The Radon-Nikodym theorem for von neumann algebras

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms. Partially supported by NSF Grant # 28976 X. Partially supported by NSF Grant # GP-28737 This revised version was published online in November 2006 with corrections to the Cover Date.     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

PRV property and the <Emphasis Type="Italic">ϕ</Emphasis>-asymptotic behavior of solutions of stochastic differential equations

In this paper, we investigate the a.s. asymptotic behavior of the solution of the stochastic differential equation dX(t) = g(X(t)) dt + σ(X(t))dW(t), X(0) ≢ 1, where g(·) and σ(·) are positive continuous functions, and W(·) is a standard Wiener process. By means of the theory of PRV functions we find conditions on g(·), σ(·), and ϕ(·) under which ϕ(X(·)) may be approximated a.s. by ϕ(μ(·)) on {X(t) → ∞}, where μ(·) is the solution of the ordinary differential equation dμ(t) = g(μ(t)) dt with μ(0) = 1. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 445–465, October–December, 2007.     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

On one-dimensional stochastic differential equations driven by stable processes

We consider the one-dimensional stochastic differential equation dX _t=b(t, X_t−) dZ _t, whereZ is a symmetric α-stable Lévy process with α ε (1, 2] andb is a Borel function. We give sufficient conditions under which the equation has a weak nonexploding solution. Partially supported by Programma Professori Visitatori of G. N. A. F. A. (Italy). Partially supported by MURST (Italy). The present research was completed while the second author was visiting the Institute of Mathematics and Informatics (Vilnius, Lithuania) in spring of 1999. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 361–385, July–September, 2000. Translated by H. Pragarauskas     

Fractional Smoothness of Some Stochastic Integrals

We study the fractional smoothness in the sense of Malliavin calculus of stochastic integrals of the form ∫0^1Ф（Xs）dXs, where Xs is a semimartingale and Ф belongs to some fractional Sobolev space over R.     

Borel reducibility as an additive property of domains

We prove that under certain requirements, if E and F are Borel equivalence relations, , is a countable union of Borel sets, and E ↾ X _n , is Borel reducible to F for all n, then E ↾ X is Borel reducible to F. Thus the property of Borel reducibility to F is countably additive as a property of domains. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 189–198.     

Almost continuous orbit equivalence for non-singular homeomorphisms

Let X and Y be Polish spaces with non-atomic Borel measures μ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, μ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III ₁ or that they are both of type III _λ, 0 < λ < 1 and, in the III _λ case, suppose in addition that both ‘topological asymptotic ranges’ (defined in the article) are log λ · ℤ. Then there exist invariant dense G _δ-subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a non-singular homeomorphism ϕ: X′ → Y′ which is an orbit equivalence between T| _X′ and S| _Y′, that is ϕ{T ⁱ x} = {S ⁱ ϕx} for all x ∈ X′. Moreover, the Radon-Nikodym derivative dν ∘ ϕ/dμ is continuous on X′ and, letting S′ = ϕ ⁻¹ Sϕ, we have T _x = S′ ^n(x) x and S′x = T ^m(x) x where n and m are continuous on X′.     

On a Conformal Gap and Finiteness Theorem for a Class of Four-Manifolds

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and make a comparison of the solutions of the σ_k equations on a degenerating family of Bach-flat metrics. The first author is supported by NSF Grant DMS–0245266. The second author is supported in part by NSF Grant DMS–0402294. The third author is supported by NSF Grant DMS–0245266. Received: August 2005 Accepted: June 2006     

Means on equivalence relations

Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth. Research partially supported by NSF Grants DMS-9987437 and DMS-0455285. Research partially supported by NSF VIGRE Grant DMS-0502315.     

Nonuniqueness ing-functions

We give an example of two distinct stationary processes {X _n} and {X′ _n} on {0, 1} for whichP[X₀=1|X₋₁=a₋₁,X₋₂=a₋₂, …]=P[X′₀=1|X′₋₁=a₋₁,X′₋₂=a₋₂, …] for all {a _i},i=−1, −2, …, even though these probabilities are bounded away from 0 and 1, and are continuous in {a _i}. Supported in part by NSF Grant DMS 89-01545. Supported in part by the US Army Research Office.     

Invariant Measure for the Markov Process Corresponding to a PDE System

In this paper, we consider the Markov process （X^∈（t）, Z^∈（t）） corresponding to a weakly coupled elliptic PDE system with a small parameter ∈ 〉 0. We first prove that （X^∈（t）, Z^∈（t）） has the Feller continuity by the coupling method, and then prove that （X^∈（t）, Z^∈（t）） has an invariant measure μ^∈（·） by the Foster-Lyapunov inequality. Finally, we establish a large deviations principle for μ^∈（·） as the small parameter e tends to zero.     

