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.
Categories: math.PR math.DG math.FA