Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

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.

then 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 ∇P t f for every f ∈ L ∞ (M ) and t > 0 along the heat flow (P t ) t≥0 with generator ∆/2, yielding its L ∞ -Lipregularization 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., ρ X x t , X y t ≤ e − t s k(X x r ,X y r )/2 dr ρ X x s , X y s holds for every s, t ≥ 0 with s ≤ t, involving the "average" k(u, v) := inf γ 1 0 k(γ r ) 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.

Main results
Let (M , g) be a smooth, geodesically complete, noncompact, connected Riemannian manifold without boundary. The metric ·, · := g(·, ·) induces the Riemannian distance ρ and the Riemannian measure m. W.r.t. ρ, we write B r (x) for the open ball of radius r > 0 around x ∈ M , Lip(M ) for the space of real-valued Lipschitz functions on M , and Lip(f ) for the Lipschitz constant of any f ∈ Lip(M ). All appearing vector spaces of functions and sections of bundles are considered as being real and, unless explicitly stated otherwise, all appearing Lebesgue and Sobolev spaces are understood w.r.t. m. With the usual abuse of notation, the fiberwise norm both on T M and T * M is | · | := ·, · 1/2 . Let ∇ be the Levi-Civita connection on M and Ric be the induced Ricci curvature. We recall that by geodesic completeness, the Laplace-Beltrami operator ∆ is an essentially self-adjoint operator in L 2 (M ) when defined initially on smooth compactly supported functions [Str83], and thus admits a unique -non-relabeled -self-adjoint extension. Let (P t ) t≥0 be the heat flow in L 2 (M ) with generator ∆/2, i.e. P t := e t∆/2 via spectral calculus. For every x ∈ M , let X x : [0, ζ x )×Ω → M be a corresponding adapted diffusion process (Brownian motion) starting at x ∈ M with lifetime ζ x , defined on a filtered probability space (Ω, F * , P), see [Elw82,Hsu02a,IW81,Wan14] for particular constructions of X x . Throughout, we fix a continuous function k : M → R. We write "Ric ≥ k on M " if The goal of this paper is to study the previous condition, where the negative part k − of k, with k − (x) := − min{k(x), 0}, obeys the integrability assumption (1.1) Our main results come in two groups. First, we study analytic and probabilistic consequences of the assumption Ric ≥ k on M if k satisfies (1.1), as described in Section 1.1 and stated in Theorem 1.1 and Theorem 1.5. Along with this, we treat an explicit class of k for which (1.1) holds, the so-called Kato decomposable ones, and highlight a general condition for k to obey the latter property, Theorem 1.3. Second, we give equivalent characterizations of the condition Ric ≥ k on M , which are summarized in Section 1.2, see Theorem 1.6 therein, and mostly do not even require (1.1).
Besides [ER + 20, GvR20], our article is among the first to systematically study analytic and probabilistic consequences of variable lower Ricci bounds -and equivalent characterizations of these -which are not uniformly bounded from below and do not underlie geometric growth conditions. We also stress our novel general sufficient condition from Theorem 1.3 to determine whether a given variable Ricci curvature lower bound is Kato decomposable, while the -albeit more general -condition (1.1) is in general hard to verify directly. Lastly, our equivalence result improves upon previously known ones especially because it involves a pathwise coupling estimate which has just recently been introduced in a slightly different framework [BHS21].

