Second order differentiation formula on $RCD^*(K,N)$ spaces

Aim of this paper is to prove the second order differentiation formula for $H^{2,2}$ functions along geodesics in $RCD^*(K,N)$ spaces with $N<\infty$. This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity. We establish this result by showing that $W_2$-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolation. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain: - equiboundedness of the densities along the entropic interpolations, - local equi-Lipschitz continuity of the Schr\"odinger potentials, - a uniform weighted $L^2$ control of the Hessian of such potentials. Finally, the techniques adopted in this paper can be used to show that in the $RCD$ setting the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality.

An example of link between Sobolev calculus and W 2 -geometry is the following result (a minor variant of a statement in [24]). It says that we can safely take one derivative of a W 1,2 (X) function along an optimal geodesic test plan π, i.e. a test plan satisfying 1 0 |γ t | 2 dt dπ(γ) = W 2 2 (e 0 ) * π, (e 1 ) * π .
Recall that on RCD(K, ∞) spaces every W 2 -geodesic (µ t ) between measures with bounded density and support is such that µ t ≤ Cm for every t ∈ [0, 1] and some C > 0 ( [49]), so that between two such measures there always exists a (unique) optimal geodesic test plan with bounded support. Thus the theorem also says that we can find 'many' C 1 functions on RCD spaces. We remark that such C 1 regularity -which was crucial in [24] -is non-trivial even if the function f is assumed to be Lipschitz and that statements about C 1 smoothness are quite rare in metric geometry.
Furthermore, projecting from π to µ t := (e t ) * π one can see that Theorem 1.1 immediately implies d dtˆf dµ t =ˆ ∇f, ∇φ t dµ t (1.1) and one might think of this identity as an 'integrated' version of the basic formula valid in the smooth framework; at the technical level the proof of the claim has to do with the fact that the geodesic (µ t ) solves the continuity equation d dt µ t + div(∇ϕ t µ t ) = 0, (1.2) where the ϕ t 's are appropriate choices of Kantorovich potentials (see also [28] in this direction), and with the fact that ∇ϕ t = ∇φ t (see Lemma A.7 below).
In [25], the first author developed a second-order calculus on RCD spaces, in particular defining the space H 2,2 (X) and for f ∈ H 2,2 (X) the Hessian Hess(f ), see [25] and the Appendix. It is then natural to ask whether an 'integrated' version of the second order differentiation formula for γ geodesic holds in this framework. In this paper we provide affirmative answer to this question, our main result being: Theorem 1.2 (Second order differentiation formula). Let (X, d, m) be an RCD * (K, N ) space, N < ∞, π an optimal geodesic test plan with bounded support and f ∈ H 2,2 (X).
Let us comment about the assumptions in Theorem 1.2 and Theorem 1.3: -The first order differentiation formula is valid on general RCD(K, ∞) spaces, while for the second order one we need to assume finite dimensionality. This is due to the strategy of our proof, which among other things uses the Li-Yau inequality.
-There exist optimal geodesic test plans without bounded support (if K = 0 or the densities of the initial and final marginals decay sufficiently fast) but in this case the functions φ t appearing in the statement(s) are not Lipschitz. As such it seems hard to have Hess(h)(∇φ t , ∇φ t ) • e t ∈ L 1 (π) and thus we cannot really hope for anything like (1.3), (1.4) to hold: this explains the need of the assumption on bounded supports.
Having at disposal such second order differentiation formula is interesting not only at the theoretical level, but also for applications to the study of the geometry of RCD spaces. For instance, the proofs of both the splitting theorem [24] and of the 'volume cone implies metric cone' [19] in this setting can be greatly simplified by using such formula (in this direction, see [58] for comments about the splitting). Also, one aspect of the theory of RCD spaces which is not yet clear is whether they have constant dimension: for Ricci-limit spaces this is known to be true by a result of Colding-Naber [18] which uses second order derivatives along geodesics in a crucial way. Thus our result is necessary to replicate Colding-Naber argument in the non-smooth setting (but not sufficient: they also use a calculus with Jacobi fields which as of today does not have a non-smooth counterpart).
Let us discuss the strategy of the proof. Our starting point is a related second order differentiation formula obtained in [25], available under proper regularity assumptions: Theorem 1.4. Let (µ t ) be a W 2 -absolutely continuous curve solving the continuity equation d dt µ t + div(X t µ t ) = 0, for some vector fields (X t ) ⊂ L 2 (T X) in the following sense: for every f ∈ W 1,2 (X) the map t →´f dµ t is absolutely continuous and it holds d dtˆf dµ t =ˆ ∇f, X t dµ t .
Assume that (i) t → X t ∈ L 2 (T X) is absolutely continuous, Then for f ∈ H 2,2 (X) the map t →´f dµ t is C 1,1 and the formula holds for a.e. t ∈ [0, 1].
If the vector fields X t are of gradient type, so that X t = ∇φ t for every t and the 'acceleration' a t is defined as d dt φ t + |∇φ t | 2 2 =: a t then (1.5) reads as d 2 dt 2ˆf dµ t =ˆHess(f )(∇φ t , ∇φ t ) dµ t +ˆ ∇f, ∇a t dµ t .
