PARTIAL REGULARITY RESULT OF SUPERQUADRATIC ELLIPTIC SYSTEMS WITH DINI CONTINUOUS COEFFICIENTS

We consider the partial regularity for weak solutions to superquadratic elliptic systems with controllable growth condition, under the assumption of Dini continuous coefficients. The proof relies upon an iteration scheme of a decay estimate for a new type of excess functional. To establish the decay estimate, we use the technique of A-harmonic approximation and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, the proof yields directly the optimal H¨older exponent for the derivative of the weak solutions on the regular set.     

A REMARK ON THE REGULARITY OF VECTOR-VALUED MAPPINGS DEPENDING ON TWO VARIABLES WHICH MINIMIZE SPLITTING-TYPE VARIATIONAL INTEGRALS

We combine the maximum principle for vector-valued mappings established by D’Ottavio, Leonetti and Musciano [7] with regularity results from [5] and prove the H¨older continuity of the first derivatives for local minimizers u:Ω→ RN of splitting-type variational integrals provided Ω is a domain in R2.     

LEAST SQUARES ESTIMATOR FOR PATH-DEPENDENT MCKEAN-VLASOV SDES VIA DISCRETE-TIME OBSERVATIONS

In this article, we are interested in least squares estimator for a class of pathdependent McK ean-Vlasov stochastic differential equations(SDEs). More precisely, we investigate the consistency and asymptotic distribution of the least squares estimator for the unknown parameters involved by establishing an appropriate contrast function. Comparing to the existing results in the literature, the innovations of this article lie in three aspects:(i) We adopt a tamed Euler-Maruyama algorithm to establish the contrast function under the monotone condition, under which the Euler-Maruyama scheme no longer works;(ii) We take the advantage of linear interpolation with respect to the discrete-time observations to approximate the functional solution;(iii) Our model is more applicable and practice as we are dealing with SDEs with irregular coefficients(for example, H¨older continuous) and pathdistribution dependent.     

HOW BIG ARE THE LAG INCREMENTS OF A TWO PARAMETER WIENER PROCESS?~(**)

The author discusses how big the lag increments of a two-parameter Wiener processare and proves the same results as in[4].These results extende and improve the results ofthe increments of a two-parameter Wiener process in[1—3].     

Quasi Sure Large Deviation for Increments of Fractional Brownian Motion in H¨older Norm

In this paper, we first prove Schilder's theorem in H¨older norm(0 ≤α 1) with respect to Cr,p-capacity. Then, based on this result, we further prove a sharpening of large deviation principle for increments of fractional Brownian motion for Cr,p-capacity in the stronger topology.     

A Liminf Result on Two—parameter Gaussian Process

In this paper,a liminf behavior is studied of a two-parameter Gaussian process which is a gener-alization of a two-parameter Wiener process.The results improve on the liminfs in [7].     

Lavrentiev's regularization method for nonlinear ill-posed equations in Banach spaces

In this paper, we deal with nonlinear ill-posed problems involving m-accretive mappings in Banach spaces. We consider a derivative and inverse free method for the implementation of Lavrentiev regularization method. Using general H¨older type source condition we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock(2005) for choosing the regularization parameter.     

ON THE ZERO-ONE LAWS FOR THE ERGODIC QUASI-INVARIANT MEASURES

Many zero-one laws for the quasi-additive functional and quasi-additive operator are proved. The results may be especially applied to the abstract Wiener measure space. As an application, we give a generalization of a remarkable result of the σ-additivity of a Gaussian cylinder probability due to L. Gross.     

H ¨older Estimate of Harmonic Functions on a Class of p.c.f. Self-Similar Sets

In this paper we establish sharp H ¨older estimates of harmonic functions on a class of connected post critically finite (p.c.f.) self-similar sets, and show that functions in the domain of Laplacian ...     

ON MULTIPLE WIENER INTEGRAL EXPANSIONS FOR D'-VALUED NONLINEAR RANDOM FUNCTIONALS

In this paper, the multiple stochastic integral with respect to a Wiener D'-process is defined. And also it is shown that for a D'-valued nonlinear random functional there exists a sequence of multiple integral kernels such that the nonlinear functional can be expanded by series of multiple Wiener integrals of the integral kernels with respect to the Wiener D'-process.     

