Differential Harnack inequalities on path space

Abstract

Recall that if (Mn,g) satisfies Ric≥0, then the Li-Yau Differential Harnack Inequality tells us for each nonnegative f:M→R+, with ft its heat flow, that Δftft−|∇ft|2ft2+n2t≥0. Our main result will be to generalize this to path space PxM of the manifold.A key point is that instead of considering infinite dimensional gradients and Laplacians on PxM we will consider, in a spirit similar to [13,8], a family of finite dimensional gradients and Laplace operators. Namely, for each H01-function φ:R+→R we will define the φ-gradient ∇φF:PxM→TxM and the φ-Laplacian ΔφF=trφHessF:PxM→R, where Hess F is the Markovian Hessian and both the gradient and the φ-trace are induced by n vector fields naturally associated to φ under stochastic parallel translation.Now let (Mn,g) satisfy Ric=0, then for each nonnegative F:PxM→R+ we will show the inequalityEx[ΔφF]Ex[F]−Ex[∇φF]2Ex[F]2+n2||φ||2≥0 for each φ, where Ex denotes the expectation with respect to the Wiener measure on PxM. By applying this to the simplest functions on path space, namely cylinder functions of one variable F(γ)≡f(γ(t)), we will see we recover the classical Li-Yau Harnack inequality exactly. We have similar estimates for Einstein manifolds, with errors depending only on the Einstein constant, as well as for general manifolds, with errors depending on the curvature. Finally, we derive generalizations of Hamilton's Matrix Harnack inequality on path space PxM. It is our understanding that these estimates are new even on the path space of Rn.