(1. 6) In the case of geodesics, the functions ϕ t appearing in (1.2) solve (in a sense which we will not make precise here) the Hamilton-Jacobi equation thus in this case the acceleration a t is identically 0. Hence if the vector fields (∇ϕ t ) satisfy the regularity requirements (i), (ii) in the last theorem we would easily be able to establish Theorem 1.2. However in general this is not the case; informally speaking this has to do with the fact that for solutions of the Hamilton-Jacobi equations we do not have sufficiently strong second order estimates. In order to establish Theorem 1.2 it is therefore natural to look for suitable 'smooth' approximation of geodesics for which we can apply Theorem 1.4 above and then pass to the limit in formula (1.5). Given that the lack of smoothness of W 2 -geodesic is related to the lack of smoothness of solutions of (1.7), also in line with the classical theory of viscous approximation for the Hamilton-Jacobi equation there is a quite natural thing to try: solve, for ε > 0, the equation where ϕ is a given, fixed, Kantorovich potential for the geodesic (µ t ), and then solve d dt µ ε t − div(∇ϕ ε t µ ε t ) = 0, µ ε 0 := µ 0 .
This plan can actually be pursued and following the ideas in this paper one can show that if the space (X, d, m) is RCD * (K, N ) and the geodesic (µ t ) is made of measures with equibounded densities, then as ε ↓ 0: i) the curves (µ ε t ) W 2 -uniformly converge to the geodesic (µ t ) and the measures µ ε t have equibounded densities.
ii) the functions ϕ ε t are equi-Lipschitz and converge both uniformly and in the W 1,2topology to the only viscous solution (ϕ t ) of (1.7) with ϕ as initial datum; in particular the continuity equation (1.2) for the limit curve holds.
These convergence results are based on Hamilton's gradient estimates and the Li-Yau inequality and are sufficient to pass to the limit in the term with the Hessian in (1.6). For these curves the acceleration is given by a ε t = − ε 2 ∆ϕ ε t and thus we are left to prove that the quantity εˆ ∇f, ∇∆ϕ ε t dµ ε t goes to 0 in some sense. However, there appears to be no hope of obtaining this by PDE estimates. The problem is that this kind of viscous approximation can produce in the limit a curve which is not a geodesic if ϕ is not c-concave: shortly said, this happens as soon as a shock appears in Hamilton-Jacobi. Since there is no hope for formula (1.4) to be true for non-geodesics, we see that there is little chance of obtaining it via such viscous approximation.
We therefore use another way of approximating geodesics: the slowing down of entropic interpolations. Let us briefly describe what this is in the familiar Euclidean setting.
Fix two probability measures µ 0 = ρ 0 L d , µ 1 = ρ 1 L d on R d . The Schrödinger functional equations are the unknown being the Borel functions f, g : where h t f is the heat flow starting at f evaluated at time t. It turns out that in great generality these equations admit a solution which is unique up to the trivial transformation (f, g) → (cf, g/c) for some constant c > 0. Such solution can be found in the following way: let R be the measure on (R d ) 2 whose density w.r.t. L 2d is given by the heat kernel r t (x, y) at time t = 1 and minimize the Boltzmann-Shannon entropy H(γ | R) among all transport plans γ from µ 0 to µ 1 . The Euler equation for the minimizer forces it to be of the form f ⊗ g R for some Borel functions f, g : where f ⊗ g(x, y) := f (x)g(y) (we shall reprove this known result in Proposition 2.1). Then the fact that f ⊗ g R is a transport plan from µ 0 to µ 1 is equivalent to (f, g) solving (1.8).
Once we have found the solution of (1.8) we can use it in conjunction with the heat flow to interpolate from ρ 0 to ρ 1 by defining This is called entropic interpolation. Now we slow down the heat flow: fix ε > 0 and by mimicking the above find f ε , g ε such that (the factor 1/2 plays no special role, but is convenient in computations). Then define The remarkable and non-trivial fact here is that as ε ↓ 0 the curves of measures (ρ ε t L d ) converge to the W 2 -geodesic from µ 0 to µ 1 .
The first connections between Schrödinger equations and optimal transport have been obtained by Mikami in [45] for the quadratic cost on R d ; later Mikami-Thieullen [46] showed that a link persists even for more general cost functions. The statement we have just made about convergence of entropic interpolations to displacement ones has been proved by Léonard in [41]. Actually, Léonard worked in much higher generality: as it is perhaps clear from the presentation, the construction of entropic interpolation can be done in great generality, as only a heat kernel is needed. He also provided a basic intuition about why such convergence is in place: the basic idea is that if the heat kernel admits the asymptotic expansion ε log r ε (x, y) ∼ − d 2 (x,y) 2 (in the sense of Large Deviations), then the rescaled entropy functionals εH(· | R ε ) converge to 1 2´d 2 (x, y) d· (in the sense of Γ-convergence). We refer to [43] for a deeper discussion of this topic, historical remarks and much more.
Starting from these intuitions and results, working in the setting of RCD * (K, N ) spaces we gain new information about the convergence of entropic interpolations to displacement ones. In order to state our results, it is convenient to introduce the Schrödinger potentials ϕ ε t , ψ ε t as ϕ ε t := ε log h tε/2 f ε ψ ε t := ε log h (1−t)ε/2 g ε . In the limit ε ↓ 0 these will converge to forward and backward Kantorovich potentials along the limit geodesic (µ t ) (see below). In this direction, it is worth to notice that while for ε > 0 there is a tight link between potentials and densities, as we trivially have in the limit this becomes the well known (weaker) relation that is in place between forward/backward Kantorovich potentials and measures (µ t ): see e.g. Remark 7.37 in [59] (paying attention to the different sign convention). By direct computation one can verify that (ϕ ε t ), (ψ ε t ) solve the Hamilton-Jacobi-Bellman equations d dt thus introducing the functions With this said, our main results about entropic interpolations can be summarized as follows.
