论文标题
热流规律性，bismut-elworthy-li的衍生式公式以及带有加藤有限的RICCI曲率的Riemannian歧管上的路线耦合
Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature
论文作者
论文摘要
我们证明，如果ricci张量$ \ mathrm {ric} $的地理上完整的riemannian歧管$ m $，并带有riemannian距离$ \ mathsf {d} $ and riemannian mather $ \ mathfrak {m} $，从下面限制在下面m \ to \ to \ mathbb {r} $，其负零件$ k^ - $满足，对于每一个$ t> 0 $，指数的集成条件\ begin \ begin {equation*} \ sup_ {x \ in m} \ mathbb {e} \ big [e} K^ - （\ MathSf {B} _r^X）/2 \，\ Mathrm {d} r} \，1 _ {\ {\ {t <q {t <ζ^x \}} \ big] <\ big] <\ ind，\ ind，\ end en e eent {qore {qore {equation*}，然后是生命的$ qubrownian $ brownian Motion $ quns $ qubrownian $ qubrownian $ qubrownian $ quams $ qummmn $ x \ m $是A.S.无限。如果$ k^ - $属于$ m $的$ k^ - $，则此假设可容纳。我们还为$ \ nabla \ mathsf {p} _tf $ in l^\ infty（m）$和$ t> 0 $沿热量流$（\ mathsf {p} _t）_ {p} $ geq 0} $ centratian/forian/for in in l^\ infty（m）$（\ mathsf {p} _t）_ { $ l^\ infty $ - $ \ mathrm {lip} $ - 正则化作为推论。 此外，考虑到$ m $的随机完整性，但是除了连续性外没有任何假设，我们证明了$ \ mathrm {ric} $ by $ k $的下降等效性与存在的$ x，y \ in m $ in m $，brhenian $ x，y \ in m $ coupling $（$ nirian coupling $（\ nathsf）$（\ mathsf {\ nathsf { $ m $以$（x，y）$开始，以至于A.S. \下划线{K}（\ Mathsf {B} _r^X，\ Mathsf {B} _r^Y）/2 \，\ Mathrm {d} r} \，\ Mathsf {d} \ big（\ big） \ end {equation*}每$ s，t \ geq 0 $都具有$ s \ leq t $，涉及“平均” $ \ upsineline {k}（k}（u，v）：= \inf_γ\ int_0^1 k（γ_r）\，\ kathrm {d} rm {d} r $的$ k $ k $ k $ use $ u $ ucodesics $ u $ ucodesices 我们的结果推广到加权的riemannian歧管，其中RICCI曲率被相应的Bakry-émeryricci张量代替。
We prove that if the Ricci tensor $\mathrm{Ric}$ of a geodesically complete Riemannian manifold $M$, endowed with the Riemannian distance $\mathsf{d}$ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous function $k\colon M\to\mathbb{R}$ whose negative part $k^-$ satisfies, for every $t>0$, the exponential integrability condition \begin{equation*} \sup_{x\in M} \mathbb{E}\big[\mathrm{e}^{\int_0^t k^-(\mathsf{b}_r^x)/2\,\mathrm{d} r}\,1_{\{t < ζ^x\}}\big] < \infty, \end{equation*} then the lifetime $ζ^x$ of Brownian motion $\mathsf{b}^x$ on $M$ starting in any $x\in M$ is a.s. infinite. This assumption on $k$ holds if $k^-$ belongs to the Kato class of $M$. We also derive a Bismut-Elworthy-Li derivative formula for $\nabla \mathsf{P}_tf$ for every $f\in L^\infty(M)$ and $t>0$ along the heat flow $(\mathsf{P}_t)_{t\geq 0}$ with generator $Δ/2$, yielding its $L^\infty$-$\mathrm{Lip}$-regularization as a corollary. Moreover, given the stochastic completeness of $M$, but without any assumption on $k$ except continuity, we prove the equivalence of lower boundedness of $\mathrm{Ric}$ by $k$ to the existence, given any $x,y\in M$, of a coupling $(\mathsf{b}^x,\mathsf{b}^y)$ of Brownian motions on $M$ starting in $(x,y)$ such that a.s., \begin{equation*} \mathsf{d}\big(\mathsf{b}_t^x,\mathsf{b}_t^y\big) \leq \mathrm{e}^{-\int_s^t \underline{k}(\mathsf{b}_r^x,\mathsf{b}_r^y)/2\,\mathrm{d} r}\,\mathsf{d}\big(\mathsf{b}_s^x,\mathsf{b}_s^y\big) \end{equation*} holds for every $s,t\geq 0$ with $s\leq t$, involving the "average" $\underline{k}(u,v) := \inf_γ\int_0^1 k(γ_r)\,\mathrm{d} r$ of $k$ along geodesics from $u$ to $v$. Our results generalize to weighted Riemannian manifolds, where the Ricci curvature is replaced by the corresponding Bakry-Émery Ricci tensor.