Consequences of variable lower Ricci bounds
To formulate our first result, given an initial point x ∈ M , let x denote the stochastic parallel transport w.r.t. ∇ along the sample paths of X x , i.e. x t : T x M → T X x t M for all t ∈ [0, ζ x ), let the process Q x : [0, ζ x ) × Ω → End(T x M ) be defined as the unique solution to the pathwise ordinary differential equation P ζ x = ∞ = 1 for every x ∈ M , (ii) for every f ∈ L ∞ (M ) and every t > 0, we have Bismut-Elworthy-Li's derivative formula where the stochastic integral inside the expectation is understood in Itô's sense, and (iii) for every t > 0, one has the L ∞ -Lip-regularization property P t : Before further commenting on Theorem 1.1 and its proof, in order to make more refined statements, we introduce the following definition.
Kato (decomposable) functions have been studied in great detail in the literature in the context of (scalar) Schrödinger operators, see [AS82,BG78,CZ95,Gün17a,SV96,Stu94] and the references therein. The survey [RS20] provides a concise overview over the use of Kato decomposability in the context of Riemannian manifolds and its connections to semigroup domination. A detailed study of the Kato class and the induced Schrödinger semigroups corresponding to a large class of Hunt processes can be found in [DvC00]. (Unfortunately, the authors of [DvC00] assume throughout that the underlying process has the Feller property, and it is not known whether this property holds in the situation of Theorem 1.1, or Theorem 1.5 below. This is why at many places we are going to rely on the results from [Gün17a] instead which are formulated for arbitrary Riemannian manifolds.) In connection with lower Ricci bounds, Kato decomposable functions have been introduced in [GP16] in the context of BV functions. They have been considered further recently in [Car19,Ros19] in the context of heat kernel, Betti number and eigenvalue estimates, and in [MO20] within the study on L p -properties of heat semigroups on forms. See also [GvR20], which treats some probabilistic and geometric aspects of molecular Schrödinger operators under Kato assumptions.
For the convenience of the reader, we have collected some important properties of Kato decomposable functions in Appendix A. In particular, note that in view of . Then k is Kato decomposable. One key feature for us about functions v ∈ K(M ) is that they always satisfy The proof of (i) in Theorem 1.1 can be found in Section 3.1, while (ii) and (iii) as well as Theorem 1.5 are studied in Section 3.2.
Let us collect some bibliographical comments on Theorem 1.1 and Theorem 1.5. In the framework of uniform bounds from below on the Ricci curvature, (i) in Theorem 1.1 is due to [Yau78]. On weighted Riemannian manifolds -on which the Ricci tensor is always replaced by the corresponding Bakry-Émery Ricci tensor, see Section 1.3 -the non-explosion for the induced diffusion processes under uniform lower Ricci bounds has been obtained by [Bak86]. In connection with (1.1), also for weighted Riemannian manifolds, the latter result has been extended by [Li94] using an approach via stochastic and Hessian flows. In fact, the corresponding condition at page 423 of [Li94] is implied by our condition (1.1). Once we have established all necessary intermediate results, our proof then closely follows the lines in [Bak86] (which is also worked towards in [Li94]). For different, more geometric non-explosion criteria in terms of distance functions, see [Wan14] and the references therein. A nonsmooth result similar to (i)however assuming a Kato-or rather a Dynkin-type [SV96,Stu94] lower bound instead of only (1.1) -has recently been treated in [ER + 20].
Formula (ii) in Theorem 1.1 has first appeared in [Bis84] in the compact case. In the noncompact case, this result, as well as Theorem 1.5, have been proven in [EL94a,EL94b] under more general assumptions than (1.1) using the slightly different technique of stochastic derivative flows. We also refer to [DT01] for similar treatises for heat semigroups over vector bundles, and also [Hsu02a,Wan14] for similar results under more geometric conditions on the lower bound of Ric. Remarkably, localized versions of the Bismut-Elworthy-Li derivative formula hold without any assumptions on the geometry of the manifold, see e.g. [Tha97,TW98,TW11].
The L ∞ -Lip-regularization (iii) from Theorem 1.1 is a corollary of (ii), thus indicating the importance of the latter in studying further regularity properties of (P t ) t≥0 . In fact, local versions of (iii) are already known even without the assumption (1.1) on k [TW98,Wan14], with slightly different estimates on Lip(P t f ) involving locally uniform lower bounds on Ric. (The proof uses the above mentioned local derivative formula.) Outside the smooth scope, a similar property as (iii) is known on RCD(K, ∞) spaces [AGS14]. This setting allows for more flexibility in the variety of spaces (metric measure spaces), but is still restricted to uniform lower Ricci bounds, formulated in a synthetic sense [LV09,Stu06].

