Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

Abstract

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 $k:M\to \mathbb{R}$ 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 ${\zeta }^x$ 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 $\nabla {\mathsf{P}}_{t}f$ for every $f\in L^{\infty }(M)$ and $t>0$ along the heat flow $({\mathsf{P}}_t{)}_{t\ge 0}$ with generator $\Delta /2$, 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 $s,t\ge 0$ with $s\le t$, involving the "average" $\underline{k}(u,v):={\inf }_{\gamma }{\int }_0^{1}k({\gamma }_r)\mathrm{d}r$ 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.

arXiv comment32 pages. Title has been changed. The case of weighted Riemannian manifolds has been included. The assumption on Dynkin decomposability has been replaced by a weaker exponential integrability condition. Theorem 1.3 has been added. The bibliography has been extended