Stochastic calculus for fractional Brownian motion with Hurst exponent H>1/4H>1/4: A rough path method by analytic extension

The $d$-dimensional fractional Brownian motion (FBM for short) $B_t=((B_t^{(1)},...,B_t^{(d)}),t\in\mathbb{R})$ with Hurst exponent $\alpha$, $\alpha\in(0,1)$, is a $d$-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_s^{(i)}B _t^{(j)}]={1/2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2 \alpha}).$ The long-standing problem of defining a stochastic integration with respect to FBM (and the related problem of solving stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either $d$ or $\alpha$. The case $\alpha={1/2}$ corresponds to the usual stochastic integration with respect to Brownian motion, while most computations become singular when $\alpha$ gets under various threshhold values, due to the growing irregularity of the trajectories as $\alpha\to0$. We provide here a new method valid for any $d$ and for $\alpha>{1/4}$ by constructing an approximation $\Gamma(\varepsilon)_t$, $\varepsilon\to0$, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process $\Gamma_z$ on the cut plane $z\in\mathbb{C}\setminus\mathbb{R}$ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see \citeCQ02) but as yet a little mysterious divergence of L\'evy's area for $\alpha\to{1/4}$.Comment: Published in at http://dx.doi.org/10.1214/08-AOP413 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics

Let $B=(B_1(t),..,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha\le 1/4$, 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 forthcoming 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 of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a {\em non-perturbative} effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read in a second time the companion article \cite{MagUnt2} (or a preliminary version arXiv:1006.1255) for the constructive proofs

The Schrödinger-Virasoro Lie group and algebra: from geometry to representation theory

This article is concerned with an extensive study of a infinite-dimensional Lie algebra \goth sv, introduced in [14] in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrödinger equation and the central charge-free Virasoro algebra Vect $(S^1)$ . We call \goth sv the Schrödinger-Virasoro Lie algebra. We choose to present \goth sv from a Newtonian geometry point of view first, and then in connection with conformal and Poisson geometry. We turn afterwards to its representation theory: realizations as Lie symmetries of field equations, coadjoint representation, coinduced representations in connection with Cartan's prolongation method (yielding analogues of the tensor density modules for Vect $(S^1)$), and finally Verma modules with a Kac determinant formula. We also present a detailed cohomogical study, providing in particular a classification of deformations and central extensions; there appears a non-local cocycle

The Schrödinger-Virasoro Lie algebra: a mathematical structure between conformal field theory and non-equilibrium dynamics

We explore the mathematical structure of the infinite-dimensional Schrödinger-Virasoro algebra, and discuss possible applications to the integrability of anisotropic or out-of-equilibrium statistical systems with a dynamical exponent $z \neq 1$ by defining several correspondences with conformal field theory

Discretizing the fractional Levy area

In this article, we give sharp bounds for the Euler- and trapezoidal discretization of the Levy area associated to a d-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean-square convergence rate of the Euler scheme. For H<3/4 the exact convergence rate is n^{-2H+1/2}, where n denotes the number of the discretization subintervals, while for H=3/4 it is n^{-1} (log(n))^{1/2} and for H>3/4 the exact rate is n^{-1}. Moreover, the trapezoidal scheme has exact convergence rate n^{-2H+1/2} for H>1/2. Finally, we also derive the asymptotic error distribution of the Euler scheme. For H lesser than 3/4 one obtains a Gaussian limit, while for H>3/4 the limit distribution is of Rosenblatt type.Comment: 28 page

The rough path associated to the multidimensional analytic fbm with any Hurst parameter

32 pagesInternational audienceIn this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion. This process has been introduced by the second author of the article, and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on the analytic fBm can be constructed for any value of the Hurst parameter in (0,1/2). This allows in particular to solve differential equations driven by this process in a neighborhood of 0 of the complex upper half-plane, thanks to a variant of the usual rough path theory due to Gubinelli

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

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

Supersymmetric extensions of Schr\"odinger-invariance

The set of dynamic symmetries of the scalar free Schr\"odinger equation in d space dimensions gives a realization of the Schr\"odinger algebra that may be extended into a representation of the conformal algebra in d+2 dimensions, which yields the set of dynamic symmetries of the same equation where the mass is not viewed as a constant, but as an additional coordinate. An analogous construction also holds for the spin-1/2 L\'evy-Leblond equation. A N=2 supersymmetric extension of these equations leads, respectively, to a `super-Schr\"odinger' model and to the (3|2)-supersymmetric model. Their dynamic supersymmetries form the Lie superalgebras osp(2|2) *_s sh(2|2) and osp(2|4), respectively. The Schr\"odinger algebra and its supersymmetric counterparts are found to be the largest finite-dimensional Lie subalgebras of a family of infinite-dimensional Lie superalgebras that are systematically constructed in a Poisson algebra setting, including the Schr\"odinger-Neveu-Schwarz algebra sns^(N) with N supercharges. Covariant two-point functions of quasiprimary superfields are calculated for several subalgebras of osp(2|4). If one includes both N=2 supercharges and time-inversions, then the sum of the scaling dimensions is restricted to a finite set of possible values.Comment: Latex 2e, 46 pages, with 3 figures include