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

Global existence for strong solutions of viscous Burgers equation. (1) The bounded case

We prove that the viscous Burgers equation has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term are smooth and bounded together with their derivatives. Such solutions may have infinite energy. The proof does not rely on energy estimates, but on a combination of the maximum principle and quantitative Schauder estimates. We obtain precise bounds on the sup norm of the solution and its derivatives, making it plain that there is no exponential increase in time. In particular, these bounds are time-independent if the forcing term is zero. To get a classical solution, it suffices to assume that the initial condition and the forcing term have bounded derivatives up to order two.Comment: 22 page

A L\'evy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index

The main tool for stochastic calculus with respect to a multidimensional process $B$ with small H\"older regularity index is rough path theory. Once $B$ has been lifted to a rough path, a stochastic calculus -- as well as solutions to stochastic differential equations driven by $B$ -- follow by standard arguments. Although such a lift has been proved to exist by abstract arguments \cite{LyoVic07}, a first general, explicit construction has been proposed in \cite{Unt09,Unt09bis} under the name of Fourier normal ordering. The purpose of this short note is to convey the main ideas of the Fourier normal ordering method in the particular case of the iterated integrals of lowest order of fractional Brownian motion with arbitrary Hurst index.Comment: 20 page

Projective Pseudodifferential Analysis and Harmonic Analysis

We consider pseudodifferential operators on functions on $\R^{n+1}$ which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. Their symbols can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space $G_n/H_n=SL(n+1,\R)/GL(n,\R)$, and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space $P_n(\R)$ : these spaces are the representation spaces of the maximal degenerate series $(\pi_{i\lambda,\epsilon})$ of $G_n$ . This new approach to the quantization of $G_n/H_n$, already considered by other authors, has several advantages: as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of $L^2(G_n/H_n)$ under the quasiregular action of $G_n$ . We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation $\pi_{i\lambda,\epsilon}$

Symbolic calculus on the time-frequency half-plane

The study concerns a special symbolic calculus of interest for signal analysis. This calculus associates functions on the time-frequency half-plane f>0 with linear operators defined on the positive-frequency signals. Full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way. The correspondence rule is detailed and the associated Wigner function is given. Formulas expressing the basic operation (star-bracket) of the Lie algebra of symbols, which is isomorphic to that of the operators, are obtained. In addition, it is shown that the resulting calculus is covariant under a three-parameter group which contains the affine group as subgroup. This observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one.Comment: 25 pages, Latex, minor changes and more references; to be published in the "Journal of Mathematical Physics" (special issue on "Wavelet and Time-Frequency Analysis"