Riemannian Path Space Research Articles

Spectral gap on Riemannian path space over static and evolving manifolds

In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber's recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space.

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.

Logarithmic Sobolev inequality on free path space over a compact Riemannian manifold

In this paper we use Bismut’s method to verify the formula of integration by parts on the free path space. By the formula, we verify the martingale representation for reference measure Pμ, the law of the Brownian motion on the base manifold with initial distribution μ. Then by the martingale representation we get the logarithmic Sobolev inequality on compact free Riemannian path space.

Flows associated to Cameron-Martin type vector fields on path spaces over a Riemannian manifold

The flow on the Wiener space associated to a tangent process constructed by Cipriano and Cruzeiro, as well as by Gong and Zhang does not allow to recover Driver’s Cameron-Martin theorem on Riemannian path space. The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver. In this paper, we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector field on the path space over a Riemannian manifold.

Deviation inequalities and the law of iterated logarithm on the path space over a loop group

A law of iterated logarithm (LIL) in small time and an asymptotic estimate of modulus of continuity are proved for Brownian motion on the loop group ℒ(G) over a compact connected Lie group G. Upper bounds are obtained via infinite-dimensional deviation inequalities for functionals on the path space ℙ(ℒ(G)) on ℒ(G), such as the supremum of Brownian motion on ℒ(G), which are proved from the Clark–Ocone formula on ℙ(ℒ(G)). The lower bounds rely on analog finite-dimensional results that are proved separately on Riemannian path space.

Horizontal lift of Ornstein–Uhlenbeck process over Riemannian path space

In this paper we give a representation formula for the heat semigroup on adapted vector fields by the horizontal lift of Ornstein–Uhlenbeck process over Riemannian path space.

Finite dimensional approximation of Riemannian path space geometry

In this paper, we construct a finite dimensional approximation for the geometry on the path space over a compact Riemannian manifold. This approximation allows to construct the horizontal lift of the Ornstein–Uhlenbeck process on the path space through the Markovian connection. We also prove a representation formula for the heat semigroup on (adapted) vector fields as well as a commutation formula for its derivative.

Stochastic calculus of variations and Harnack inequality on Riemannian path spaces

We describe the tangent space of Riemannian path space as a space of tangent processes localized on Brownian sheets; the bundle of adapted frames above a Riemannian path space and its structural equation are given. The stochastic calculus of variations allows us to derive Harnack–Bismut inequality for the Norris semigroup. To cite this article: A.-B. Cruzeiro, P. Malliavin, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 817–820.

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

We 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.

Markovian Connection, Curvature and Weitzenböck Formula on Riemannian Path Spaces

We shall consider on a Riemannian path space Pmo(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 Weitzenböck formula will be established. To this end, we shall introduce an “inner product” between the tangent processes and simple vector fields.

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

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 Weitzenböck formula restricted to adapted vector fileds: the Ricci-tensor is shown to be equal to the identity.

Frame Bundle of Riemannian Path Space and Ricci Tensor in Adapted Differential Geometry

The vanishing of the renormalized Ricci tensor of the path space above a Ricci flat Riemannian manifold is discussed.

Integration by Parts Formula and Logarithmic Sobolev Inequality on the Path Space Over Loop Groups

The geometric stochastic analysis on the Riemannian path space developed recently gives rise to the concept of tangent processes. Roughly speaking, it is the infinitesimal version of the Girsanov theorem. Using this concept, we shall establish a formula of integration by parts on the path space over a loop group. Following the martingale method developed in Capitaine, Hsu and Ledoux, we shall prove that the logarithmic Sobolev inequality holds on the full paths. As a particular case of our result, we obtain the Driver–Lohrenz’s heat kernel logarithmic Sobolev inequalities over loop groups. The stochastic parallel transport introduced by Driver will play a crucial role.