Weak Levi-Civita Connection for the Damped Metric on the Riemannian Path Space and Vanishing of Ricci Tensor in Adapted Differential Geometry

Abstract We shall establish in the context of adapted differential geometry on the path space P m o ( M ) a Weitzenbock formula which generalizes that in (A. B. Cruzeiro and P. Malliavin, J. Funct. Anal . 177 (2000), 219–253), without hypothesis on the Ricci tensor. The renormalized Ricci tensor will be vanished. The connection introduced in (A. B. Cruzeiro and S. Fang, 1997, J. Funct. Anal. 143 , 400–414) will play a central role.

Additive functionals and push forward measures under Veretennikov's flow

16 pages; In this work, we will be interested in the push forward measure $(\vf_t)_*\gamma$, where $\vf_t$ is defined by the stochastic differential equation \begin{equation*} d\vf_t(x)=dW_t + \ba(\vf_t(x))dt, \quad \vf_0(x)=x\in\mbR^m, \end{equation*} and $\gamma$ is the standard Gaussian measure. We will prove the existence of density under the hypothesis that the divergence $\div(\ba)$ is not a function, but a signed measure belonging to a Kato class; the density will be expressed with help of the additive functional associated to $\div(\ba)$.

On the Estimate for Commutators in DiPerna–Lions Theory

In this note, we will exhibit the estimates on the commutator of semi-groups, motivated by commutator estimates in Di Perna-Lions theory.

Analysis on free Riemannian path spaces

Abstract The gradient operator is defined on the free path space with reference measure P μ , the law of the Brownian motion on the base manifold with initial distribution μ, where μ has strictly positive density w.r.t. the volume measure. The formula of integration by parts is established for the underlying directional derivatives, which implies the closability of the gradient operator so that it induces a conservative Dirichlet form on the free path space. The log-Sobolev inequality for this Dirichlet form is established and, consequently, the transportation cost inequality is obtained for the associated intrinsic distance.

Transportation-cost inequality on path spaces with uniform distance

Abstract Let M be a complete Riemannian manifold and μ the distribution of the diffusion process generated by 1 2 ( Δ + Z ) where Z is a C 1 -vector field. When Ric − ∇ Z is bounded below and Z has, for instance, linear growth, the transportation-cost inequality with respect to the uniform distance is established for μ on the path space over M . A simple example is given to show the optimality of the condition.

Flow of Homeomorphisms and Stochastic Transport Equations

Abstract We consider Stratonovich stochastic differential equations with drift coefficient A 0 satisfying only the condition of continuity where r is a positive C 1 function defined on a neighborhood ]0, c 0] of 0 such that (Osgood condition), and s → r(s) is decreasing while s → sr(s 2) is increasing. We prove that the equation defines a flow of homeomorphisms if the diffusion coefficients A 1,…, A N are in . If , we prove limit theorems for Wong–Zakai approximation as well as for regularizing the drift A 0. As an application, we solve a class of stochastic transport equations.

Sobolev estimates for optimal transport maps on Gaussian spaces

We will study variations in Sobolev spaces of optimal transport maps with the standard Gaussian measure as the reference measure. Some dimension free inequalities will be obtained. As application, we construct solutions to Monge-Ampere equations in finite dimension, as well as on the Wiener space.

Heat semi-group and generalized flows on complete Riemannian manifolds

Abstract We will use the heat semi-group to regularize functions and vector fields on Riemannian manifolds in order to develop Di Perna–Lions theory in this setting. Malliavinʼs point of view of the bundle of orthonormal frames on Brownian motions will play a fundamental role. As a byproduct we will construct diffusion processes associated to an elliptic operator with singular drift.

Markovian Connection, Curvature and Weitzenböck Formula on Riemannian Path Spaces

Abstract We shall consider on a Riemannian path space P m o ( M ) the Cruzeiro–Malliavin's Markovian connection. The Laplace operator will be defined as the divergence of the gradient. We shall compute explicitly the associated curvature tensor. A Weitzenbock formula will be established. To this end, we shall introduce an “inner product” between the tangent processes and simple vector fields.

De Rham–Hodge–Kodaira Operator on Loop Groups

AbstractWe consider a based loop group Le(G) over a compact Lie groupG, endowed with its pinned Wiener measureν(the law of the Brownian bridge onG) and we shall calculate the Ricci curvature for differentialn-forms over Le(G). A type of Bochner–Weitzenböck formula for general differentialn-forms (or Shigekawa identity) will be established.

Transportation cost inequalities on path and loop groups

AbstractLet G be a connected Lie group with the Lie algebra G. The action of Cameron–Martin space H(G) on the path space Pe(G) introduced by L. Gross (Illinois J. Math. 36 (1992) 447) is free. Using this fact, we define the H-distance on Pe(G), which enables us to establish a transportation cost inequality on Pe(G). This method will be generalized to the path space over the loop group Le(G), so that we obtain a transportation cost inequality for heat measures on Le(G).

