Random rotations of the Wiener path

Summary Let (W, H, ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyhH. It is shown that (R(w)h) is again a Gaussian (0, |h| _H ² )-random variable. Consequently, if (e _i ,i)W ^* is a complete, orthonormal basis ofH, then defines a measure preserving transformation, a rotation, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, (R(w+k)h) is (0, |h| _H ² )-Gaussian for allk, hH, thenR is an isometry and Rh is quasi-nilpotent for allHH. The relation between the stochastic calculi for these Wiener pathsw and , as well as the conditions of the inverbibility of the map are discussed and the problem of the absolute continuity of the image of the Wiener measure under Euclidean motion on the Wiener space (i.e. composed with a shift) is studied.The research of the second author was supported by the Fund for the Promotion of Research at the TechnionDedicated to the memory of Albert Badrikian     

Some relations among classes of σ-fields on Wiener space

Summary In this work we study sigma fields and their tangent spaces on the Wiener space which are invariant in some sense with respect to the basic operators of the Malliavin Calculus.     

Differentiation formulas for stochastic integrals in the plane

For a one-parameter process of the form X_t=X₀+∫^t₀φ_sdW_s+∫^t₀ψ_sds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(X_t) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process.Let {W₂, z∈R²₊} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals ∫ φdWand ∫ψdWdW, as well as mixed integrals ∫h dz dW and ∫gdW dz. Now let X_z be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(X_z) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(X_z) on increasing paths in $\text{R}{}^2{}_{\mathrm{+}}$.     

Hydrogen generation from weak acids: electrochemical and computational studies of a diiron hydrogenase mimic

Extended investigation of electrocatalytic generation of dihydrogen using [(mu-1,2-benzenedithiolato)][Fe(CO)3]2 has revealed that weak acids, such as acetic acid, can be used. The catalytic reduction producing dihydrogen occurs at approximately -2 V for several carboxylic acids and phenols resulting in overpotentials of only -0.44 to -0.71 V depending on the weak acid used. This unusual catalytic reduction occurs at a potential at which the starting material, in the absence of a proton source, does not show a reduction peak. The mechanism for this process and structures for the intermediates have been discerned by electrochemical and computational analysis. These studies reveal that the catalyst is the monoanion of the starting material and an ECEC mechanism occurs.     

Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space

Let (W, H, μ) be the classical Wiener space. Assume that U = I _W  + u is an adapted perturbation of identity, i.e., u : W → H is adapted to the canonical filtration of W. We give some sufficient analytic conditions on u which imply the invertibility of the map U. In particular it is shown that if ${u\in {\rm ID}_{p,1}(H)}$ is adapted and if ${\exp(\frac{1}{2}\|\nabla u\|_2^2-\delta u)\in L^q(\mu)}$ , where p ^?1 + q ^?1 = 1, then I _W  + u is almost surely invertible. With the help of this result it is shown that if ${\nabla u\in L^\infty(\mu,H\otimes H)}$ , then the Girsanov exponential of u times the Wiener measure satisfies the logarithmic Sobolev inequality and this implies the invertibility of U = I _W  +  u . As a consequence, if, there exists an integer k ≥  1 such that ${\|\nabla^k u\|_{H^{\otimes(k+1)}}\in L^\infty(\mu)}$ , then I _W  +  u is again almost surely invertible under the almost sure continuity hypothesis of ${t\to\nabla^i \dot{u}_t}$ for i ≤  k ? 1.     

The Si-Si effect on ionization of beta-disilanyl sulfides and selenides

The ionization energies of conformationally constrained, newly synthesized beta-disilanyl sulfides and selenides were determined by photoelectron spectroscopy. These ionization energies reflect substantial (0.53-0.75 eV) orbital destabilizations. The basis for these destabilizations was investigated by theoretical calculations, which reveal geometry-dependent interaction between sulfur or selenium lone pair orbitals and sigma-orbitals, especially Si-Si sigma-orbitals. These results presage facile redox chemistry for these compounds and significantly extend the concept of sigma-stabilization of electron-deficient centers.     

Transformation of Wiener measure under anticipative flows

Summary LetT()=+F() be a transformation from the Wiener space to itself with the range ofF() assumed to be in the Cameron-Martin space. The absolute continuity and the density function associated withT is considered;T is assumed to be embedded in or defined through a parameterizationT _t =+F _t () andF _t is assumed to be differentiable int. The paper deals first with the case where the range of thet-derivative ofF _t () is also in the Cameron-Martin space and new representations for the Radon-Nikodym derivative and the Carleman-Fredholm determinant are derived. The case where thet-derivative ofF _t is not in the Cameron-Martin space is considered next and results on the absolute continuity and the density function, under conditions which are considerably weaker than previously known conditions, are presented.The work of the second author was supported by the fund for promotion of research at the Technion