Ana Bela Cruzeiro

Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids

We construct a family of probability spacesΩℱ,Pγ), γ<0 associated with the Euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow φt (time evolution) leavingPγ globally invariant. φt is obtained as the limit of Galerkin approximations associated with Euler equations.Pγ is also in invariant measure for a stochastic process associated with a Navier-Stokes equation with viscosity, γ, stochastically perturbed by a white noise force.

Equations différentielles sur l'espace de Wiener et formules de Cameron-Martin non-linéaires

Proprietes algebriques des approximations sur R n . Une formule de perturbation des flots et convergence des approximations. Formule de changement de variable sur X et equation differentielle

Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus

Abstract We give a rigorous version of the functional calculus developed by R. Feynman in relation to his path integral formulation of Quantum Mechanics. Our approach is Euclidean but distinct from the one founded on the Feynman-Kac formula. It uses two basic ingredients: a new probabilistic interpretation of the classical heat equation, introduced recently in the framework of Euclidean Quantum Mechanics, and an infinite dimensional differential calculus adapted to functionals of the diffusion processes relevant for this interpretation.

Non perturbative construction of invariant measure through confinement by curvature

Abstract A curvature criterium, computable by infinitesimal differential geometry, insures the existence of invariant probability measure for an a priori given elliptic operator; uniqueness, reversibility, formula of integration by part are discussed in this context; existence of invariant measure for some non linear OU operator on an Hilbert space.

Stochastic Euler-Poincaré reduction

We prove a Euler-Poincare reduction theorem for stochastic processes taking values on a Lie group, which is a generalization of the reduction argument for the deterministic case [J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, (Texts in Applied Mathematics). (Springer, 2003)]. We also show examples of its application to SO(3) and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.

Bernstein Processes Associated with a Markov Process

A general description of Bernstein processes, a class of diffusion processes, relevant to the probabilistic counterpart of quantum theory known as Euclidean Quantum Mechanics, is given. It is compatible with finite or infinite dimensional state spaces and singular interactions. Although the relations with statistical physics concepts (Gibbs measure, entropy,…) is stressed here, recent developments requiring Feynman’s quantum mechanical tools (action functional, path integrals, Noether’s Theorem,…) are also mentioned and suggest new research directions, especially in the geometrical structure of our approach.

Momentum Maps and Stochastic Clebsch Action Principles

We derive stochastic differential equations whose solutions follow the flow of a stochastic nonlinear Lie algebra operation on a configuration manifold. For this purpose, we develop a stochastic Clebsch action principle, in which the noise couples to the phase space variables through a momentum map. This special coupling simplifies the structure of the resulting stochastic Hamilton equations for the momentum map. In particular, these stochastic Hamilton equations collectivize for Hamiltonians that depend only on the momentum map variable. The Stratonovich equations are derived from the Clebsch variational principle and then converted into Itô form. In comparing the Stratonovich and Itô forms of the stochastic dynamical equations governing the components of the momentum map, we find that the Itô contraction term turns out to be a double Poisson bracket. Finally, we present the stochastic Hamiltonian formulation of the collectivized momentum map dynamics and derive the corresponding Kolmogorov forward and backward equations.

A class of anticipative tangent processes on the Wiener space

Abstract We prove a representation and an integration by parts formula for a class of anticipative tangent processes on the Wiener space and give some applications.

Stochastic analysis and mathematical physics

Preface * 1. Functorial Analysis in Geometric Probability Theory (H. Airault/P. Malliavin) * 2. Stochastic Volterra Equations with Singular Kernels (L. Coutin/L. Decreusefond) * 3. Stochastic Diffeology and Homotopy (R. Leandre) * 4. Some Results on Entropic Projections (C. Leonard) * 5. Mehler-type Semigroups on Hilbert Spaces and Their Generators (P. Lescot) * 6. Singular Limiting Behavior in Nonlinear Stochastic Wave Equations (M. Oberguggenberger/F. Russo) * 7. Complete Positivity and Open Quantum Systems (R. Rebolledo) * 8. Properties of Measure-preserving Shifts on the Wiener Space (A.S. Ustunel) * 9. Martingale and Markov Uniqueness of Infinite Dimensional Nelson Diffusions (L. Wu)

Energy Identities and Estimates for Anticipative Stochastic Integrals on a Riemannian Manifold

If x denotes an ℝd-valued P τ-adapted Brownian motion where P τ is the usual past filtration, the following energy identity