Xicheng Zhang

Finite dimensional approximation of Riemannian path space geometry

Abstract 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.

A Littlewood-Paley Type Inequality on the Path Space

Considering adapted differential geometry on the path space of a Riemannian manifold (in the spirit of [6]) we prove a corresponding L p Littlewood-Paley inequality using Meyer’s methodology.

Ornstein-Uhlenbeck semigroups on Riemannian path spaces

where μ denotes the Wiener measure. This corresponds to an extension to finite dimensions of the Mehler’s formula. There are other ways to introduce this semigroup, notably through its action on the finite dimensional Wiener chaos or by associating the semigroup to the generator, the so-called Ornstein-Uhlenbeck operator, and constructing the correspondent diffusion. For this last approach one can proceed at least in two different ways: using Dirichlet form theory ([?][?][?]) or defining a two-parameter diffusion (i.e., a stochastic process with values on the Wiener space) as a perturbation of a two-parameter Brownian motion ([?]). The Wiener measure is invariant for the Ornstein-Uhlenbeck semigroup, which is a positive self-adjoint contraction operator on the spaces L(X, μ) for any p ≥ 1. Nelson’s hypercontractivity also holds true, namely