Gaussian estimates for elliptic operators with unbounded drift

We consider a strictly elliptic operator

     

Weak Type Inequality for a Family of Singular Integral Operators Related with the Gaussian Measure

In this paper we study a family of singular integral operators that generalizes the higher order Gaussian Riesz Transforms and find the right weight w to make them continuous from L ¹(wdγ) into L ^1, ∞(dγ), being Some boundedness properties of these operators had already been derived by Urbina (Ann Scuola Norm Sup Pisa Cl Sci 17(4):531–567, 1990) and Pérez (J Geom Anal 11(3):491–507, 2001).      

Strong mixing coefficients for non-commutative Gaussian processes

Bounds for non-commutative versions of two classical strong mixing coefficients for -Gaussian processes are found in terms of the angle between the underlying Hilbert spaces. As a consequence, we construct a -mixing -Gaussian stationary sequence with growth conditions on variances of partial sums. If classical processes with analogous properties were to exist, they would provide a counter-example to the Ibragimov conjecture.

     

A statistically important Gaussian Process

It is well known, that under the condition LAN and some more regularity conditions, the process of log-likelihood functions converges weakly to a degenerate Gaussian process (the trajectories of which are straight lines). In the non-regular case considered by several authors [1, 9] the limiting process is non-degenerate and characterized by the covariance function $\text{R(t,s) =}\text{1}\text{2}\text{(|t|}{}^{\mathrm{p}}\text{+ |s|}{}^{\mathrm{p}}\text{? |t ? s|}{}^{\mathrm{p}}\text{)}$. In the present paper, we derive sever properties of this process with relevance to applications in statistics.In particular, a bound for the risk of equivariant estimates is given and the maximum likelihood estimate (MLE) is shown to be well defined. The calculation of the exact distribution of the MLE is left as an open problem.     

Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients

Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on . In particular, in the case when they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations.

     

A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations

We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations.     

Inequalities for the Gaussian hypergeometric function

we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 c 5d/6 ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.     

Comparison results for the lower tail of Gaussian seminorms

Let =( _n ) be i.i.d.N(0, 1) random variables andq(x), q(x):R [0, ) be seminorms. We investigate necessary and sufficient conditions that the ratio ofP(q()<) andP(q()<) goes to a positive constant as 0⁺. We give satisfactory answers forl ₂-norms and also some results for sup-norms andl _p-norms. Some applications are given to the rate of escape of infinite dimensional Brownian motion, and we give the lower tail of the Ornstein-Uhlenbeck process and a weighted Brownian bridge under theL ₂-norms.     

关于对称拟凸多项式的不相连定理

假定μn为Rn上的标准高斯测度,X为Rn上的随机向量,分布为μn.不相连猜测说的是:如果f与g为Rn上的两个多项式,而且f(X)与g(X)相互独立,则存在Rn上的正交变换Y = LX及整数k使得f o L-1为(y1,y2,…,yk)的函数,goL-1为(yk+1,yk+2,…,yn)的函数.此时,称f与g不相连.在这...     

Error estimates for approximate approximations with Gaussian kernels on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact intervals, only. This method, which is based on an approximate partition of unity, was introduced by Maz’ya in 1991 and has mainly been used for functions defined on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed.In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L₁-estimates are given, where all the constants are determined explicitly.     

局部广义高斯序列的若干新结果

给出了局部广义高斯序列部分和的一类绝对矩不等式及大偏差,并利用这些不等式证明了局部广义高斯序列的大数定律,同时建立了局部广义高斯序列的重对数收敛速度.     

Multilevel Minimal Residual Methods for Nonsymmetric Elliptic Problems

The subject of this paper is to study the performance of multilevel preconditioning for nonsymmetric elliptic boundary value problems. In particular, a minimal residual method with respect to an appropriately scaled norm, measuring the size of the residual projections on all levels, is studied. This norm, induced by the multilevel splitting, is also the basis for a proper stopping criterion. Our analysis shows that the convergence rate of this minimal residual method using the multilevel preconditioner by Bramble, pasciak and Xu is bounded independently of the mesh-size. However, the convergence rate deteriorates with increasing size of the skew-symmetric part. Our numerical results show that by incorporating this into a multilevel cycle starting on the coarsest level, one can save fine-level-iterations and, therefore, computational work.     

The Gaussian Isoperimetric Inequality and Transportation

Any probability measure on ^d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that t . This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let (dx) = e ^–(x) dx be a probability measure on ^d, where is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities.     

On Reverses of the Blaschke-Santalo Inequality for Convex Bodies

Reisner proved a reverse of the Blaschke-Santal5 inequality for zonoid bodies, Bourgain and Milman showed another reverse of the Blaschke-Santal5 inequality for centered convex bodies. In this paper, two reverses of the Blaschke-Santal5 inequality for convex bodies are given by the Petty projection inequality and above two reverses. Further, using above methods, we also obtain two analogues of the Petty＇s conjecture for projection bodies, respectively.     

On Restarting the Arnoldi Method for Large Nonsymmetric Eigenvalue Problems

The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues are desired. We analyze several approaches to restarting and show why Sorensen's implicit QR approach is generally far superior to the others. Ritz vectors are combined in precisely the right way for an effective new starting vector. Also, a new method for restarting Arnoldi is presented. It is mathematically equivalent to the Sorensen approach but has additional uses.

     

The ABC Conjecture Implies Vojta's Height Inequality for Curves

Following Elkies (Internat. Math. Res. Notices7 (1991) 99-109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyǐ function. We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Roth's theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two.     

Carleman不等式的一种推广

本文对 Ｃａｒｌｅｍ ａｎ 不等式作出一种推广．     

Tail dependence of the Gaussian copula revisited

Tail dependence refers to clustering of extreme events. In the context of financial risk management, the clustering of high-severity risks has a devastating effect on the well-being of firms and is thus of pivotal importance in risk analysis.When it comes to quantifying the extent of tail dependence, it is generally agreed that measures of tail dependence must be independent of the marginal distributions of the risks but rather solely copula-dependent. Indeed, all classical measures of tail dependence are such, but they investigate the amount of tail dependence along the main diagonal of copulas, which has often little in common with the concentration of extremes in the copulas’ domain of definition.In this paper we urge that the classical measures of tail dependence may underestimate the level of tail dependence in copulas. For the Gaussian copula, however, we prove that the classical measures are maximal. The implication of the result is two-fold: On the one hand, it means that in the Gaussian case, the (weak) measures of tail dependence that have been reported and used are of utmost prudence, which must be a reassuring news for practitioners. On the other hand, it further encourages substitution of the Gaussian copula with other copulas that are more tail dependent.     

A bilinear approach to the restriction and Kakeya conjectures

Bilinear restriction estimates have appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the spherical restriction theorem of Wolff from to , and also obtain a sharp spherical restriction theorem for .