-convergence For every sequence ε n ↓ 0 there is a subsequence -not relabeledsuch that for any t ∈ (0, 1] the functions ϕ ε t converge both locally uniformly and in W 1,2 loc (X) to a function ϕ t such that −tϕ t is a Kantorovich potential from µ t to µ 0 . Similarly for the ψ's.
-Second order For every δ ∈ (0, 1/2) we have (1.11) Notice that since in general the Laplacian is not the trace of the Hessian, there is no direct link between these two bounds.
With the exception of the convergence ρ ε t m → µ t , all these results are new even on smooth manifolds (in fact, even on R d ).
The zeroth and first order bounds are both consequences of the Hamilton-Jacobi-Bellman equations (1.9) satisfied by the ϕ's and ψ's and can be obtained from Hamilton's gradient estimate and the Li-Yau inequality. The facts that the limit curve is the W 2 -geodesic and that the limit potentials are Kantorovich potentials are consequence of the fact that we can pass to the limit in the continuity equation (1.10) and that the limit potentials satisfy the Hamilton-Jacobi equation. In this regard it is key that we approximate at the same time both the 'forward' potentials ψ and the 'backward' one ϕ: see the proof of Proposition 5.4 and recall that the simple viscous approximation may converge to curves which are not W 2 -geodesics.
Notice that these zeroth and first order convergences are sufficient to pass to the limit in the term with the Hessian in (1.6). As said, also the viscous approximation could produce the same kind of convergence.
The crucial advantage of dealing with entropic interpolations (which has no counterpart in viscous approximation) is thus in the second order bounds and convergence results which show that the term with the acceleration in (1.6) vanishes in the limit and thus eventually allows us to prove our main result Theorem 1.2. In this direction, we informally point out that being the geodesic equation a second order one, in searching for an approximation procedure it is natural to look for one producing some sort of second order convergence.
The limiting property (1.12) is mostly a consequence -although perhaps non-trivial -of the bound (1.11) (see in particular Lemma 4.10 and the proof of Theorem 5.12), thus let us focus on how to get (1.11). The starting point here is a formula due to Léonard [39], who realized that there is a connection between entropic interpolation and lower Ricci bounds: he computed the second order derivative of the entropy along entropic interpolations and in this direction our contribution has been the rigorous proof in the RCD framework of his formal computations, thus getting where Γ 2 is the 'iterated carré du champ' operator defined as (in the setting of RCD spaces some care is needed when handling this object, but let us neglect this issue here). (1.14) If we assume for simplicity that K = 0 we have Γ 2 ≥ 0, so that (1.13) tells in particular that t → H(µ ε t | m) is convex for any ε > 0, and if we also assume that m(X) = 1 such function is non-negative. Therefore (1.14) gives that for any δ ∈ (0, 1/2) it holds Recalling the Bochner inequalities ( [20], [9], [25]) we see that (1.11) follows from (1.15). Then with some work (see Lemma 4.10 and Theorem 5.12 for the details) starting from (1.11) we can deduce (1.12) which in turn ensures that the term with the acceleration in (1.6) vanishes in the limit ε ↓ 0, thus leading to our main result Theorem 1.2.

The Schrödinger problem
Let (X, τ ) be a Polish space, µ 0 , µ 1 ∈ P(X) and R be a non-negative Radon measure on X 2 . Recall that γ ∈ P(X 2 ) is called transport plan for µ 0 , µ 1 provided π 0 * γ = µ 0 and π 1 * γ = µ 1 , where π 0 , π 1 : X 2 → X are the canonical projections. We are interested in finding a transport plan of the form . As we shall see in this short section, in great generality this problem can be solved in a unique way and the plan γ can be found as the minimum of among all transport plans from µ 0 to µ 1 , where H( · | ·) is the Boltzmann-Shannon entropy. For appropriate choice of the reference measure R (which will also be our choice in the following), this minimization problem is called Schrödinger problem, we refer to [43] for a survey on the topic.
Let us first recall the definition of the relative entropy functional in the case of a reference measure with possibly infinite mass (see [42] for more details). Given a σ-finite measure ν on a Polish space (Y, τ ′ ), there exists a measurable function W : Y → [0, ∞) such that Introducing the probability measure ν W := z −1 W e −W ν, for any σ ∈ P(Y) such that´W dσ < +∞ the Boltzmann-Shannon entropy is defined as for allν ∈ P(Y); notice that Jensen's inequality and the fact thatν ∈ P(Y) grant that ρ log(ρ) dν is well defined and non-negative, in particular the definition makes sense. The definition is meaningful, because if´W ′ dσ < +∞ for another function W ′ such that z W ′ < +∞, then Hence H( · | ν) is well defined for all σ ∈ P(Y) such that´W dσ < +∞ for some non-negative measurable function W with z W < +∞.
The following proposition collects the basic properties of the minimizer of the Schrödinger problem; we emphasize that point (i) of the statement is already known in the literature on the subject (see in particular [40], [11] and [50]) and there are similarities between point (ii) and some results in [11]. A complete proof has already been presented in [33] for the compact case; here we adapt the arguments to our more general case. Notice that Radon measures on Polish spaces are always σ-finite, hence the above discussion about the Boltzmann-Shannon entropy applies. Proposition 2.1. Let (X, τ, m) be a Polish space equipped with a non-negative Radon measure m and let R be a non-negative Radon measure on X 2 such that π 0 Let µ 0 = ρ 0 m and µ 1 = ρ 1 m be Borel probability measures and assume that there exists a Borel function B : X → [0, ∞) such that Then the following holds. Then: i-a) There exists a unique minimizer γ of H( · | R) among all transport plans from µ 0 to µ 1 .
ii-b) The functions f, g given by point ) and γ is the only transport plan which can be written as proof (i-a) Existence follows by the direct method of calculus of variations: the class of transport plans is not empty, narrowly compact (see e.g. [4]) and H( · | R) is well defined therein; indeed by assumption´W dσ < +∞ with W (x, y) := B(x) + B(y) for all transport plan σ. Moreover by (2.1) we have that so that H( · | R) is narrowly lower semicontinuous on the class of transport plans.