Characterizations of variable lower Ricci bounds
We now come to our second main result, i.e. several equivalent characterizations of lower Ricci bounds, which we shortly introduce.
The closest characterization of Ric ≥ k on M is the L 1 -Bochner inequality which is related to the Ricci curvature of M by the following well-known Bochner formula: given We also derive a one-to-one connection between lower Ricci bounds by k and the existence of certain couplings of Brownian motions on M . Here, if M is stochastically complete, then given x, y ∈ M , by a coupling of Brownian motions starting in (x, y), we understand an M × M -valued stochastic process (X x , X y ) : [0, ∞) × Ω → M × M on some filtered probability space (Ω, F * , P) such that X x and X y are Brownian motions on M starting in x and y, respectively. To formulate an appropriate pathwise coupling estimate, we denote by Geo(M ) the set of minimizing geodesics γ : [0, 1] → M , and define the lower semicontinuous function k : (1.4) Observe that k can be recovered from the diagonal values of k, i.e. k(u) = k(u, u) for every u ∈ M . The key feature about k is that it provides a way to avoid cut-loci.
We prove "(ii) =⇒ (i)" in Section 4.1, "(i) =⇒ (iii)" in Section 4.2 and "(iii) =⇒ (ii)" in Section 4.3. For Kato decomposable functions k, another equivalent characterization of Ric ≥ k on M in terms of the L 1 -gradient estimate is discussed in Section A.1.
Again, some bibliographical comments are in order.
In the abstract framework of [ER + 20], the equivalence "(i) ⇐⇒ (ii)" -with (ii) in a weak formulation -together with their equivalence to (a nonsmooth version of) the L 1 -gradient estimate from Theorem A.1 has been shown independently.
The pathwise estimate appearing in (iii), as well as the equivalence of (iii) to lower Ricci bounds, extends similar results from [BHS21], where analogous equivalences have been established in the synthetic framework of (infinitesimally Hilbertian) CD(k, ∞) spaces with lower semicontinuous, lower bounded variable Ricci bounds k : M → R (see also [Stu15]). Even for the smooth case, the stated pathwise inequality involving the function k has been firstly introduced in [BHS21]. (Although it is quite straightforward to detect the place where k enters from the construction of the coupling, see Section 4.2, the function k was seemingly never mentioned explicitly in the literature before [BHS21].) In the Riemannian case, Theorem 1.6 establishes a similar result in full generality without any lower boundedness assumption on k. We point out that, in contrast to [BHS21], the coupling technique on manifolds does not require any notion of "Wasserstein contractivity" for the dual heat flow to (P t ) t≥0 on the space of Borel probability measures on M . It is rather provided in a direct way by the method of coupling by parallel displacement [Cra91,Ken86], see [ACT11] for a treatise in the case of constant k. Let us also point out [Vey11], which claims the existence of a coupling (X x , X y ) of Brownian motions, possibly with drift, such that for every t > 0, even holds on the event that X x r and X y r do not belong to each other's cut-locus for every r ∈ [0, t]. The real-valued function κ, the so-called "coarse curvature" of M , is defined outside the diagonal of M × M and is slightly larger than k.

Extensions to possible other settings
Apart from well-known geometric and topological applications [Bis86,Bue99,Li94], recent results for molecular Schrödinger operators [GvR20] suggest a detailed study of weighted Riemannian manifolds having Kato-type lower bounds on their Bakry-Émery Ricci tensor. In this context, we note that Theorem 1.1, Theorem 1.5 and Theorem 1.6 remain valid if for some Φ ∈ C 2 (M ), we replace • m by the weighted measure e −2Φ m, • ∆ by the drift Laplacian ∆ − 2 ∇Φ, ∇· , • Ric by the Bakry-Émery Ricci tensor Ric + 2 Hess Φ, • (P t ) t≥0 by the semigroup generated by ∆/2 − ∇Φ, ∇· , noting that the latter is again essentially self-adjoint [Li92, • X x by the diffusion generated by the operator ∆/2 − ∇Φ, ∇· , see e.g. [Wan14, Chapter 3] for the particular form of the corresponding stochastic differential equation and the construction of its solution, and • I by the weighted index form stated in Remark 4.4.
Other appropriate changes compared to the non-weighted setting, if needed, will always be indicated in the sequel. It would also be interesting to study Theorem 1.1, Theorem 1.5 and Theorem 1.6 in the context of lower bounds on the Bakry-Émery Ricci curvature Ric Z := Ric + 2 ∇Z which is associated to a C 1 -vector field Z on M not necessarily of gradient-type. See [Wan05,Wan14] and the references therein for a summary of similar statements under different, more geometric conditions. Given appropriate interpretations of the involved analytic objects, see [Wan05,Wan14] for details, some of the results immediately carry over with trivial modifications (for instance, the chain "(iii) =⇒ (ii) ⇐⇒ (i)" in Theorem 1.6). On the other hand, many of our arguments, e.g. Theorem 2.1 below and thus (i) in Theorem 1.1, or Theorem A.1, are implicitly based on self-adjointness of the semigroup (P t ) t≥0 and the heat flow on 1-forms. The latter properties lack in this generality, which is why we restricted ourselves to gradient vector fields.
Finally, a further possible (but highly nontrivial) direction of investigation is the case of manifolds with boundary, taking the heat flow with Neumann boundary conditions. See [CF12,Wan14] and the references therein for an account on diffusion processes on these. The key difficulty in this context will be to take into account the local time of the boundary appropriately. Results that are relevant in this context have been obtained in [Air75, AL17, DZ05, Hsu02b, IW81, Mér79, Wan14].

