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We study the canonical heat flow (H ) ≥0 on the cotangent module 2 ( * ) over an RCD( , ∞) space ( , d, m), ∈ ℝ. We show Hess-Schrader-Uhlenbrock's inequality and, if ( , d, m) is also an RCD * ( , ) space, ∈ (1, ∞), Bakry-Ledoux's inequality for (H ) ≥0 w.r.t. the heat flow (P ) ≥0 on 2 ( ). A crucial tool is that the dimensional vector 2-Bochner inequality is self-improving, entailing a dimensional vector 1-Bochner inequality-a version of which is also available in the dimension-free case-as a byproduct. Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for 1-forms, the previous inequalities yield various -properties of (H ) ≥0 , ∈ [1, ∞].Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian ⃗ Δ, of the negative functional Laplacian −Δ, and of the Schrödinger operator −Δ + .In the RCD * ( , ) case, we prove compactness of ⃗ Δ −1 if is compact, and the independence of the -spectrum of ⃗ Δ on ∈ [1, ∞] under a volume growth condition.We terminate by giving an appropriate interpretation of a heat kernel for (H ) ≥0 . We show its existence in full generality without any local compactness or doubling assumptions, and derive fundamental estimates and properties of it.

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We prove that if the Ricci curvature of a geodesically complete Riemannian manifold X, endowed with the Riemannian distance ρ and the Riemannian volume measure m, is bounded from below by a Dynkin decomposable function k : X → R, then X is stochastically complete. This assumption on k is satisfied if its negative part k − belongs to the Kato class of X. In addition, given f ∈ L p (X) for sufficiently large p in a range depending only on k − , we derive a global Bismut derivative formula for ∇P t f for every t > 0 along the heat flow (P t ) t≥0 , whose L ∞ -Lip-regularization we obtain as a corollary.Moreover, for such functions k, we show that the Ricci curvature of X is bounded from below by k if and only if X supports the L 1 -gradient estimate w.r.t. k, i.e. for every f ∈ W 1,2 (X) and t ≥ 0, |∇P t f | ≤ P k t |∇f | holds m-a.e., (P k t ) t≥0 being the Schrödinger semigroup in L 2 (X) generated by −(∆− k)/2. If k is additionally lower semicontinuous, another equivalent characterization of lower Ricci bounds by k is proven to be the existence, given any x, y ∈ X, of a Markovian coupling (b x , b y ) of Brownian motions on X starting in (x, y) such that a.s. for every s, t ∈ [0, ∞) with s ≤ t, one has the pathwise estimate