Since H( · | R) is strictly convex, uniqueness is equivalent to the existence of a γ ∈ Adm(µ 0 , µ 1 ) with finite entropy w.r.t. R and by (2.3) we conclude.
Let us now pick h ∈ L ∞ (X 2 , γ) such that π 0 * (hγ) = π 1 * (hγ) = 0 and ε ∈ (0, h −1 L ∞ (γ) ). Then (1 + εh)γ is a transport plan from µ 0 to µ 1 and noticing that hp is well defined R-a.e. we have so that u((1 + εh)p) ∈ L 1 (R). Then again by the monotone convergence theorem we get By the minimality of γ we know that the left-hand side in this last identity is non-negative, thus after running the same computation with −h in place of h and noticing that the choice of h grants that´hp dR =´h dγ = 0 we obtain hp log(p) dR = 0 ∀h ∈ L ∞ (γ) such that π 0 * (hγ) = π 1 * (hγ) = 0. (2.7) The rest of the argument is better understood by introducing the spaces V, ⊥ W ⊂ L 1 (γ) and V ⊥ , W ⊂ L ∞ (γ) as follows where here and in the following the function ϕ ⊕ ψ is defined as ϕ ⊕ ψ(x, y) := ϕ(x) + ψ(y). Notice that the Euler equation (2.7) reads as log(p) ∈ ⊥ W and our thesis as log(p) ∈ V ; hence to conclude it is sufficient to show that ⊥ W ⊂ V . Claim 1: V is a closed subspace of L 1 (γ).
We start claiming that f ∈ V if and only if f ∈ L 1 (γ) and Indeed the 'only if' follows trivially from γ ≪ m ⊗ m and the definition of V . For the 'if' we apply Fubini's theorem to get the existence of x ′ ∈ P 0 and y ′ ∈ P 1 such that e. x, y ∈ P 0 × P 1 .
Let h ∈ L ∞ (γ) \ W , so that either the first or second marginal of hγ is non-zero. Say the first. Thus since π 0 * γ = µ 0 we have π 0 Then the function f := f 0 ⊕ 0 = f 0 • π 0 belongs to V and we havê Let f ∈ L 1 (γ) \ V , use the fact that V is closed and the Hahn-Banach theorem to find h ∈ L ∞ (γ) ∼ L 1 (γ) * such that´f h dγ = 0 and´f h dγ = 0 for everyf ∈ V . Thus h ∈ V ⊥ and hence by the previous step h ∈ W . The fact that´f h dγ = 0 shows that f / ∈ ⊥ W , as desired.
(ii-a) The bounds (2.2) and (2.4) ) is finite, hence the claim follows by direct computations: (ii-b) Then let σ be a transport plan from µ 0 to µ 1 such that σ = f ′ ⊗ g ′ R for suitable non-negative Borel functions f ′ , g ′ . We claim that in this case it holds f ′ , g ′ ∈ L ∞ (m), leading in particular to the claim in the statement about γ. By disintegrating R w.r.t. π 0 , from π 0 * (f ′ ⊗ g ′ R) = ρ 0 m and R 0 = m we get that x. Notice then that the sets where f ′ and g ′ are positive must coincide with P 0 and P 1 respectively, up to m-negligible sets, so that nothing changes in (2.9) if we restrict the integral to P 1 . Moreover, since from (2.4) we have that R x ≥ cm in P 1 for m-a.e. x ∈ P 0 , we see that g ′ ∈ L 1 (m) with c g ′ L 1 (m) ≤ˆg ′ (y) dR x (y) for m-a.e. x ∈ P 0 and thus (2.9) yields , m-a.e. in P 0 , which is the first in (2.5), because in X \ P 0 we already know that f ′ vanishes m-a.e. By interchanging the roles of f ′ and g ′ , the same conclusion follows for g ′ .
For the uniqueness of γ, put ϕ := log f ′ , ψ := log g ′ and notice that, by what we have just proved, they are bounded from above. On the other hand because, as already remarked in the proof of (i), (2.2) implies that H(· | R) is well defined on Adm(µ 0 , µ 1 ). From these two facts we infer that ϕ • π 0 , ψ • π 1 ∈ L 1 (σ). (2.10) Putting for brevity p ′ := f ′ ⊗ g ′ and arguing as before to justify the passage to the limit inside the integral we get (because σ and γ have the same marginals) = 0.
This equality and the convexity of H( · | R) yield H(σ | R) ≤ H(γ | R) and being γ the unique minimum of H( · | R) among transport plans from µ 0 to µ 1 , we conclude that σ = γ.
The above result is valid in the very general framework of Polish spaces. We shall now restate it in the form we shall need in the context of RCD spaces.
Recall that on a finite-dimensional RCD * (K, N ) space (X, d, m), m satisfies the volume growth condition (A. 20), so that we can choose W = d 2 (·,x) for anyx ∈ X in (2.1). Setting z :=´e −d 2 (·,x) dm andm and this shows that H( · | m) is well defined on P 2 (X) and W 2 -lower semincontinuous. Let us also remind that on RCD spaces there is a well defined heat kernel r ε [x](y) (see (A.3) and (A.4)). The choice of working with r ε/2 in the following statement is convenient for the computations we will do later on.
