Constantin and Iyer's representation formula for the Navier--Stokes equations on manifolds

The purpose of this paper is to establish a probabilistic representation formula for the Navier--Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case of $\mathbb R^n$ or of $\mathbb T^n$. On a Riemannian manifold, however, there are several different choices of Laplacian operators acting on vector fields. In this paper, we shall use the de Rham--Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy--Le Jan--Li's idea to decompose it as a sum of the square of Lie derivatives.Comment: 26 pages. We add Section 4 discussing the Killing vector fields on Riemannian symmetric spaces which satisfy the conditions in Section

Remarks on spectral gaps on the Riemannian path space

In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces

Stochastic differential equtions with non-lipschitz coefficients:II. Dependence with respect to initial values

The existence of the unique strong solution for a class of stochastic differential equations with non-Lipschitz coefficients was established recently. In this paper, we shall investigate the dependence with respect to the initial values. We shall prove that the non confluence of solutions holds under our general conditions. To obtain a continuous version, the modulus of continuity of coefficients is assumed to be less than \dis |x-y|\log{1\over|x-y|}. In this case, it will give rise to a flow of homeomorphisms if the coefficients are compactly supported.Comment: 14 page

Stochastic differential equations with non-lipschitz coefficients: I. Pathwise uniqueness and large deviation

We study a class of stochastic differential equations with non-Lipschitzian coefficients.A unique strong solution is obtained and a large deviation principle of Freidln-Wentzell type has been established.Comment: A short version will be published in C. R. Acad. Pari

Transport equations and quasi-invariant flows on the Wiener space

AbstractWe 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 LlogL. An explicit expression for the density is also given

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

AbstractWe shall establish in the context of adapted differential geometry on the path space Pmo(M) a Weitzenböck 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

AnL2Estimate for Riemannian Anticipative Stochastic Integrals

AbstractWe define a metric and a Markovian connection on the path space of a Riemannian manifold which are different from those introduced in [CM] and prove a corresponding Weitzenböck formula. AnL2inequality for the divergence is obtained as a consequence

Stochastic differential equations and stochastic parallel translations in the Wasserstein space

We will develop some elements in stochastic analysis in the Wasserstein space $\mathbb{P}_2(M)$ over a compact Riemannian manifold $M$, such as intrinsic It$\^o$ formulae, stochastic regular curves and parallel translations along them. We will establish the existence of parallel translations along regular curves, or stochastic regular curves in case of $\mathbb{P}_2(\mathbb{T})$. Surprisingly enough, in this last case, the equation defining stochastic parallel translations is a SDE on a Hilbert space, instead of a SPDE