Preliminaries
For more details on the heat flows on functions and on 1-forms collected in this chapter, we refer the reader to [Dav89, Gri09, Gün17a, Hsu02a, Ros97, Str83] and the references therein. For details on their connection with the underlying stochastic processes, see [IW81,Mal97,Wan14]. Moreover, all objects and results presented here have counterparts in the weighted case outlined in Section 1.3: the heat flow on functions [Gri09], Brownian motion (or rather the corresponding Ornstein-Uhlenbeck process) [IW81,Li92,Wan14], and the heat flow on 1-forms [Li92].
Heat flow on functions The operator ∆/2 is the generator of the symmetric strongly local, regular Dirichlet form E : Note that under our standing assumption on M , The heat semigroup or heat flow (P t ) t≥0 introduced in the beginning of this article is directly linked to E by spectral calculus and is a strongly continuous, positivity preserving contraction semigroup of linear, self-adjoint operators in L 2 (M ). Powerful L 2 -L ∞ -regularization properties of the heat flow on relatively compact subsets of M , an exhaustion procedure and bootstrapping of regularity imply the existence of the socalled minimal heat kernel p ∈ C ∞ (0, ∞) × M × M ; (0, ∞) on M , the smallest positive fundamental solution to the heat operator ∂/∂t−∆/2. It has the property that for every f ∈ L 2 (M ) and t > 0, (a version of) P t f can be represented by Actually, (P t ) t≥0 extends to a contraction semigroup of linear operators from L p (M ) into L p (M ) for every p ∈ [1, ∞] which is strongly continuous if p < ∞ and weak * continuous if p = ∞. Moreover, the previous representation formula is still valid for every p ∈ [1, ∞] and f ∈ L p (M ). For such f , the above properties of the heat kernel show that in the classical sense. In addition, we have . Given a point x ∈ M , any stochastic process X with sample paths in C([0, ∞); M ∂ ) which is defined on a probability space (Ω, F , P) (i.e. the map t → X t (ω) belongs to C([0, ∞); M ∂ ) for all ω ∈ Ω) is termed Brownian motion on M starting in x if its law equals the Wiener measure P x on C([0, ∞); M ∂ ) concentrated at paths starting in x. (Usually we want to underline the dependency of X from its starting point x, whence we shall often write X x .) Recall that P x is the uniquely determined probability measure on C([0, ∞); M ∂ ) with (ev 0 ) ♯ P x = δ x (where ev 0 (γ) := γ 0 is the evaluation map at 0) and whose transition density is given by the function q :

Brownian motion
with the usual convention that inf ∅ := ∞. Since the Wiener measure is concentrated on paths having ∂ as a trap, for every p ∈ [1, ∞] and f ∈ L p (M ) one has If (Ω, F * , P) is filtered and X x adapted to the given filtration, then X x is called an adapted Brownian motion. In this case, X x is a semimartingale on M in the sense that for every f ∈ C ∞ (M ), the real-valued process f • X x is a semimartingale up to the explosion time ζ x . The stochastic parallel transport along X x w.r.t. ∇ started in x ∈ M is a process x constructed in terms of the horizontal lift of X x to the orthonormal frame bundle over M , making the linear map x t : Heat flow on 1-forms In the sequel, Borel equivalence classes of 1-forms on M having a certain regularity R are denoted by Γ R (T * M ), and similarly Γ R (T M ) for Borel equivalence classes of vector fields with regularity R. For instance, given p ∈ [1, ∞], we get the Banach space Γ L p (T * M ) given by all Borel equivalence classes ω of , by geodesic completeness this operator has a unique self-adjoint extension in the Hilbert space Γ L 2 (T * M ), which will be denoted with the same symbol again. Note our sign convention: ∆ is nonnegative, while ∆ is nonpositive. The heat semigroup ( P t ) t≥0 on 1-forms given by P t := e −t ∆/2 in Γ L 2 (T * M ) is smooth, in the sense for every ω ∈ Γ L 2 (T * M ) one has a jointly smooth representative P · ω which solves the heat equation on 1-forms with initial condition ω (and in [0, ∞) × M if ω is also smooth). On exact forms, P t can be represented by the heat operator P t ; more precisely, for every f ∈ W 1,2 (M ) one has [DT01,Li92] P t df = dP t f for every t ≥ 0. (2.4) If one drops geodesic completeness of M , such a commutation relation becomes subtle (cf. [Tha98] for a negative and [Gün17a] for a positive result in this direction). Lastly, a key result is the Feynman-Kac formula, for which we recall the process Q x from (1.2). Compare with Section A.1. Note that the last asserted estimate in the theorem follows from Gronwall's inequality, cf. e.g. (3.1) below. See also [EL94b,Mal74] for the compact case.
holds, and in particular where C t is defined in (1.1).

Remark 2.2.
On weighted Riemannian manifolds, in the notation of Section 1.3 one has to replace (

Stochastic completeness
A key tool for proving stochastic completeness under geodesic completeness, already used in [Bak86], are sequences of first-order cutoff-functions [Str83, Chapter 2]. Their existence is equivalent to the geodesic completeness of M ( [PS14], see also [Gün16]). Proof of (i) in Theorem 1.1. We show (2.3), i.e. that P t ½ M = ½ M for every t > 0. Let φ ∈ C ∞ c (M ), and let (ψ n ) n∈N be a sequence of first-order cutoff functions provided by Lemma 3.1. Then Theorem 2.1 applied to the 1-form ω := dψ n for every n ∈ N gives Since P · ψ n solves the heat equation (2.1), also using Fubini's theorem, integration by parts as well as the commutation rule (2.4) we arrive at Therefore, we obtain Since φ was arbitrary, this proves the claim.