Then there exist and are uniquely m-a.e. determined (up to multiplicative constants) two Borel non-negative functions f ε , g ε : X → [0, ∞) such that f ε ⊗g ε R ε/2 is a transport plan from µ 0 to µ 1 . In addition, f ε , g ε belong to L ∞ (X) and their supports are included in supp(µ 0 ) and supp(µ 1 ) respectively.
proof Start observing that R ε/2 0 = R ε/2 1 = m and if we set B := d 2 (·,x) withx ∈ X arbitrarily chosen, then the second and third in (2.2) are authomatically satisfied; for the first one notice thatˆX is a probability measure and recall (A.20). Hence Proposition 2.1, the fact that the Gaussian estimates (A.5) on the heat kernel grant that there are constants 0 < c ε ≤ C ε < +∞ such that in P 0 × P 1 and the fact that f ε ⊗ g ε R ε/2 is a transport plan from µ 0 to µ 1 provide us with the conclusion.

Hamilton's and Li-Yau's estimates
Here we recall Hamilton's gradient estimate and Li-Yau Laplacian estimates for log h t u, where u is a non-negative function.
Let us start with the following result, which we shall frequently use later on without explicit mention: and u 0 ∈ L 2 ∩ L ∞ (X) be non-negative and not identically zero. Put u t := h t u 0 .
Then log u t ∈ Test ∞ loc (X). proof By (A.7) u t ∈ Test ∞ (X) and by (A.5) u t is locally bounded away from 0. Taking into account the fact that log is smooth on (0, ∞), the conclusion easily follows from (A.6).
We now recall Hamilton's gradient estimate on RCD(K, ∞) spaces, which is known to be true from [37]: proof In [37] this result has been stated for proper RCD(K, ∞) spaces; still, the assumption that bounded sets are relatively compact is never used so that the proof works in general RCD spaces. We remark that in [37] the authors refer to [27], [3], [6] and [51] for the various calculus rules and that in these latter references no properness assumption is made.
In the finite-dimensional case, thanks to the Gaussian estimates for the heat kernel we can easily obtain a bound independent of the L ∞ norm of the initial datum: Then there is a constant C depending on K, N only such that for any u 0 ∈ L 1 (X) non-negative, not identically 0 and with bounded support the inequality holds for all t > 0, where u t := h t u and In particular, for every 0 < δ ≤ T < ∞ andx ∈ X there is a constant C δ,T > 0 depending on K, N, δ, T,x and the diameter of supp(u 0 ) such that for every ε ∈ (0, 1) it holds proof Recall the representation formula (A.4) and that for the transition probability densities r t [y](x) we have the Gaussian estimates (A.5), which can be simplified as for appropriate constants C 0 , C 1 , C 2 depending only on K, N . Therefore, we have By the fact that m is uniformly locally doubling we know that it holds where C 3 , C 4 only depend on K, N . As a consequence, the above yields We now apply Proposition 3.2 with u t in place of u 0 (notice that the assumptions are fulfilled) to get m-a.e., which is (equivalent to) the bound (3.1). The last statement is now obvious, noticing that for anyx ∈ supp(u 0 ).
A further result that we shall need soon is the Li-Yau inequality in the form proved by Baudoin and Garofalo (see [22] for the case of finite mass and [35] for the general one).
We restate such inequality in the form that we shall use: . Then for every 0 < δ ≤ T < ∞ andx ∈ X there exists a constant C δ,T > 0 depending on K, N, δ, T,x and the diameter of supp(u 0 ) such that the following holds. For any u 0 ∈ L 1 (X) non-negative, not identically zero and with bounded support and for any ε ∈ (0, 1) it holds and use Hamilton's gradient estimate (3.2) to control |∇ log u t | 2 in the right-hand side.  For any ε > 0 we consider the couple (f ε , g ε ) given by Theorem 2.2 normalized in such a way thatˆl for t ∈ (0, 1) In order to investigate the time behaviour of the functions just defined, let us introduce the weighted L 2 and W 1,2 spaces. The weight we will always consider is e −V with V = M d 2 (·,x); because of (A.20), e −V m has finite mass for every M > 0. For L 2 (X, e −V m) no comments are required. The weighted Sobolev space is denoted and defined as where |Df | is the local minimal weak upper gradient already introduced. Since V is locally bounded, W 1,2 (X, e −V m) turns out to coincide with the Sobolev space built over the metric measure space (X, d, e −V m), thus motivating the choice of the notation. The advantage of dealing with L 2 (X, e −V m) and W 1,2 (X, e −V m) is the fact that they are Hilbert spaces, unlike L 2 loc (X) and W 1,2 loc (X). After this premise, let us begin with a couple of quantitative estimates for f ε t , g ε t and ρ ε t .
Lemma 4.2. With the same assumptions and notation as in Setting 4.1 and defining for anyx ∈ X there exist positive constants C 1 , ..., C 9 depending on K, N , ρ 0 , ρ 1 ,x only such that the following bounds hold: i) For any ε > 0 and t ∈ (0, 1] we have 3) and analogously for g ε t and t ∈ [0, 1).
ii) For any ε ∈ (0, 1) and t ∈ [0, 1] we have proof (i) Direct consequence of the representation formula (A.4), the Gaussian estimates (A.5) and the fact that ρ 0 and f ε have the same support (ii) We shall indicate by C a constant depending only on K, N , ρ 0 , ρ 1 ,x whose value might change in the various instances it appears. Start from Noticing that the the Bishop-Gromov inequality (A.18) ensures that for every s ∈ [0, 1] it holds V s ≤ Cm(B 1 (x)) and v s ≥ Cm(B 1 (x))s N/2 , we obtain the claim for t ∈ [1/2, 1]. The case t ∈ [0, 1/2] follows by a symmetric argument.
