Differential Harnack Inequalities on Path Space

Abstract

Recall that if $(M^n,g)$ satisfies $\mathrm{Ric}\geq 0$, then the Li-Yau Differential Harnack Inequality tells us for each nonnegative $f:M\to \mathbb{R}^+$, with $f_t$ its heat flow, that $\frac{\Delta f_t}{f_t}-\frac{|\nabla f_t|^2}{f_t^2} +\frac{n}{2t}\geq 0.$ 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 a family of finite dimensional gradients and Laplace operators. Namely, for each $H^1_0$-function $\varphi:\mathbb{R}^+\to \mathbb{R}$ we will define the $\varphi$-gradient $\nabla_\varphi F: P_xM\to T_xM$ and the $\varphi$-Laplacian $\Delta_\varphi F =\text{tr}_\varphi\mathrm{Hess} F:P_xM\to \mathbb{R}$, where $\mathrm{Hess} F$ is the Markovian Hessian and both the gradient and the $\varphi$-trace are induced by $n$ vector fields naturally associated to $\varphi$ under stochastic parallel translation. Now let $(M^n,g)$ satisfy $\mathrm{Ric}=0$, then for each nonnegative $F:P_xM\to \mathbb{R}^+$ we will show the inequality $$\frac{E_x [\Delta_\varphi F]}{E_x [F]}-\frac{E_x [\nabla_\varphi F]^2}{E_x [F]^2} +\frac{n}{2}|| \varphi ||^2\geq 0$$ for each $\varphi$, 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 $F(\gamma) \equiv f(\gamma(t))$, 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 $\mathbb{R}^n$.