Bismut-Elworthy-Li's derivative formula and the Lipschitz smoothing property
In view of proving Bismut-Elworthy-Li's derivative formula and the L ∞ -Lip-regularization property of (P t ) t≥0 , for convenience we state the following version of the Burkholder-Davis-Gundy inequality for q ∈ [1, ∞) (although we only need the upper bounds, respectively), which improves the classically known constants to better ones.  Recall the process Q x defined by (1.2) and taking values in T x M .
Proof of (ii) in Theorem 1.1. Fix x ∈ M , t > 0 and ξ ∈ T x M . It suffices to assume |ξ| ≤ 1. We first assume that f ∈ C ∞ c (M ). By [DT01, Proposition 3.2] and keeping in mind that ζ x = ∞ a.s., the process N x given by , is a local martingale. We show that under the given assumption (1.1) on k, this process is even a martingale. As already indicated in Theorem 2.1, given any s ≥ 0, it follows from Gronwall's inequality and Ric ≥ k on M that a.s. we have Hence, for every q ∈ [1, ∞), by Lemma 3.2 we obtain (This estimate will only be needed for q = 1 in this proof, but is derived for arbitrary q as above for later convenience.) Now, estimating |Q x r | as in (3.1) above and using the commutation relation (2.4) as well as Theorem 2.1, for all r ∈ [0, t] one a.s. has The first summand on the right-hand side is finite thanks to (1.1). Estimating the second summand by (3.2) above for q = 1, also the second summand is finite again by (1.1). It follows that N x is a true martingale, and thus The claimed equality for bounded f ∈ C ∞ (M ) follows by replacing f by ψ n f in (3.3) for every n ∈ N, where (ψ n ) n∈N is as in Lemma 3.1, and letting n → ∞ (together with the dominated convergence theorem on the right-hand side). In turn, if only f ∈ L ∞ (M ), a similar procedure works by replacing f by P ε f in (3.3), where ε > 0, and letting ε → 0.
Proof of (iii) in Theorem 1.1. Using the previous Bismut-Elworthy-Li formula and (3.2) above for q = 1, for every x ∈ M , t > 0 and ξ ∈ T x M with |ξ| ≤ 1, we get and duality gives Now we assume Kato decomposability of k in the rest of this chapter, devoting ourselves to the proof of Theorem 1.5. In this situation, one has to guarantee that the right-hand side of Bismut-Elworthy-Li's formula is well-defined for f ∈ L p (M ), where p ∈ (1, ∞), which is essentially the content of the following lemma.  Proof. Let V ∈ Γ L ∞ (T M ) and f ∈ L ∞ (M ), for which we assume without loss of generality that |V | L ∞ ≤ 1 and f L ∞ ≤ 1. The inequality (3.2) for q = 1 and Lemma 1.4 directly show the claimed integrability of f (X x t ) t 0 Q x s V (x), dW x s , and they also show that E V t is a bounded linear operator from L ∞ (M ) into L ∞ (M ). If p ∈ (1, ∞), successively using Hölder's inequality, (3.2) for q = p/(p − 1), Lemma 1.4 again and mass preservation of (P t ) t≥0 , we infer the existence of a finite constant C > 0 depending only on k − , t and p such that for every f ∈ L p (M ) ∩ L ∞ (M ), We conclude the statement by a standard approximation argument.
Proof of Theorem 1.5. Trivially, L ∞ (M ) ∩ L p (M ) is dense in L p (M ). Note that, given p ∈ (1, ∞), and f ∈ L p (M ), it follows from the divergence theorem as well as Lemma 3.3 -replacing ξ by an appropriate smooth and bounded vector field V ∈ Γ(T M ) such that V (x) = ξ -that both sides of (3.3) are continuous in f w.r.t. convergence in L p (M ). In particular, the desired pointwise identity follows.

Proof of Theorem 1.6
We turn to characterizations of continuous lower Ricci curvature bounds in terms of functional inequalities and existence of couplings. Throughout this chapter, we assume that k : M → R is continuous, and only state explicitly if we need (1.1).

From the L 1 -Bochner inequality to lower Ricci bounds
Proof of "(ii) =⇒ (i)" in Theorem 1.6. Let x ∈ M , and let ξ ∈ T x M obey |ξ| = 1. In the notation from Lemma 4.1, consider the function f := ρ ± H provided therein. By Lemma 4.1, Bochner's formula (1.3) and the chain rule for ∆, we have The arbitrariness of ξ concludes the proof.

