial objects. From this observation one easily deduces a converse to the result of Bezhanishvili, Morandi, and Olberding: if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.     

The Newton Method for Operators with Hölder Continuous First Derivative

We analyze the convergence of the Newton method when the first Fréchet derivative of the operator involved is Hölder continuous. We calculate also the R-order of convergence and provide some a priori error bounds. Based on this study, we give some results on the existence and uniqueness of the solution for a nonlinear Hammerstein integral equation of the second kind.     

Hölder type quasicontinuity

It is proved that a functionuL ^m,p (R ⁿ ) (which coincides with the Sobolev spaceW ^1,p (R ⁿ ) ifm=1) coincides with a Hölder continuous functionw outside a set of smallm,q-capacity, whereq<p. Moreover, ifm=1, then the functionw can be chosen to be close tou in theW ^1,p -norm.     

On the Compactness of the Hypercomplex Commutator in Hölder Continuous Functions Spaces

This paper is devoted to the compactness of the hypercomplex commutator S _γ M _a ? M _a S _γ, where S _γ is the Cauchy singular integral operator (in the Douglis sense), a is a Hölder continuous hypercomplex function and M _a is the multiplication operator given by M _a f = a f. We extend a known compactness sufficient condition for the commutator of the Cauchy singular integral operator to the frame of the hypercomplex analysis, where γ is merely required to be an arbitrary regular closed Jordan curve.     

Dimension (In)equalities and Hölder Continuous Curves in Fractal Percolation

We relate various concepts of fractal dimension of the limiting set $\mathcal{C}$ in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in  $\mathcal{C}$ (the “dust”). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of non-trivial Hölder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.     

Hölder rigidity for matrices

It is proved that a (C ₁, C ₂)-Hölder valuation is (2, α)-equivalent to classical valuation on the set of matrices over a skew field and on the set of cubic matrices over a field. These results provide an extension of the Garcia theorem.     

Nachruf auf Otto Hölder

Ohne Zusammenfassung     

Directional Hölder Metric Regularity

This paper studies the first-order behavior of the value function of a parametric optimal control problem with nonconvex cost functions and control constraints. By establishing an abstract result on the Fréchet subdifferential of the value function of a parametric minimization problem, we derive a formula for computing the Fréchet subdifferential of the value function to a parametric optimal control problem. The obtained results improve and extend some previous results.     

Attractors for a Damped Stochastic Wave Equation of Sine–Gordon Type with Sublinear Multiplicative Noise

Abstract

The existence of compact random attractors is proved for a damped stochastic wave equation of Sine–Gordon type with sublinear multiplicative noise under homogeneous Dirichlet boundary condition. To be important, in this note a precise estimate of upper bound of Hausdorff dimension of the random attractors is obtained in lower dimension.     

Estimate of the Hölder Exponent Based on the $$epsilon$$ -Complexity of Continuous Functions

Mathematical Notes -     

Hölder Error Bounds and Hölder Calmness with Applications to Convex Semi-infinite Optimization

Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Hölder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Hölder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Hölder calmness modulus of the argmin mapping in the framework of linear programming.