论文标题
路径空间上的差异竖琴不平等现象
Differential Harnack Inequalities on Path Space
论文作者
论文摘要
回想一下，如果$（m^n，g）$满足$ \ mathrm {ric} \ geq 0 $，那么li-yau差异harnack不平等现象会告诉我们每个非负$ f：m \ to \ mathbb {r} $ \ frac {δf_T} {f_t} - \ frac {| \ nabla f_t |^2} {f_t^2} +\ frac {n} {n} {2t} \ geq 0. $我们的主要结果将是该概述的lace $ p_xm $的途径。 关键点是，而不是考虑$ p_xm $上的无限尺寸梯度和拉普拉斯人，而是考虑一个有限的尺寸梯度和拉普拉斯操作员的家族。即，对于每个$ h^1_0 $ -Function $φ：\ Mathbb {r}^+\ to \ mathbb {r} $，我们将定义$φ$ -gradient $ \nabla_φf：p_xm = \ text {tr}_φ\ mathrm {hess} f：p_xm \ to \ mathbb {r} $，其中$ \ mathrm {hess} f $是马克维亚·海森（Markovian Hessian），梯度和$φ$ - the $φ$ - 自然相关的$ n $ trace $ n $ trace pare $ n $ vector partor to $ $ $ $ $ $φ$ $φ 现在让$（m^n，g）$满足$ \ mathrm {ric} = 0 $，然后，对于每个非负$ f：p_xm \ to \ mathbb {r}^+$，我们将显示不平等$ \ frac \ frac {e_x { [\nabla_φf]^2} {e_x [f]^2} +\ frac {n} {2} || φ||^2 \ geq 0 $$对于每种$φ$，其中$ e_x $表示相对于$ p_xm $上的维也纳措施的期望。通过将其应用于路径空间上最简单的函数，即一个变量$ f（γ）\ equiv f（γ（t））$的气缸函数，我们将看到我们恢复了经典的li-yau harnack不平等。我们对爱因斯坦歧管有类似的估计值，仅取决于爱因斯坦常数以及一般歧管，并且误差取决于曲率。最后，我们得出了汉密尔顿矩阵harnack在路径空间上的概括$ p_xm $。我们的理解是，即使在$ \ mathbb {r}^n $的路径空间上，这些估计是新的。
Recall that if $(M^n,g)$ satisfies $\mathrm{Ric}\geq 0$, then the Li-Yau Differential Harnack Inequality tells us for each nonnegative $f:M\to \mathbb{R}^+$, with $f_t$ its heat flow, that $\frac{Δf_t}{f_t}-\frac{|\nabla f_t|^2}{f_t^2} +\frac{n}{2t}\geq 0.$ Our main result will be to generalize this to path space $P_xM$ of the manifold. A key point is that instead of considering infinite dimensional gradients and Laplacians on $P_xM$ we will consider a family of finite dimensional gradients and Laplace operators. Namely, for each $H^1_0$-function $φ:\mathbb{R}^+\to \mathbb{R}$ we will define the $φ$-gradient $\nabla_φF: P_xM\to T_xM$ and the $φ$-Laplacian $Δ_φF =\text{tr}_φ\mathrm{Hess} F:P_xM\to \mathbb{R}$, where $\mathrm{Hess} F$ is the Markovian Hessian and both the gradient and the $φ$-trace are induced by $n$ vector fields naturally associated to $φ$ under stochastic parallel translation. Now let $(M^n,g)$ satisfy $\mathrm{Ric}=0$, then for each nonnegative $F:P_xM\to \mathbb{R}^+$ we will show the inequality $$\frac{E_x [Δ_φF]}{E_x [F]}-\frac{E_x [\nabla_φF]^2}{E_x [F]^2} +\frac{n}{2}|| φ||^2\geq 0$$ 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 $F(γ) \equiv f(γ(t))$, 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 $\mathbb{R}^n$.