Remark 4.2.
In the weighted setting outlined in Section 1.3 -retaining the notation therein -the only essential change needed to modify the previous proof is to replace the unweighted Bochner identity (1.3) by its weighted counterpart where ∆ Φ := ∆ − 2 ∇Φ, ∇· and Ric Φ := Ric + 2 Hess Φ. The latter follows from (1.3), the definition of Hess Φ and metric compatibility of ∇, see e.g. page 28 in [Pet06]: The chain rule for ∆ Φ is analogous to the one for ∆.

From lower Ricci bounds to pathwise couplings
We start with the existence of a suitable coupling of Brownian motions under the inequality Ric ≥ k on M , also assuming (1.1) in this section. (Note that the stochastic completeness of M is already known by Theorem 1.1.) The coupling technique is wellknown and called coupling by parallel displacement, see [Cra91,Ken86,Wan05,Wan14] and the references therein. See also [Wan94] for a "local" treatise on regular subdomains. We first collect some notation. Denote by cut The construction of this coupling is thoroughly carried out in Theorem 2.1.1 and (the proof of) Proposition 2.5.1 in [Wan05], see also [Wan14, Theorem 2.3.2]. The key to deduce "(i) =⇒ (iii)" in Theorem 1.6 from Theorem 4.3 now is to construct an appropriate function I ′ ∈ C(M ×M ) with I ′ ≥ I outside diag ∪cut, hence circumventing cut-locus issues. This is the place where the definition of k enters. (4.1) As k is lower-semicontinuous, a well-known consequence of Baire's theorem yields the existence of a pointwise increasing sequence (k n ) n∈N in C(M × M ) converging pointwise to k. Applying Theorem 4.3 with I ′ replaced by I ′ n ∈ C(M × M ) given by I ′ n (u, v) := −ρ(u, v) k n (u, v) and integrating the resulting differential inequality, a.s. we have for every s, t ≥ 0 with s ≤ t and for every n ∈ N. (Recall that X x and X y coincide past their coupling time.) Letting n → ∞ with the aid of the monotone convergence theorem, we obtain the desired pathwise estimate.

Remark 4.4.
In the weighted case from Section 1.3, the quantity I has to be replaced by its weighted counterpart