The following proposition collects the basic properties of the functions defined in Setting 4.1 and the respective 'PDEs' solved: With the same assumptions and notation as in Setting 4.1, the following holds.
All the functions are well defined and for any ε > 0: where I is the respective domain of definition (for (ρ ε t ) we pick I = (0, 1)) and V = M d 2 (·,x); their time derivatives are given by the following expressions for a.e. t ∈ [0, 1]: Moreover, for every ε > 0 we have: These identities, (4.5) for (f ε t ), estimate (A.8) and (4.3) imply that for anyx there is M > 0 such that for V := M d 2 (·,x) the bound (4.6) for (ϕ ε t ) holds, as claimed. Similarly, we see that . The expressions for ∇ϕ ε t , ∆ϕ ε t and the equation for (f ε t ) also grant that m-a.e. it holds d dt for a.e. t and since we have seen that the right-hand side belongs to L 2 but as already noticed in the proof of Theorem 2.2, these are equivalent to the fact that f ε ⊗ g ε R ε/2 is a transport plan from µ 0 to µ 1 ; hence, (4.8) holds by the very choice of (f ε , g ε ) made.
Finally, the fact that (ϑ ε t ) belongs to AC loc ((0, 1), W 1,2 (X, e −V m)) and satisfies the bound (4.6) is a direct consequence of the analogous property for (ϕ ε t ), (ψ ε t ). The equation for its time derivative comes by direct computation: hence the proof is complete.
Using the terminology adopted in the literature (see [43]) we shall refer to: • ϕ ε t and ψ ε t as Schrödinger potentials, in connection with Kantorovich ones; • (µ ε t ) t∈[0,1] as entropic interpolation, in analogy with displacement one.

Uniform estimates for the densities and the potentials
We start collecting information about quantities which remain bounded as ε ↓ 0. For all δ ∈ (0, 1) andx ∈ X there exists C > 0 which only depends on K, N, δ,x such that for every t ∈ [δ, 1] and ε ∈ (0, 1). Analogous bounds hold for the ψ ε t 's in the time interval proof Fix δ ∈ (0, 1) andx ∈ X as in the statement and notice that the bound (3.2) yields Thus recalling the Sobolev-to-Lipschitz property (A.10) we obtain the bound (4.9a). The bound (4.9b) is a restatement of (3.4). Finally, let M > 0 and χ a 1-Lipschitz cut-off function with bounded support; notice that |h| = h + 2h − whencê The gradient estimates that we just obtained together with the Gaussian bounds on f ε Then H ε t ∈ L 1 (X) for every ε, t ∈ (0, 1). Moreover for every δ ∈ (0, 1/2) we have Finally, (0, 1) ∋ t →´H ε t dm is continuous. proof General considerations We shall repeatedly use the fact that if h 1 has Gaussian decay and h 2 polynomial growth, i.e.
for some c 1 , . . . , c 4 > 0,x ∈ X, then their product h 1 h 2 belongs to L 1 ∩ L ∞ (X): the L ∞ bound is obvious, the one for the L 1 -norm is a direct consequence of the volume growth (A.20) and explicit computations.
For what concerns the continuity of (0, 1) ∋ t →´H ε t dm, notice that Proposition 4.3 yields that all the maps (0, 1) ∋ t → |∇h ε and notice that the term ρ ε t |∇ϕ ε t ||∇ψ ε t | can be handled as before and that by (A.7) and the maximum principle for the heat flow we have that Hence the conclusion follows from the Gaussian bounds (4.3).
The last term in the right-hand side is dominated locally uniformly in t ∈ (0, 1) by what already proved. Similarly, the term g ε t |∇ϕ ε t | 4 is, locally in t, dominated thanks to the Gaussian bounds on g ε t , a domination for g ε t |∆f ε t ||∇ϕ ε t | 2 then follows using (4.13). Writing ∇∆f ε t = ∇h t−δ ∆f ε δ for any t ≥ δ > 0 and using (A.7) and the Bakry-Émery estimates (A.9) we see that |∇∆f ε t | is, locally in t, uniformly bounded in L ∞ (X), (4.14) thus a local uniform domination for εg ε t |∇ϕ ε t ||∇∆f ε t | follows. It remains to consider the term g ε t |Hess(f ε t )| 2 HS : we know from (A.13) that |Hess(f ε t )| HS ∈ L 2 (X) and from (4.3) that g ε t ∈ L ∞ (X). This is sufficient to conclude that ρ ε t ∇h ε t , ∇∆h ε t ∈ L 1 (X). To prove (4.12), thanks to the dominations previously obtained, it is enough to prove that where for any R > 0 the function χ R is a cut-off given by Lemma A.2. From (A.16) we havê , both locally uniformly in t ∈ (0, 1). Hence (4.15) follows from the fact that ( It remains to prove that t →´ρ ε t is bounded from below on supp(χ R ) by a positive constant depending continuously on t also taking into account what previously proved we see that the integrand in the right-hand side of (4.16) is continuous as map with values in L 0 (X, m | supp(χ R ) ) and, locally in t, uniformly dominated by an L 1 (X)-function. This is sufficient to conclude. For everyx ∈ X there exist constants C, C ′ > 0 which depends on K, N,x, ρ 0 , ρ 1 such that for every t ∈ [0, 1] and for every ε ∈ (0, 1).