Precise Asymptotics for Lévy Processes

Let {X(t), t ≥ 0} be a Lévy process with EX(1) = 0 and EX ²(1) < ∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t ≥ 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables. This work is supported by the National Natural Science Foundation (Grant No. 10671188) and Special Foundation of USTC     

Polynomial Szemerédi theorems for countable modules over integral domains and finite fields

Given a pair of vector spacesV andW over a countable fieldF and a probability spaceX, one defines apolynomial measure preserving action ofV onX to be a compositionT o ϕ, where ϕ:V→W is a polynomial mapping andT is a measure preserving action ofW onX. We show that the known structure theory of measure preserving group actions extends to polynomial actions and establish a Furstenberg-style multiple recurrence theorem for such actions. Among the combinatorial corollaries of this result are a polynomial Szemerédi theorem for sets of positive density in finite rank modules over integral domains, as well as the following fact:Let be a finite family of polynomials with integer coefficients and zero constant term. For any α>0, there exists N ∈ ℕ such that whenever F is a field with |F|≥N and E ⊆F with |E|/|F|≥α, there exist u∈F, u≠0, and w∈E such that w+ϕ(u)∈E for all ϕ∈ . The first two authros are supported by NSF, grant DMS-0070566 and DMS-0245350. The second author was supported by the A. Sloan Foundation. The third author is supported by NSF, grant DMS-0200700.     

Homogenous projective factors for actions of semi-simple Lie groups

We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ _K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q. Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999     

Biorthogonal multiple wavelets generated by vector refinement equation

Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis.In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form (?)(x)=(?)a(α)(?)(Mx-α),x∈R~s, where the vector of functions(?)=((?)_1,...,(?)_r)~T is in(L_2(R~s))~r,a=:(a(α))_(α∈Z~s)is a finitely supported sequence of r×r matrices called the refinement mask,and M is an s×s integer matrix such that lim_(n→∞)M~(-n)=0.Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.     

Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires

Résumé Soit X un processus gaussien stationnaire non dérivable. Nous étudions le nombre de passages en zéro du processus régularisé par convolution. Sous des hypothèses peu restrictives sur X, cette variable convenablement normalisée, converge au sens de L ² quand la taille du filtre tend vers zéro. Lorsque X admet un temps local continu, la limite obtenue est le temps local.

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Summary Let {X(t)} be a stationary non differentiable Gaussian process and let ϕ_ɛ(u=ɛ⁻¹ ϕ(u/ɛ) be an approximate identity. Setting X _ɛ(t)=X*ϕ_ɛ(t) and letting N _ɛ(T) be the number of zeros of X _ɛ in the interval [0, T] it is shown that under weak technical conditions there are constants C(ɛ) so that C(ɛ) N _ɛ(T) converges in L ² as ɛ→0. When X admits a continuous local time, the limit is the local time L(0, T) at zero of X(t).

     

Linear functional differential equations of neutral type asymptotically autonomous

Summary It is studied the relationship between the solutions of the linear functional differential equations(1) (d/dx) D(x_t)=L(x_t) and its perturbed equation(2) [(d/dx) D(x_t)−G(t, x_t)]= =L(x_t)+F(t, x_t) and is proved, under certain hypotheses which will be precised bellow that, if μ is a simple characteristic root of(1), then there exist a σ > 0 and a non zero vector a such that system(2) has a solution satisfying where δ(t)=αd{F(t, ϕ_μ)+μG(t, ϕ_μ)+F(t, X₀G(t, ϕ_μ))}, ϕ_μ(θ)=c·exp (μθ), −r⩾θ⩾0 and α, d, X₀ are given constants. Entrata in Redazione il 5 gennaio 1972.     

Density estimation for linear processes

Summary LetX _t , ...,X _n be random variables forming a realization from a linear process where {Z _t } is a sequence of independent and identically distributed random variables with E|Z _t |<∞ for some ε>0, andg _r →0 asr→∞ at some specified rate. LetX ₁ have a probability density functionf. It is then established that for every realx, the standard kernel type estimator based onX _t (1≦t≦n) is, under some general regularity conditions, asymptotically normal and converges a.s. tof(x) asn→∞. Research was supported in part by the Air Force Office of Scientific Research Grant No. AFOSR-81-0058.     

Transfer of absolute continuity by a flow generated by a stochastic equation with reflection

Let ϕ_t(x), x ∈ ℝ₊ be a value taken at time t ≥ 0 by a solution of a stochastic equation with normal reflection from a hyperplane starting at initial time from x. We characterize the absolutely continuous (with respect to Lebesgue measure) component and the singular component of a stochastic measure-valued process μ_t = μ ○ ϕ _t ⁻¹ that is the image of a certain absolutely continuous measure μ under random mapping ϕ_t(·). We prove that the restriction of the Hausdorff measure H ^d−1 to the support of the singular component is σ-finite and give sufficient conditions guaranteeing that the singular component is absolutely continuous with respect to H ^d−1. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1663–1673, December, 2006.