Differential Harnack inequalities on path space

Recall that if (Mⁿ,g) satisfies Ric≥0, then the Li-Yau Differential Harnack Inequality tells us for each nonnegative f:M→R⁺, with f_t its heat flow, that [Formula presented]. Our main result will be to generalize this to path space P_xM of the manifold. A key point is that instead of considering infinite dimensional gradients and Laplacians on P_xM 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:P_xM→T_xM and the φ-Laplacian Δ_φF=tr_φHessF:P_xM→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:P_xM→R⁺ we will show the inequality [Formula presented] for each φ, where E_x denotes the expectation with respect to the Wiener measure on P_xM. 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 P_xM. It is our understanding that these estimates are new even on the path space of Rⁿ.