  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 < \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 $\nabla \mathsf{P}_tf$ for every $f\in L^\infty(M)$ and
$t>0$ along the heat flow $(\mathsf{P}_t)_{t\geq 0}$ with generator $\Delta/2$,
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 $s,t\geq 0$ with $s\leq t$, 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-\'Emery Ricci tensor.