From pathwise couplings to the L 1 -Bochner inequality
Even if k is smooth, as implicitly discovered in Section 4.2 above, the function k from (1.4) is in general only lower semicontinuous. In the current section, we shall need to bypass this lack of continuity by approximation through Lipschitz functions. To this aim, the following fact, in which Lipschitz continuity on M × M is understood w.r.t. the product metric ρ 2 given by ρ 2 2 (x, y), (x ′ , y ′ ) := ρ 2 (x, x ′ ) + ρ 2 (y, y ′ ), is helpful. A purely metric construction of the claimed sequence can be found e.g. at page 107 in [AGS05], replacing (X, d) by (D × D, ρ 2 ) therein. In particular, note that since k(x) = k(x, x) for every x ∈ M , in D, k is the pointwise limit of a pointwise increasing sequence of functions in Lip b (M ) which are everywhere not smaller than inf k(D).
The step from the pathwise coupling property w.r.t. k towards (1.5) requires a nontrivial extension of the arguments for [BHS21, Theorem 5.17] (which adapt the duality argument from [Kuw10] to the case of synthetic variable Ricci bounds and make crucial use of uniform lower boundedness of the Ricci curvature) for short times instead of fixed ones. This kind of localization argument was indeed used in [BHS21] in different variants at different instances. For this, a certain short-time behavior of Brownian motion as subsequently recorded plays a crucial role.
Given any x ∈ M and ε > 0, let τ x ε be the first exit time of Brownian motion starting in a fixed x ∈ M from B ε (x). The following estimate for τ x ε is a variant of [Wan14, Lemma 2.1.4], noting that by Laplacian comparison, compare with [Hsu02a, Corollary 3.4.4, Corollary 3.4.5, Theorem 3.6.1], the constant c 1 therein can be chosen uniformly in x off its respective cut-locus as long as long as x belongs to a compact subset of M . (An analogous version of Lemma 4.6 holds for general gradient diffusions, taking -in the notation of Section 1.3 -into account the continuity of ∇Φ.) Lemma 4.6. For every compact D ⊂ M and every ε > 0, there exists a constant c > 0 such that P τ x ε ≤ t ≤ e −c/t for every x ∈ D, t ∈ (0, 1]. Proof of "(iii) =⇒ (ii)" in Theorem 1.6.
Step 1. Initial preparations. Let f ∈ C ∞ c (M ) and x ∈ M with |∇f (x)| = 0 be arbitrary. Let ε ∈ (0, 1/4], which is kept fixed throughout this proof, be such that |∇f | is bounded away from zero -in particular smooth -on B 4ε (x). Moreover, let γ be the unique geodesic starting in x with initial velocityγ 0 = ∇f (x)/|∇f (x)|. The continuity of k yields k ≥ K on B 6 (x) for some negative real number K. Define the set of points in M with distance at most 1 to γ by By the definition (1.4) of k and since ρ(x, γ s ) ≤ 1 for every s ∈ [0, 1], we have (4.2) Finally, let ℓ ∈ Lip b (M × M ) be any function obeying K ≤ ℓ ≤ k on D × D as provided by Lemma 4.5.
Step 2. Rewriting the quantities to consider. The key idea to derive the L 1 -Bochner inequality (1.5) for f from the given pathwise coupling estimates is to consider certain difference quotients of the map (t, s) → P t f (γ s ) near (0, 0), and to express the involved heat semigroups in terms of coupled Brownian motions. To address the first point, we note that, given t > 0, by the smoothness of s → P t f (γ s ) on [0, ∞), Taylor's theorem in its mean value remainder form and the geodesic equation for γ, given any s > 0 there exists ν ∈ [0, s] such that Hess P t f (γ ν ) γ ν ,γ ν .
Dividing by s, subtracting ∇f (x),γ 0 and dividing by t, respectively, yields To now invoke the coupled Brownian motions, given s ∈ (0, 1] let us denote by (X x , X γs ) a process starting in (x, γ s ) given by the pathwise coupling property w.r.t. k. Let τ x ε and τ γs ε denote the first exit times of the marginal Brownian motions X x and X γs from B ε (x) and B ε (γ s ), respectively. Since B ε (x), B ε (γ s ) ⊂ D, for every s ∈ [0, 1] a.s. on the event τ x ε > t and τ γs ε > t we have Step 3. Estimating (4.3) via coupled Brownian motions. Given s ∈ 0, e −1/2 , define t s := −c/log s 2 ∈ (0, c], where c > 0 is the constant from Lemma 4.6 associated to D and ε. Note that t s → 0 and s α /t s → 0 as s → 0 for α ∈ {1/2, 1}. Consider the events A s := τ x ε > t s and τ γs ε > t s , Now we estimate the contributions of the events defined in (4.5) separately.
Step 3.1. The contribution of A c s becomes negligible thanks to lim sup where the last inequality is granted by Lemma 4.6.
Step 3.2. Furthermore, by (4.4) and (4.2), the contribution of V s is controlled by Step 3.3. In a similar way, we can ignore the influence of W s by lim sup Step 3.4. Finally we turn to the most delicate part, namely the study of the effect of U s . To this aim, we first note that, defining the function ℓ ∈ Lip b (M ) by ℓ( Together with (4.4) and since ρ(X x ts , X γs ts ) < s 1/2 on A s ∩ V c s , we thus obtain lim sup For small enough s > 0, on the event A s we have B 2s 1/2 (X x ts ) ⊂ B 2ε (x). In this case, by applying the mean value theorem twice, Let ψ ∈ C ∞ c (M ) be nonnegative with ψ = |∇f | on B 2ε (x). Invoking the dominated convergence theorem, (2.2) and the smoothness of the heat semigroup up to zero, the terms containing s 1/2 above become negligible as s ↓ 0, and we are left with lim sup where in second last identity, we used that the marginal law of X x is independent of s.

A Kato decomposable lower Ricci bounds and their Schrödinger semigroups
A.
Proof. If k obeys Ric ≥ k on M , then the claimed L 1 -gradient estimate is just a restatement of Theorem 2.1 for exact 1-forms together with (2.4).
Conversely, assume the L 1 -gradient estimate. A similar argument as in the proof of (i) in Theorem 1.1 in Section 3.1 -directly employing (A.1) instead of Theorem 2.1shows that M is stochastically complete. Let f ∈ C ∞ c (M ) and x ∈ M with |∇f (x)| = 0. Let ε > 0 such that |∇f | is bounded away from zero -in particular smooth -on B ε (x). By Kato's inequality for the Bochner Laplacian [HSU80, Proposition 2.2], we have |∇f | ∈ W 1,2 (M ). Thus, given a nonnegative φ ∈ C ∞ c (M ) with support in B ε (x), by the chain rule, (2.2) and a standard representation of the quadratic form E in terms of (P t ) t≥0 , see e.g. page 15 in [Dav89], It remains to estimate the latter limit. Let τ y ε be the first exit time of X y from B ε (y). Since k is bounded on the bounded set y∈Bε(x) B ε (y), and by continuity of Brownian sample paths, the dominated convergence theorem gives lim sup Using the Cauchy-Schwarz inequality, Lemma 1.4 and Lemma 4.6, for arbitrary T > 0 we obtain lim sup  [Gün12] in a more general context, and of course it also follows from Theorem 2.1 by approximating forms in Γ L 2 (T * M ) by elements of Γ C ∞ c (T * M ) using Lemma A.4.) In view of the Cauchy-Schwarz inequality it seems unlikely that the Feynman-Kac formula on 1-forms holds for all square integrable 1forms under the weaker assumption (1.1), although we are not aware of a counterexample (which would be interesting to have). We refer the reader also to the recent [BG21], where Feynman-Kac formulas for general perturbations of order no larger than 1rather than just zeroth order perturbations -of Bochner Laplacians on vector bundles have been treated. Remark A.3. Somewhat in line with the previous remark, assume that k satisfies (1.1) instead of Kato decomposability. Of course, if Ric ≥ k on M , the L 1 -gradient estimate from Theorem A.1 then still holds by virtue of Theorem 2.1. However, as it becomes apparent from the above proof, the converse implication seems to be more involved and to require at least some higher order exponential integrability of k − .

