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

We prove that if the Ricci tensor $\mathrm{R}\mathrm{i}\mathrm{c}$ 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 $k\text{\hspace{0.17em}⁣}:M\to \mathbb{R}$ 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 < \zeta^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 $\mathrm{\nabla }{\mathsf{P}}_{t}f$ for every $f\in L^{\mathrm{\infty }}(M)$ and $t>0$ along the heat flow $({\mathsf{P}}_t{)}_{t\ge 0}$ with generator $\mathrm{\Delta }\mathrm{/}2$, yielding its $L^{\mathrm{\infty }}$-$\mathrm{L}\mathrm{i}\mathrm{p}$-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 $\mathrm{R}\mathrm{i}\mathrm{c}$ 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$alonggeodesicsfrom$u$to$v$. Our results generalize to weighted Riemannian manifolds, where the Ricci curvature is replaced by the corresponding Bakry-'Emery Ricci tensor.