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 Ric of a geodesically complete Riemannian manifold M , endowed with the Riemannian distance ρ and the Riemannian measure m, is bounded from below by a continuous function k : M → R whose negative part k − satisfies, for every t > 0, the exponential integrability conditionthen the lifetime ζ x of Brownian motion X x on M starting in any x ∈ 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 ∇Ptf for every f ∈ L ∞ (M ) and t > 0 along the heat flow (Pt) t≥0 with generator ∆/2, yielding its L ∞ -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 Ric by k to the existence, given any x, y ∈ M , of a coupling (X x , X y ) of Brownian motions on M starting in (x, y) such that a.s.,holds for every s, t ≥ 0 with s ≤ t, involving the "average" k(u, v) := infγ 1 0 k(γr) dr 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.

Torus stability under Kato bounds on the Ricci curvature

“…The first occurrence of $$\mbox{k}_T(M^n,g)$$ in the study of Riemannian manifolds seems to be [23]. The geometric and analytic consequences of a bound on $$\mbox{k}_T(M^n,g)$$ have been extensively studied since then, see, for example, [1, 4, 12, 13, 15, 32, 34, 36]. It is useful to note that if $$\mbox{Ric}\geqslant -\kappa ^2 g$$, then $$\mbox{k}_T(M^n,g)\leqslant \kappa ^2 T$$; hence, a smallness assumption on $$\mbox{k}_T(M^n,g)$$ should be understood as a control on the part of the manifold where $$\mbox{Ric}_{\mbox{{-}}}\gtrsim T^{-1}$$.…”

“…The first occurrence of k 𝑇 (𝑀 𝑛 , g) in the study of Riemannian manifolds seems to be [23]. The geometric and analytic consequences of a bound on k 𝑇 (𝑀 𝑛 , g) have been extensively studied since then, see, for example, [1,4,12,13,15,32,34,36]. It is useful to note that if Ric ⩾ −𝜅 2 g, then k 𝑇 (𝑀 𝑛 , g) ⩽ 𝜅 2 𝑇; hence, a smallness assumption on k 𝑇 (𝑀 𝑛 , g) should be understood as a control on the part of the manifold where Ric -≳ 𝑇 −1 .…”