TY - JOUR

T1 - Differential Harnack inequalities on path space

AU - Haslhofer, Robert

AU - Kopfer, Eva

AU - Naber, Aaron

N1 - Funding Information:
The first author has been supported by NSERC grant RGPIN-2016-04331 and a Sloan Research Fellowship . The second author has been supported by the German Research Foundation through the Hausdorff Center for Mathematics and the Collaborative Research Center 1060 . The third author has been supported by NSF grant DMS-1809011 . All three authors thank the Fields Institute in Toronto for support during the thematic program on Geometric Analysis.
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/12/3

Y1 - 2022/12/3

N2 - 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 [Formula presented]. 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 inequality [Formula presented] 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.

AB - 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 [Formula presented]. 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 inequality [Formula presented] 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.

KW - Harnack inequality

KW - Path space

KW - Ricci curvature

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U2 - 10.1016/j.aim.2022.108714

DO - 10.1016/j.aim.2022.108714

M3 - Article

AN - SCOPUS:85139194142

SN - 0001-8708

VL - 410

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108714

ER -