Large deviations for rough paths of the fractional Brownian motion

Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by Coutin and Qian (2002), we prove a large deviation principle in the space of geometric rough paths, extending classical results on Gaussian processes. As a by-product, geometric rough paths associated to elements of the reproducing kernel Hilbert space of the fractional Brownian motion are obtained and an explicit integral representation is given.Comment: 32 page

White Noise Space Analysis and Multiplicative Change of Measures

In this paper, we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogs of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations

White Noise Space Analysis and Multiplicative Change of Measures

In this paper we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogues of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces, and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory, and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations

Krein's spectral theory and the Paley-Wiener expansion for fractional Brownian motion

In this paper we develop the spectral theory of the fractional Brownian motion (fBm) using the ideas of Krein's work on continuous analogous of orthogonal polynomials on the unit circle. We exhibit the functions which are orthogonal with respect to the spectral measure of the fBm and obtain an explicit reproducing kernel in the frequency domain. We use these results to derive an extension of the classical Paley-Wiener expansion of the ordinary Brownian motion to the fractional case.Comment: Published at http://dx.doi.org/10.1214/009117904000000955 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the equivalence of probability spaces

For a general class of Gaussian processes $W$, indexed by a sigma-algebra $\mathscr F$ of a general measure space $(M,\mathscr F, \sigma)$, we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering $\sigma(A)$, for $A\in\mathscr F$, as a quadratic variation of $W$ over $A$. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: $(i)$ a computation of generalized Ito-integrals; and $(ii)$ a proof of an explicit, and measure-theoretic equivalence formula, realizing an equivalence between the two approaches to Gaussian processes, one where the choice of sample space is the traditional path-space, and the other where it is Schwartz' space of tempered distributions.Comment: To appear in Journal of Theoretical Probabilit