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 ρ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous function $k:\mathit{M}\to \mathbb{R}$ whose negative part $k^{-}$ satisfies, for every $t>0$, the exponential integrability condition

$$\underset{x\in \mathit{M}}{sup}\mathbb{E}\left[{\mathrm{e}}^{{\int }_0^t{k}^{-}(X_r^x)∕2\mathrm{d}r}{\mathbb{1}}_{\{t<{\mathit{\zeta }}^x\}}\right]<\mathrm{\infty },$$

then the lifetime ${\mathit{\zeta }}^x$ of Brownian motion $X^x$ on M starting in any $x\in \mathit{M}$ 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 $f\in L^{\mathrm{\infty }}(\mathit{M})$ and $t>0$ along the heat flow ${({\mathsf{P}}_t)}_{t\ge 0}$ with generator $\mathrm{\Delta }∕2$, yielding its ${\mathit{L}}^{\mathrm{\infty }}$-$\mathrm{Lip}$-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 $x,y\in \mathit{M}$, of a coupling $(X^x,X^y)$ of Brownian motions on M starting in $(x,y)$ such that a.s.,

$$\mathit{\rho }(X_t^x,X_t^y)\le {\mathrm{e}}^{-{\int }_s^t\underset{\_}{k}(X_r^x,X_r^y)∕2\mathrm{d}r}\mathit{\rho }(X_s^x,X_s^y)$$

holds for every $s,t\ge 0$ with $s\le t$, involving the “average” $\underset{\_}{k}(u,v):={inf}_{\mathit{\gamma }}{\int }_0^{1}k({\mathit{\gamma }}_r)\mathrm{d}r$ 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.