Differential Harnack inequalities on path space
Differential Harnack inequalities on path space
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路径空间上的微分 Harnack 不等式
DOI：
10.1016/j.aim.2022.108714
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发表时间：
2022
影响因子：
1.7
通讯作者：
Naber, Aaron
中科院分区：
文献类型：
--
作者：
Haslhofer, Robert;Kopfer, Eva;Naber, Aaron
回想一下，如果(Mn，g)满足Ric≥0，那么Li-Yau微分Harnack不等式告诉我们，对于每个非负f：M→R+，当f t它的热流时，Δf t f t−|∇f t|2 f t 2+n 2 t≥0。我们的主要结果是将其推广到流形的路径空间P×M。一个关键点是，与考虑P×M上的无限维梯度和拉普拉斯算子不同，我们将以类似于[13]、[8]的精神来考虑一族有限维梯度和拉普拉斯算子。也就是说，对于每个H0 1-函数φ：R+→R，我们将定义φ-梯度∇φF：P x M→T x M和φ-LaplianΔφF=trφHess F：P x M→R，其中Hess F是马尔可夫Hessian，梯度和φ迹都是由在随机平行平移下与φ自然相关的n个向量场诱导的。现在设(Mn，g)满足Ric=0，则对于每个非负F：P x M→R+，我们将证明对每个Δφ的−E x[∇φF]Ex[F]φE x[≥F]2 E x[F]2+n 2||φ||2γ)≡0，其中Ex表示关于Px M上的Wiener测度的期望。通过将它应用于路径空间上最简单的函数，即一元柱面函数F(γf(T))，我们将看到我们精确地恢复了经典的Li-Yau Harnack不等式。我们对爱因斯坦流形也有类似的估计，误差只取决于爱因斯坦常数，以及一般流形，误差取决于曲率。最后，我们得到了哈密尔顿矩阵Harnack不等式在路径空间PxM上的推广。我们的理解是，这些估计即使在Rn的路径空间上也是新的。
Recall that if (M n, 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 Δ f t f t−|∇ f t| 2 f t 2+ n 2 t≥ 0. Our main result will be to generalize this to path space P x M of the manifold. A key point is that instead of considering infinite dimensional gradients and Laplacians on P x M we will consider, in a spirit similar to [13],[8], a family of finite dimensional gradients and Laplace operators. Namely, for each H 0 1-function φ: R+→ R we will define the φ-gradient∇ φ F: P x M→ T x M and the φ-Laplacian Δ φ F= tr φ Hess F: P x M→ 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 (M n, g) satisfy Ric= 0, then for each nonnegative F: P x M→ R+ we will show the inequality E x [Δ φ F] E x [F]− E x [∇ φ F] 2 E x [F] 2+ n 2|| φ|| 2≥ 0 for each φ, where E x denotes the expectation with respect to the Wiener measure on P x M. 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 x M. It is our understanding that these estimates are new even on the path space of R n.
黎曼预期随机积分的 AnL2 估计
DOI：
--
发表时间：
1997
期刊：
影响因子：
--
作者：
A. Cruzeiro;S. Fang
通讯作者：
S. Fang