PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE S_Q(T)(1≤Q≤∞)

Probabilistic linear （N, δ）-widths and p-average linear N-widths of Sobolev space W2^r（T）, equipped with a Gaussian probability measure #, are studied in the metric of Sq （T） （1 ≤ Q ≤∞）, and determined the asymptotic equalities：λN,δ（W2^r（T）,μ,Sq（T））={（N^-1）^r＋p/2-1/q√1＋1/N·ln1/δ, 1≤q≤2, （N^-1）^r＋p/2-1/q（1＋N^-1/q√ln1/δ）,2〈q〈∞, （N^-1）^r＋p/2√lnN/δ, q=∞,and λN^（a）（W2^r（T）,μ,Sq（T））p={（N^-1）^r＋p/2-1/q, 1≤q〈∞, （N^-1）^r＋p/2-1/q√lnN, q=∞,where 0 〈 p 〈 ∞, δ∈ （0, 1/2], ρ 〉 1, and Sq（T） is a subspace of L1（T）, in which the Fourier series is absolutely convergent in lq sense.

Optimal Recovery of Functions on the Sphere on a Sobolev Spaces with a Gaussian Measure in the Average Case Setting

In this paper, we study optimal recovery （reconstruction） of functions on the sphere in the average case setting. We obtain the asymptotic orders of average sampling numbers of a Sobolev space on the sphere with a Gaussian measure in the Lq （S^d-1） metric for 1 ≤ q ≤ ∞, and show that some worst-case asymptotically optimal algorithms are also asymptotically optimal in the average case setting in the Lq （S^d-1） metric for 1 ≤ q ≤ ∞.

A Note on the Difference of Gaussian Measure of Two Balls in Hilbert Spaces

Let H be a separable Hilbert space, μ be a symmetric Gaussian measure on H. Applying the meathod of sum of independent random variables. A finer estimate of the difference of Gaussian measure of two balls is obtained.

Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

In this paper,the authors establish the two-weight boundedness of the local fractional maximal operators and local fractional integrals on Gaussian measure spaces associated with the local weights.More precisely,the authors first obtain the two-weight weak-type estimate for the locala fractional maximal operators of orderαfrom L^(p)(v)to L^(q,∞)(u)with 1≤p≤q<∞under a condition of(u,v)∈∪b>a A_(p,q,a)^(b') ,and then obtain the two-weight weak-type estimate for the local fractional integrals.In addition,the authors obtain the two-weight strong-type boundedness of the local fractional maximal operators under a condition of(u,v)∈M_(p,q,a)^(6a+9√da^2) and the two-weight strong-type boundedness of the local fractional integrals.These estimates are established by the radialization method and dyadic approach.

Stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise

The current paper is devoted to the study of the stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise. First, the dynamics of stochastic FitzHugh-Nagumo systems are studied. Then, the existence and uniqueness of their invariant measures, which mix exponentially are proved. Finally, the asymptotic behaviors of invariant measures when size of noise gets to zero are investigated.

On Existence of the Even L_(p)Gaussian Minkowski Problem for p>n

This paper concerns the even L_(p)Gaussian Minkowski problem in n-dimensional Euclidean space R^(n).The existence of the solution to the even L_(p)Guassian Minkowski problem for p>n is obtained.

NORM CALCULATIONS OF A CLASS OF INTEGRAL OPERATORS

In this paper we calculate the norm of a special class of integral operators acting on LP （C^n, dvs）, where dvs is the Gaussian measure on C^n.

GAUSSIAN OPERATORS AND SKOROHOD INTEGRALS OF RANDOM FIELDS(Ⅱ)

The present note is a continuation of [1]. We will give a method for calculating the Skorohod integrals and the concept of stochastic derivatives for random fields. Based on them, some kind of generalization of It formula is presented here. The readers are referred to [1] for necessary definitions and notations.

Continuity of the Solution to the L_(p) Minkowski Problem in Gaussian Probability Space

In this paper,it is proved that the weak convergence of the L_(p) Gaussian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p≥1.Moreover,continuity of the solution to the L_(p) Gaussian Minkowski problem with respect to p is obtained.

On a Right Inverse of a Polynomial of the Laplace in the Weighted Hilbert Space L^(2)(R^(n),e^(−|x|^(2)))

Let P(∆)be a polynomial of the Laplace operator∆=n∑j=1∂^(2)∂x^(2)_(j) on R^(n).We prove the existence of a bounded right inverse of the differential operator P(∆)in the weighted Hilbert space with the Gaussian measure,i.e.,L^(2)(R^(n),e^(−|x|^(2))).

THE PERTURBATION THEORY FOR THE SUMMABILITY OF SELFADJOINT OPERATORS

Let(E,H,μ)be an abstract Wiener spacein the sense of L.Gross.It is proved that if u is a measurable map from E to H such that u∈W 2.1(H,μ)and there exists a constantα,0<α<1,such that either∑n‖D nu(w)‖2 Hα2 a.s.or‖u(w+h)-u(w)‖Hα‖h‖H a.s.for every h∈H and E exp108(1-α)2∑‖D n u‖H)<∞,then the measureμT-1 is equivalent toμ,where T(w)=w+u(w)for w∈E.And the explicit expression of the Radon-Nikodym derivative(cf.Theorem 2.1)is given.