proof From (4.4) and direct manipulation we see that there are constants c, c ′ , r > 0 depending on K, N,x, ρ 0 , ρ 1 only such that (4.18) hence to conclude it is sufficient to show that there exists a constant M > 0 depending on K, N,x, ρ 0 , ρ 1 only such that For later purposes it will be useful to observe that from (4.18) and the volume growth estimate (A. 20) it follows that there is R > r such that Now fix ε > 0. We know from Proposition 4.3 that (ρ ε t ) ∈ C([0, 1], L 2 (X))∩AC loc ((0, 1), L 2 (X)) and by the maximum principle for the heat equation ρ ε t ≤ C ε for all t ∈ [0, 1], thus for any 1)). An application of the dominated convergence theorem grants that its derivative can be computed passing the limit inside the integral, obtaining Then the definition of ϑ ε t , (4.9a), (4.9b) and (4.4) allow to justify the integration by parts, whence and recalling that ϑ ε t = ψ ε t − ε 2 log ρ ε t we obtain (for the same reasons as above, the integrals are well defined) that Now notice (the same arguments as above justify integration by parts) that (ρ ε and recalling (4.20) we get Passing to the p-th roots, writing Switching the roles of ρ 0 and ρ 1 we get the analogous control for t ∈ [ 1 2 , 1], whence the claim

The entropy along entropic interpolations
In [39] Léonard computed the first and second derivatives of the relative entropy along entropic interpolations: here we are going to show that his computations are fully justifiable in our setting. As we shall see later on, these formulas will be the crucial tool for showing that the acceleration of the entropic interpolation goes to 0 in the suitable weak sense.
We start by noticing that a form of Bochner inequality for the Schrödinger potentials can be deduced. Observe that in general the object Γ 2 (ϕ ε Now we are in position for motivating Léonard's computations, thus getting the formulas for the first and second derivative of the entropy along entropic interpolations. For any ε > 0 the map t → H(µ ε t | m) belongs to C([0, 1]) ∩ C 2 (0, 1) and for every t ∈ (0, 1) it holds proof By Lemma 4.5 we know that the central and right-hand sides in (4.24a) and (4.24b) exist, are finite and continuously depend on t ∈ (0, 1). Also, the equality between the central and right-hand sides follows trivially from the relations 1]) and that the identities (4.24a) and (4.24b) hold for a.e. t ∈ (0, 1). Lemma 4.5 ensures that t → H(µ ε and notice that for t ∈ [0, 1/2] the bound (4.3) ensures that the function g ε t is uniformly bounded above by a Gaussian and that log g ε t has a quadratic growth. On the other hand, we know by Theorem 2.2 that f ε 0 = f ε is in L ∞ , thus the maximum principle for the heat flow and the fact that z → z log z is bounded from below give that the L ∞ norms of f ε t , f ε t log f ε t are uniformly bounded in t ∈ [0, 1/2]. As discussed in the proof of Lemma 4.5, this is sufficient to conclude and a similar arguments yields the desired bound for t ∈ [1/2, 1]. Now fix ε > 0 and for R > 0 let χ R ∈ Test ∞ (X) be a cut-off function as given by Lemma A.2. Notice that Lemma 4.5 grants that Also, Proposition 4.3 tells that (ρ ε t ) ∈ AC loc ((0, 1), L 2 (X)) and that it is, locally in t ∈ (0, 1) and in space, uniformly bounded away from 0 and ∞. Therefore, for u(z) := z log z we have that (0, 1) ∋ t → χ R u(ρ ε t ) ∈ L 2 (X) is absolutely continuous. In particular, so is´χ R u(ρ ε t ) dm and it is then clear that Using the formula for d dt ρ ε t provided by Proposition 4.3 we then get Since |∇χ R | is uniformly bounded and identically 0 on B R (x), Lemma 4.5 grants that the last expression in the above identity converges to´ ∇ρ ε t , ∇ϑ ε t dm as R → ∞ locally uniformly in t ∈ (0, 1). This fact, (4.25) and the initial discussion give C 1 ((0, 1)) regularity for t → H(µ ε t | m) and (4.24a). For (4.24b), notice that from Proposition 4.3 we know that (ρ ε t ) ∈ AC loc ((0, 1), W 1,2 (X)) and (ϑ ε t ) ∈ AC loc ((0, 1), W 1,2 (X, e −V m)) with V = M d 2 (·,x), for somex ∈ X and M > 0 sufficiently large. Hence (0, 1) ∋ t → χ R ∇ρ ε t , ∇ϑ ε t ∈ L 2 (X) is absolutely continuous. In particular, so is´χ R ∇ρ ε t , ∇ϑ ε t dm and Thus from the formulas for d dt ρ ε t , d dt ϑ ε t provided in Proposition 4.3 we obtain .

Now notice that a few integration by parts and the Leibniz rule give
Since |∇χ R | is uniformly bounded and identically 0 on B R (x), Lemma 4.5 gives that

26c)
and for any δ ∈ (0, 1 2 ) proof (4.26a) follows from (4.17) and the volume growth (A.20). As regards (4.26b), notice that (4.26a) and (2.11) give a uniform lower bound on H(µ ε t | m); for the upper bound notice that (4.17) implies a uniform quadratic growth on log ρ ε t . Let us now pass to (4.26c) and observe that Proposition 4.4 together with (4.26a) grants As a second step, notice that (4.24a) gives so that taking into account (4.26b) and (4.28) we see that the right-hand side is uniformly bounded for ε ∈ (0, 1). Using again (4.28) we deduce that A symmetric argument provides the analogous bound for (ψ ε t ) and thus recalling that ϑ ε t = 1 2 (ψ ε t − ϕ ε t ) and ε log ρ ε t = ψ ε t + ϕ ε t we obtain (4.26c). Now use the fact that ϑ ε t = −ϕ ε t + ε 2 log ρ ε t in conjunction with (4.24a) to get Recalling the lower bound (4.9b) and (4.26a), we get that for some constant C δ independent on ε it holds d dt and an analogous argument starting from The bounds (4.27a) and (4.27b) then come from this last inequality used in conjunction with (4.24b), (4.26c) and the weighted Bochner inequalities (4.22a) and (4.22b) respectively.
