# Differential Harnack Inequalities on Path Space

@article{Haslhofer2020DifferentialHI, title={Differential Harnack Inequalities on Path Space}, author={Robert Haslhofer and Eva Kopfer and Aaron Naber}, journal={arXiv: Differential Geometry}, year={2020} }

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…

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