Lectures on stochastic differential equations and Malliavin calculus

This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject. Table of Contents: Background material; Sobolev spaces on $\mathbb{R}^n$; Differentiable measures on linear spaces; Some classes of differentiable measures; Subspaces of differentiability of measures; Integration by parts and logarithmic derivatives; Logarithmic gradients; Sobolev classes on infinite dimensional spaces; The Malliavin calculus; Infinite dimensional transformations; Measures on manifolds; Applications; References; Subject index. (Surv/164)

Differentiable Measures and the Malliavin Calculus

A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold

This work is involved with the short-time asymptotics of diffusion semigroups in a general setting. A generalization of Fang's version of Varadhan's formula is proven for general Dirichlet spaces that are local and conservative. The intrinsic metric appearing in the formula is characterized by pointwise distance for canonical Dirichlet spaces on loop groups.

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Small-time Gaussian behavior of symmetric diffusion semi-groups

Using probabilistic methods, we prove new rigidity results for groups and pseudo-groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class ({\em i.e.} between $C^1$ and $C^2$). In particular, we demonstrate some generalizations of Denjoy's Theorem and the classical Kopell's Lemma for Abelian groups. After that, these techniques are applied to the study of codimension 1 foliations. We obtain for instance several generalized versions of Sacksteder's Theorem in class $C^1$. We conclude with some remarks about the stationary measure.

Sur la dynamique unidimensionnelle en r\'egularit\'e interm\'ediaire

Integration on loop groups. I. Quasi invariant measures

Integration on loop group. III : Asymptotic Peter-Weyl orthogonality

The group of difisomorphisms of a riemannian manifold of dimension >1 is very “ramified”. The existence of a “reasonnable” quasi invariant measure seems therefore very doubtfull. The case of the group of the diffeomorphisms of the circle S1 is quite different. We shall in this paper identify it with a loop space which carries a natural Wiener measure.

An Infinitesimally Quasi Invariant Measure on the Group of Diffeomorphisms of the Circle

Le syst~me de de Rham-Hodge au dessus d'une vari~t~ Riemannienne V ne se r~duit pas ~ un syst~me diagonal dans la cas non homog~ne ; en effet les formules de la moyenne associ~es s'obtiennent en r~solvant des ~quations diff~rentielles lin~aires ~ coefficients variables le long des trajectoires de la diffusion relev~e [3] .Dans le cas o~ l'espace V est homog~ne les ~quations diff~rentielles associ~es sont alors ~ coefficient constants et se r~solvent ainsi par des exponentielles e g~ sur une d~composition adapt~e de l'espaee, On se propose d'explieit~ le plus possible cette diagonalisation, y compris le calcul des exposants ~ . (cf Th~or~me 16). Un th~or~me d'annulation de cohomologie au dessus de V se ram~nera (cf th~or~me II) ainsi enti~rement~ a) ~ un calcul des exposants ~ calcul qui d~pendra uniquement d'op~rations effectu~es dans l'alg~bre de Lie. b) ~ la d~termination de la borne sup~rieure du spectre d'op~rateurs de Casimir operant sur des fonctions ~quivariantes. Un th~or~me d'annulation sera alors ~quivalent ~ la r~solution de h~ probl~me qui rel~ve des m~thodes de l'analyse (cf [2] qui constitue une premiere tentative dans cette direction). Soit G un groupe de Lie que l'on supposera unimodulaire, H un sous-groupe compact, V = G/H et p : G ~ G/H la projection. On a par hypoth~se

Marie-Paule Malliavin

Papers

Integration on loop groups. I. Quasi invariant measures

Integration on loop group. III : Asymptotic Peter-Weyl orthogonality

An Infinitesimally Quasi Invariant Measure on the Group of Diffeomorphisms of the Circle

Diagonalisation du système de de Rham-Hodge au dessus d’un espace Riemannien homogène

Grade et theoreme d'intersection en algebre non commutative