From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L\'evy area of fractional Brownian motion with Hurst index α∈(1/8,1/4)\alpha\in(1/8,1/4)

{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \cite{MagUnt1}, this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as L\'evy area

Adsorption structure of glycine on TiO2(1 1 0): a photoelectron diffraction determination

High-resolution core-level photoemission and scanned-energy mode photoelectron diffraction (PhD) of the O 1s and N 1s states have been used to investigate the interaction of glycine with the rutile TiO2(1 1 0) surface. Whilst there is clear evidence for the presence of the zwitterion View the MathML sourceCH2COO− with multilayer deposition, at low coverage only the deprotonated glycinate species, NH2CH2COO is present. Multiple-scattering simulations of the O 1s PhD data show the glycinate is bonded to the surface through the two carboxylate O atoms which occupy near-atop sites above the five-fold-coordinated surface Ti atoms, with a Ti–O bondlength of 2.12 ± 0.06 Å. Atomic hydrogen arising from the deprotonation is coadsorbed to form hydroxyl species at the bridging oxygen sites with an associated Ti–O bondlength of 2.01 ± 0.03 Å. Absence of any significant PhD modulations of the N 1s emission is consistent with the amino N atom not being involved in the surface bonding, unlike the case of glycinate on Cu(1 1 0) and Cu(1 0 0)

Characterization of SU(1,1) coherent states in terms of affine group wavelets

The Perelomov coherent states of SU(1,1) are labeled by elements of the quotient of SU(1,1) by the compact subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parameterize the coherent states by elements of that group or equivalently by points in the half-plane. Such a formulation permits to find new properties of the SU(1,1) coherent states and to relate them to affine wavelets.Comment: 11 pages, latex, to be published in J. Phys. A : Math. Ge

Quantitative adsorbate structure determination under catalytic reaction conditions

Current methods allow quantitative local structure determination of adsorbate geometries on surfaces in ultrahigh vacuum (UHV) but are incompatible with the higher pressures required for a steady-state catalytic reactions. Here we show that photoelectron diffraction can be used to determine the structure of the methoxy and formate reaction intermediates during the steady-state oxidation of methanol over Cu(110) by taking advantage of recent instrumental developments to allow near-ambient pressure x-ray photoelectron spectroscopy. The local methoxy site differs from that under static UHV conditions, attributed to the increased surface mobility and dynamic nature of the surface under reaction conditions

Photoelectron diffraction investigation of the structure of the clean TiO2(110)(1×1) surface

The surface relaxations of the rutile TiO2(110)(1×1) clean surface have been determined by O 1 s and Ti 2p3∕2 scanned-energy mode photoelectron diffraction. The results are in excellent agreement with recent low-energy electron diffraction (LEED) and medium energy ion scattering (MEIS) results, but in conflict with the results of some earlier investigations including one by surface x-ray diffraction. In particular, the bridging O atoms at the surface are found to relax outward, rather than inward, relative to the underlying bulk. Combined with the recent LEED and MEIS results, a consistent picture of the structure of this surface is provided. While the results of the most recent theoretical total-energy calculations are qualitatively consistent with this experimental consensus, significant quantitative differences remain

H\"older-continuous rough paths by Fourier normal ordering

We construct in this article an explicit geometric rough path over arbitrary $d$-dimensional paths with finite $1/\alpha$-variation for any $\alpha\in(0,1)$. The method may be coined as 'Fourier normal ordering', since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest Fourier frequencies. In doing so, there appear non-trivial tree combinatorics, which are best understood by using the structure of the Hopf algebra of decorated rooted trees (in connection with the Chen or multiplicative property) and of the Hopf shuffle algebra (in connection with the shuffle or geometric property). H\"older continuity is proved by using Besov norms. The method is well-suited in particular in view of applications to probability theory (see the companion article \cite{Unt09} for the construction of a rough path over multidimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or \cite{Unt09ter} for a short survey in that case).Comment: 50 pages, 6 figure

An explicit formula for the Berezin star product

We prove an explicit formula of the Berezin star product on Kaehler manifolds. The formula is expressed as a summation over certain strongly connected digraphs. The proof relies on a combinatorial interpretation of Englis' work on the asymptotic expansion of the Laplace integral.Comment: 19 pages, to appear in Lett. Math. Phy

The local structure of OH species on the V2O3(0 0 0 1) surface: a scanned-energy mode photoelectron diffraction study

Scanned-energy mode photoelectron diffraction (PhD), using O 1s photoemission, together with multiple-scattering simulations, have been used to investigate the structure of the hydroxyl species, OH, adsorbed on a V2O3(0 0 0 1) surface. Surface OH species were obtained by two alternative methods; reaction with molecular water and exposure to atomic H resulted in closely similar PhD spectra. Both qualitative assessment and the results of multiple-scattering calculations are consistent with a model in which only the O atoms of outermost layer of the oxide surface are hydroxylated. These results specifically exclude significant coverage of OH species atop the outermost V atoms, i.e. in vanadyl O atom sites. Ab initio density-functional theory cluster calculations provide partial rationalisation of this result, which is discussed the context of the general understanding of this system

Ageing in the critical contact process: a Monte Carlo study

The long-time dynamics of the critical contact process which is brought suddenly out of an uncorrelated initial state undergoes ageing in close analogy with quenched magnetic systems. In particular, we show through Monte Carlo simulations in one and two dimensions and through mean-field theory that time-translation invariance is broken and that dynamical scaling holds. We find that the autocorrelation and autoresponse exponents lambda_{Gamma} and lambda_R are equal but, in contrast to systems relaxing to equilibrium, the ageing exponents a and b are distinct. A recent proposal to define a non-equilibrium temperature through the short-time limit of the fluctuation-dissipation ratio is therefore not applicable.Comment: 18 pages, 7 figures, Latex2e with IOP macros; final for