Gradient estimates for parabolic systems from composite material

In this paper, we derive W~(1,∞) and piecewise C~(1,α) estimates for solutions, and their t-derivatives, of divergence form parabolic systems with coefficients piecewise H¨older continuous in space variables x and smooth in t. This is an extension to parabolic systems of results of Li and Nirenberg [Comm Pure Appl Math, 2003, 56:892–925] on elliptic systems. These estimates depend on the shape and the size of the surfaces of discontinuity of the coefficients, but are independent of the distance between these surfaces.     

The Riemann-Hilbert problem for mixed complex equations of first order with degenerate rank 0

This article deals with the Riemann-Hilbert boundary value problem for quasilinear mixed(elliptichyperbolic) complex equations of first order with degenerate rank 0. Firstly, we give the representation theorem and prove the uniqueness of solutions for the boundary value problem. Afterwards, by using the method of successive iteration, the existence and estimates of solutions for the boundary value problem are verified. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. In this article, the proof of H¨older continuity of a singular double integer is very difficult and interesting as in this Section 4 below.     

Hausdorff measures of the image, graph and level set of bifractional Brownian motion

Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.     

Estimations of directional Hlder regularity by shearlets

This paper studies directional Hlder regularity of two-variable functions by their shearlet coefficients, where the shearlets are defined by Guo and Labate(2013). We provide necessary conditions for a function possessing some directional H¨older regularity and the corresponding sufficient conditions, motivated by the work of Sampo and Sumetkijakan(2009) and Lakhonchai et al.(2010).     

A GENERAL FORM OF THE INCREMENTS OF A TWO-PARAMETER WIENER PROCESS

In this paper,we consider a general form of the increments for a two-parameter Wiener process.Both the Csorgo-Revesz‘s increments and a class of the lag increments are the special cases of this general form of increments.Our results imply the theorem that have been given by Csorgo and Revesz(1978),and some of their conditions are removed.     

Markov process functionals in finance and insurance

The Markov property of Markov process functionals which are frequently used in economy, finance, engineering and statistic analysis is studied. The conditions to judge Markov property of some important Markov process functionals are presented, the following conclusions are obtained： the multidimensional process with independent increments is a multidimensional Markov process; the functional in the form of path integral of process with independent increments is a Markov process; the surplus process with the doubly stochastic Poisson process is a vector Markov process. The conditions for linear transformation of vector Markov process being still a Markov process are given.     

Ground State Solutions for Schr¨odinger-Poisson Systems

This paper deals with a class of Schr¨odinger-Poisson systems. Under some con-ditions, we prove that there exists a ground state solution of the system. The proof is based on the compactness lemma for the system. Our results here improve some existing results in the literature.     

Polar functions and intersections of the random string processes

Let {us(x) : s ≥ 0 , x ∈R} be a random string taking values in Rd . The main goal of this paper is to discuss the characteristics of the polar functions of {u s(x) : s ≥ 0 , x∈R} . The relationship between a class of continuous functions satisfying the H¨older condition and a class of polar-functions of {us (x) : s ≥ 0 , x ∈R} is presented. The Hausdorff and packing dimensions of the set that the string intersects a given non-polar continuous function are determined. The upper and lower bounds are obtained for the probability that the string intersects a given function in terms of respectively Hausdorff measure and capacity.     

STOCHASTIC ANALYSIS OF THE STOCHASTIC FUNCTIONAL ON THE BASIC SPACE

The concept of tho basic space is introduced in this paper, which contains tho white noise space and tho abstract Wiener space. Many results of stochastic analysis of the stochastic functional on white noiso space ([4]) and abstract Wiener space ([12]) are generalisod into this space, for example, Fourier transformation, Fourier-Wiener transformation and Malliavin's calculus, etc.     

Ground State Solutions for Schrdinger-Poisson Systems

This paper deals with a class of Schr¨odinger-Poisson systems. Under some conditions, we prove that there exists a ground state solution of the system. The proof is based on the compactness lemma for the system. Our results here improve some existing results in the literature.