Stochastic differential equations with coefficients in Sobolev spaces

We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\delta(A_0)$ w.r.t. the Gaussian measure $\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\text{loc}^{1,p'}$ for some $p'>1$. Assume $\int_{\R^d} \exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \d\gamma_d0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-f…

Ricci Tensors on Some Infinite Dimensional Lie Algebras

Abstract The Ricci tensor has been computed in several infinite dimensional situations. In this work, we shall be interested in the case of the central extension of loop groups and in the asymptotic behaviour of the Ricci tensor on free loop groups as the Riemannian metric varies.

Solving stochastic differential equations on Homeo(S1)

Abstract The Brownian motion with respect to the metric H 3/2 on Diff( S 1 ) has been constructed. It is realized on the group of homeomorphisms Homeo( S 1 ). In this work, we shall resolve the stochastic differential equations on Homeo( S 1 ) for a given drift Z .

Corrigendum to “Solving stochastic differential equations on Homeo(S1)” [J. Funct. Anal. 216 (2004) 22–46]

Stochastic anticipative calculus on the path space over a compact Riemannian manifold

Abstract In this paper, we shall first give another expression for Cruzeiro-Malliavin structure equation, by means of the Skorohod integral. The torsion tensor with respect to the Markovian connection used in [CF] is computed. This is the key step to establish a Stroock-like formula of commutation on the derivative of the Skorohod integral, which enables us to prove an Ito formula. As an application, we shall give a maximal inequality for Skorohod integrals following [AN2].

A Weitzenböck formula for the damped Ornstein–Uhlenbeck operator in adapted differential geometry

Abstract On the Riemannian path space we consider the Ornstein–Uhlenbeck operator associated to the Dirichlet form E (f,g)=E〈 ∇ f, ∇ g〉 H , where ∇ is the damped gradient and 〈·,·〉 H the scalar product of the Cameron–Martin space H . We prove a corresponding Weitzenbock formula restricted to adapted vector fileds: the Ricci-tensor is shown to be equal to the identity.

Hydrodynamics and Stochastic Differential Equation with Sobolev Coefficients

In this chapter, we will explain how the Brenier’s relaxed variational principle for Euler equation makes involved the ordinary differential equations with Sobolev coefficients and how the investigation on stochastic differential equations (SDE) with Sobolev coefficients is useful to establish variational principles for Navier–Stokes equations. We will survey recent results on this topic.

Fokker–Planck equation with respect to heat measures on loop groups

Abstract The Dirichlet form on the loop group L e ( G ) with respect to the heat measure defines a Laplacian Δ DM on L e ( G ) . In this note, we will use Wasserstein distance variational method to solve the associated heat equation for a given data of finite entropy.

Transport equations and quasi-invariant flows on the Wiener space

Abstract We shall investigate on vector fields of low regularity on the Wiener space, with divergence having low exponential integrability. We prove that the vector field generates a flow of quasi-invariant measurable maps with density belonging to the space L log L . An explicit expression for the density is also given.

Connexion markovienne, courbure et formule de Weitzenböck sur l'espace des chemins riemanniens

Resume Nous considerons la connexion markovienne sur l'espace des chemins riemanniens. Le tenseur de courbure est calcule explicitement et une formula de Weitzenbock est etablie.

A class of stochastic differential equations with non-Lipschitzian coefficients: pathwise uniqueness and no explosion

Abstract A new result for the pathwise uniqueness of solutions of stochastic differential equations with non-Lipschitzian coefficients is established. Furthermore, we prove that the solution has no explosion under the growth ξlogξ. To cite this article: S. Fang, T. Zhang, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Integration by parts for heat measures over loop groups

Abstract The formula of integration by parts for heat measures over a loop group established by B. Driver is revesited through an alternative approach to this result. We shall first establish directly the integration by parts formula over an unimodular Lie group (which will be the finite product of a compact Lie group with a correlated metric), using the concept of tangent processes. A new expression for Ricci tensor will enable us the passage to the limit.

Isotropic stochastic flow of homeomorphisms on Sd for the critical Sobolev exponent

Abstract In this work, we shall deal with the critical Sobolev isotropic Brownian flows on the sphere S d . Based on previous works by O. Raimond and LeJan and Raimond (see [O. Raimond, Ann. Inst. H. Poincare 35 (1999) 313–354] and [Y. LeJan, O. Raimond, Ann. of Prob. 30 (2002) 826–873], we prove that the associated flows are flows of homeomorphisms.

Canonical Brownian Motion on the Diffeomorphism Group of the Circle

AbstractFor infinitesimal data given on the group of diffeomorphism of the circle with respect to the metric H3/2, the associated Brownian motion has been constructed by Malliavin (C.R. Acad. Sci. Parist.329 (1999), 325–329). In this work, we shall give another approach and prove the invariance of heat measures under the adjoint action of S1.