Covering numbers of different points in Dvoretzky covering

Consider the Dvoretzky random covering on the circle T with a decreasing length sequence {?n}_n?1 such that . We study, for a given β?0, the set Fβ of points which are asymptotically covered by a number βLn of the first n randomly placed intervals where . Three typical situations arise, delimited by two “phase transitions”, according to is zero, positive-finite or infinite, where . More precisely, if ?n tends to zero rapidly enough so that then, with probability one, dimHFβ=1 for all β?0; if ?n is moderate so that then, with probability one, we have for and Fβ=∅ for where and is the interval consisting of β's such that ; eventually, if ?n is so slow that then, with probability one, F₁=T. This solves a problem raised by L. Carleson in a rather satisfactory fashion.Analogous results are obtained for the Poisson covering of the line, which is studied as a tool.     

Quasi sure analysis of local times of anticipating smooth semimartingales

Let be the anticipating smooth semimartingale and be its generalized local time. In this paper, we give some estimates about the quasi sure property of Xt and its quadratic variation process t〈X〉. We also study the fractional smoothness of and prove that the quadratic variation process of can be constructed as the quasi sure limit of the form , where is a sequence of subdivisions of [a,b], , i=0,1,…,n².     

Spatial asymptotic behavior of homeomorphic global flows for non-Lipschitz SDEs

Let x→?s,t(x) be a -valued stochastic homeomorphic flow produced by non-Lipschitz stochastic differential equation , where W=(W¹,W²,…) is an infinite sequence of independent standard Brownian motions. We first give some estimates of modulus of continuity of {?s,t(⋅)}, then prove that the flow ?s,t(x), when x nears infinity, grows slower than for some constant c>0 and integrable random variable Z via lemma of Garsia-Rodemich-Rumsey Lemma (abbreviated as GRR Lemma) improved by Arnold and Imkeller [L. Arnold, P. Imkeller, Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory, Stochastic Process. Appl. 62 (1996) 19-54] and moment estimates for one- and two-point motions.     

On the topological Helly theorem

The main result of this paper is the following theorem, related to the missing link in the proof of the topological version of the classical result of Helly: Let be any family of simply connected compact subsets of R² such that for every i,j∈{0,1,2} the intersections Xi∩Xj are path connected and is nonempty. Then for every two points in the intersection there exists a cell-like compactum connecting these two points, in particular the intersection is a connected set.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

On a type of stochastic differential equations driven by countably many Brownian motions

Consider the one-dimensional SDE , where Wⁱ is an infinite sequence of independent standard Brownian motions, i=1,2,3,…. We prove two theorems on the existence and uniqueness of solutions with non-Lipschitz coefficients, and give a non-contact property and a strong comparison theorem for solutions.     

Classification of quasifinite modules over the Lie algebras of Weyl type

For a nondegenerate additive subgroup Γ of the n-dimensional vector space over an algebraically closed field of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type spanned by all differential operators uD₁^m₁?D_n^m_n for (the group algebra), and m₁,…,m_n?0, where D₁,…,D_n are degree operators. In this paper, it is proved that an irreducible quasifinite -module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly bounded -modules is completely given. It is also proved that an irreducible quasifinite -module is a module of the intermediate series and a complete classification of quasifinite -modules is also given, if Γ is not isomorphic to .     

Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters H=(H₁,H₂), and (²[0,1],B(²[0,1]),μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in ²[0,1], and four types of stochastic surface integrals: , i=1,2, , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H₁,H₂∈(1/4,1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.     

On the Clark Ocone formula for the abstract Wiener space

Fix an abstract Wiener space where is a separable Hilbert space densely embedded into a Banach space . A pathwise construction of the Itô integral as a continuous square integrable martingale is given, where the integrands are -valued processes and the integrator is a -valued Brownian motion. We use this approach to the vector integral to prove that each Malliavin differentiable functional ? defined on the space of continuous -valued functions on [0,1], endowed with the Wiener measure, can be decomposed into the sum of the expected value of ? and the Itô integral of the conditional expectation of the Malliavin derivative of ? with respect to the Brownian filtration. The Malliavin derivative of ? is an -valued stochastic process. In a second application, it is shown that the iterated Itô integral, defined as a process on , is a continuous square integrable martingale.     

Polynomial identities and noncommutative versal torsors

To any cleft Hopf Galois object, i.e., any algebra obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” and . The algebra is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, is a cleft H-Galois extension of a “big” commutative algebra . Any “form” of can be obtained from by a specialization of and vice versa. If the algebra is simple, then is an Azumaya algebra with center . The algebra is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of are satisfied. We construct an embedding of into ; this embedding maps the center of into when the algebra is simple. In this case, under an additional assumption, , thus turning into a central localization of . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.     

The symmetry of the CS condition on one-sided ideals in a prime ring

Let R be a prime ring and e∈R be an idempotent. We show that eR_R is nonsingular, CS and if and only if is nonsingular, CS and .     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.