A.2 Schrödinger semigroups
For Kato decomposable k, the right-hand side of (A.1) has a more analytic interpretation in terms of the Schrödinger semigroup associated to k, which is briefly discussed now. Assume in this section that k is a (not necessarily continuous) function which is Kato decomposable and in L 2 loc (M ). Then ∆ − k is essentially self-adjont in L 2 (M ) [Gün17b], and the Schrödinger semigroup (P k t ) t≥0 is defined to be P k t := e t(∆−k)/2 via spectral calculus. This is a strongly continuous semigroup of bounded linear operators in L 2 (M ). As k is Kato decomposable, (P k t ) t≥0 has a pointwise well-defined version which, for every f ∈ L 2 (M ), can be expressed [Gün17a] via Brownian motion X x on M in terms of Up to dealing with the small additional difficulty of stochastic incompleteness, the proof given in [GP16] follows from a careful examination of the classical Euclidean proof given in [AS82] (which however states a bound of the form C 1 e C 2 t ). The above stronger bound has played a crucial role in the context of total variation considered in [GP16]. Analogous statements for general Hunt processes having the Feller property can also be found in [DvC00]. the dominated convergence theorem shows that the pointwise limit P k t f of (P k t f n ) n∈N as n → ∞ is well-defined. This definition does not depend on the choice of (f n ) n∈N as long as sup n∈N f n L ∞ < ∞. It is also clear that this procedure preserves (A.4).
To show strong continuity of (P k t ) t≥0 in L p (M ) for p < ∞, by approximation and (A.4), it suffices to show continuity of t → P k t f on [0, ∞) in L p (M ) for f ∈ L 2 (M ) ∩ L p (M ) ∩ L ∞ (M ). By the semigroup property, we may and will restrict ourselves to the proof of continuity at t = 0. Given any x ∈ M , note that a.s., we have t 0 k(X x r ) dr → 0 as t ↓ 0 since k ∈ L 2 loc (M ), and that e − t 0 k(X x r )/2 dr − 1 ≤ e T 0 k − (X x r )/2 dr + 1 (A.5) Example A.7. Assume that M is a model manifold in the sense of [Gri09], meaning that M = R d as a manifold with d ≥ 2, and that the Riemannian metric ·, · is given in polar coordinates as dr 2 + ψ(r) dθ 2 , where ψ is a smooth positive function on (0, ∞). The volume of balls on such manifolds does not depend on the center, and the Ricci curvature behaves in the radial direction like ψ ′′ /ψ − (d − 1)(ψ ′ ) 2 /ψ 2 , see e.g. page 266 in [Bes87]. Assume now ψ ′′ /ψ − (d − 1)(ψ ′ ) 2 /ψ 2 − ∈ L p ψ d−1 ((0, ∞)) + L ∞ ((0, ∞)) for some p > d/2, where L p ψ d−1 ((0, ∞)) is the L p -space of functions w.r.t. ½ (0,∞) ψ d−1 L 1 . As the volume measure behaves in the radial direction as ψ d−1 (r) dr, it follows that the Ricci curvature is bounded from below by a function with negative part in L p (M ) + L ∞ (M ).
Indeed, a. guarantees that there exists a complete metric g 0 on M which -in polar coordinates -is written as g 0 = dr 2 + ψ 0 (r) dθ 2 . Assumption b. guarantees that the Ricci curvature associated to g 0 is bounded from below by a constant, and c. implies that g is quasi-isometric to g 0 . For instance, one can take the Euclidean metric corresponding to ψ 0 (r) := r or the hyperbolic metric corresponding to ψ 0 (r) = sinh(r) as reference metrics.