Denis Feyel

On Fractional Brownian Processes

We use Liouville spaces in order to prove the existence of some different fractional α-Brownian motion ( 0 < α ≤ 1 ), or fractional ( α, β )-Brownian sheets. There are also applications to the Wiener stochastic integral with respect to these α-Brownian.

Hausdorff measures on the Wiener space

We construct a ‘Hausdorff measure’ of finite co-dimension on the Wiener space. We then extend the Federer co-area Formula to this Wiener space for functions with the sole condition that they belong to the first Sobolev space. An explicit formula for the density of the images of the Wiener measure under such functions follows naturally from this. As a corollary, this yields a new and easy proof of the Krée-Watanabe theorem concerning the regularity of the images of the Wiener measure.

Opérateurs linéaires gaussiens

We extend operators from the Cameron-Martin space to Gaussian Lusinian locally convex space. We then are allowed to give sense to the Mehler formula for every such bounded operator. An application is made to Hilbert-Schmidt operators. Next we show that capacities asociated to second quantizations of operators are tight on compact sets, and this is a general result even if the underlying space is not a Banach space.

Hausdorff-Gauss Measures

According to a result of Bouleau-Hirsch, the law of an ℝ n -valued Wiener functional belonging to the Gaussian-Dirichlet space W 1,2has a density with respect to Lebesgue measure λn. On the other hand there exists a classical formula for changing variables, the coarea Federer formula for Lipschitz functions. The goal of this chapter is to compute the exact value of this density by means of an extension of the coarea formula to the Wiener space, or more generally to an abstract Wiener space.

Brownian processes in infinite dimension

AbstractLetE be a locally convex space endowed with a centered gaussian measure ξ. We construct a continuousE-valued brownian motionWt with covariance ξ. The main goal is to solve the SDE of Langevin type dXt=

Harmonic analysis in infinite dimension

\sqrt {2a}

Curvilinear Integrals Along Enriched Paths

dWt−AXt wherea andA are unbounded operators of the Cameron-Martin space of (E, ξ). It appears as the unique linear measurable extension of the solution of the classical Cauchy problemv′(t)=