Malliavin calculus in abstract Wiener space using infinitesimals

Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space by replacing with any abstract Wiener space .We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space is the standard part of a ∗finite-dimensional Brownian motion B. Then we define iterated Itô integrals as standard parts of internal iterated Itô integrals. The integrator of the internal integrals is B and the values of the integrands are multilinear forms on , where is a ∗finite-dimensional linear space over between the Hilbert space and its ∗-extension .In the first part we prove a chaos decomposition theorem for L²-functionals on Ω that are measurable with respect to the σ-algebra generated by b. This result yields a chaos decomposition of L²-functionals with respect to the Wiener measure on the standard space of -valued continuous functions on [0,1]. In the second part we define the Malliavin derivative and the Skorohod integral as standard parts of internal operators defined on ∗finite-dimensional spaces. In an application we use the transformation rule for finite-dimensional Euclidean spaces to study time anticipating and non-anticipating shifts of Brownian motion by Bochner integrals (Girsanov transformations).     

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.     

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.     

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.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

Polynomials of Meixner's type in infinite dimensions—Jacobi fields and orthogonality measures

The classical polynomials of Meixner's type—Hermite, Charlier, Laguerre, Meixner, and Meixner-Pollaczek polynomials—are distinguished through a special form of their generating function, which involves the Laplace transform of their orthogonality measure. In this paper, we study analogs of the latter three classes of polynomials in infinite dimensions. We fix as an underlying space a (non-compact) Riemannian manifold X and an intensity measure σ on it. We consider a Jacobi field in the extended Fock space over L²(X;σ), whose field operator at a point x∈X is of the form , where λ is a real parameter. Here, ∂_x and are, respectively, the annihilation and creation operators at the point x. We then realize the field operators as multiplication operators in , where is the dual of , and μ_λ is the spectral measure of the Jacobi field. We show that μ_λ is a gamma measure for |λ|=2, a Pascal measure for |λ|>2, and a Meixner measure for |λ|<2. In all the cases, μ_λ is a Lévy noise measure. The isomorphism between the extended Fock space and is carried out by infinite-dimensional polynomials of Meixner's type. We find the generating function of these polynomials and using it, we study the action of the operators ∂_x and in the functional realization.     

On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define

     

Homeomorphic property of solutions of SDE driven by countably many Brownian motions with non-Lipschitzian coefficients

We study m-dimensional SDE , where {Wi}_i?1 is an infinite sequence of independent standard d-dimensional Brownian motions. The existence and pathwise uniqueness of strong solutions to the SDE was established recently in [Z. Liang, Stochastic differential equations driven by countably many Brownian motions with non-Lipschitzian coefficients, Preprint, 2004]. We will show that the unique strong solution produces a stochastic flow of homeomorphisms if the modulus of continuity of coefficients is less than , ?∈[0,1) with ?(−1)=1, and the coefficients are compactly supported.     

Ratio asymptotic of Hermite-Padé orthogonal polynomials for Nikishin systems. II

We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures N(σ₁,…,σm) such that for each k, σk has constant sign on its support consisting on an interval , on which almost everywhere, and a set without accumulation points in .     

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.