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- W3214943162 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 condition supx∈ME[e∫0tk−(X rx)∕2dr1{t<ζx}]<∞, then the lifetime ζx of Brownian motion Xx 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 Δ∕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 (Xx,Xy) of Brownian motions on M starting in (x,y) such that a.s., ρ(Xtx,X ty)≤e−∫stk_(X rx,X ry)∕2drρ(Xsx,X sy) holds for every s,t≥0 with s≤t, involving the “average” k_(u,v):=infγ ∫01k(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." @default.
- W3214943162 created "2021-12-06" @default.
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- W3214943162 date "2021-01-01" @default.
- W3214943162 modified "2023-09-30" @default.
- W3214943162 title "Heat flow regularity, Bismut–Elworthy–Li’s derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature" @default.
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- W3214943162 doi "https://doi.org/10.1214/21-ejp703" @default.
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