With the help of the previous lemma we can now prove that some crucial quantities vanish in the limit ε ↓ 0; as we shall see in the proof of our main Theorem 5.12, this is what we will need to prove that the acceleration of the entropic interpolations goes to 0 as ε goes to zero.
For (4.29c) we observe that and use the fact that the first square root in the right-hand side is bounded (by (4.27b)) and the second one goes to 0 (by (4.29b)).
To prove (4.29d) we start again from the identity ρ ε t |∇ log ρ ε t | 2 = −ρ ε t ∆ log ρ ε t + ∆ρ ε t to geẗ After a multiplication by ε 2 we see that the first integral on the right-hand side vanishes as ε ↓ 0 thanks to (4.29c). For the second we start noticing that an application of the dominated convergence theorem ensures thaẗ then we observe that for every η > 0 the map as well. Thus by the chain rule for gradients, the Leibniz rule (A.15) and also using a cut-off argument in conjunction with Lemma 4.5 to justify integration by parts, we see that it holds and being this true for any η > 0, from (4.30) we obtain In this last expression the first square root is uniformly bounded in ε ∈ (0, 1) by (4.27a), while the second one vanishes as ε ↓ 0 thanks to (4.29b). The last claim follows from the fact that under the stated additional regularity properties of ρ 0 , ρ 1 we can take δ = 0 in (4.27a), (4.27b). Then we argue as before.
Lemma 4.11. Let (X, d, m) be an RCD * (K, ∞) space with K ∈ R endowed with a Borel nonnegative measure m which is finite on bounded sets and h ∈ D(∆) ∩ L 1 (X) with ∆h ∈ L 1 (X). Thenˆ∆ hdm = 0 proof Letx ∈ X, R > 0 and χ R ∈ Test ∞ (X) be a cut-off function as given by Lemma A.2. Then hdm.
Since Lemma A.2 ensures that ∆χ R L ∞ (X) is uniformly bounded in R, the conclusion follows letting R → ∞ in the above.

Compactness
Starting from the uniform estimates discussed in Section 4, let us first prove that when we pass to the limit as ε ↓ 0, up to subsequences Schrödinger potentials and entropic interpolations converge in a suitable sense to limit potentials and interpolations.
(iv) We know that for any x ∈ X it holds lip ϕ t (x) ≤ lim r↓0 Lip(ϕ t | Br(x) ) and, since X is geodesic, that Lip(ϕ ε t | Br(x) ) = sup Br(x) lipϕ ε t . Thus the claim follows from the bound (4.9a) and the fact that Lip(ϕ t | Br(x) ) ≤ lim n→∞ Lip(ϕ εn t | Br(x) ), which in turn is a trivial consequence of the local uniform convergence we already proved. Similarly for the ψ's.
(v) For the first in (5.5) we pass to the limit in the identity recalling the uniform bound (4.19). To get the second in (5.5) we multiply both sides of (5.10) by ρ ε t and integrate to obtainˆ( Letting ε = ε n ↓ 0 we see that the right-hand side goes to 0 by (4.26b); then we use the fact that W 2 (µ εn t , µ t ) → 0, that the functions ϕ ε t , ψ ε t have uniform quadratic growth and converge locally uniformly to ϕ t , ψ t respectively to obtain that the left-hand side goes to´ϕ t + ψ t dµ t . This is sufficient to conclude.

Identification of the limit curve and potentials
We now show that the limit interpolation is the geodesic from µ 0 to µ 1 and the limit potentials are Kantorovich potentials. We shall make use of the following simple lemma valid on general metric measure spaces: be a complete separable metric measure space endowed with a non-negative measure m Y which is finite on bounded sets and assume that W 1,2 (Y) is separable. Let π be a test plan and f ∈ W 1,2 (Y). Then t →´f • e t dπ is absolutely continuous and d dtˆf • e t dπ ≤ˆ|df |(γ t )|γ t | dπ(γ) a.e. t ∈ [0, 1], (5.11) where the exceptional set can be chosen to be independent on f .
, then the map t →´f t • e t dπ is also absolutely continuous and d ds proof The absolute continuity of t →´f • e t dπ and the bound (5.11) are trivial consequences of the definitions of test plans and Sobolev functions. The fact that the exceptional set can be chosen independently on f follows from the separability of W 1,2 (Y) and standard approximation procedures, carried out, for instance, in [25]. For the second part, we start noticing that the second derivative in the right-hand side exists for a.e. t thanks to what we have just proved, so that the claim makes sense. The absolute continuity follows from the fact that for any t 0 , t 1 ∈ [0, 1], t 0 < t 1 it holds ˆf and our assumptions on (f t ) and π. Now fix a point t of differentiability for (f t ) and observe that the fact that f t+h −ft h strongly converges in L 2 (Y) to d dt f t and (e t+h ) * π weakly converges to (e t ) * π as h → 0 and the densities are equibounded is sufficient to get Hence the conclusion comes dividing by h the trivial identitŷ and letting h → 0.
We now prove that in the limit the potentials evolve according to the Hopf-Lax semigroup (recall formula (A.25)).

Proposition 5.4 (Limit curve and potentials).
With the same assumptions and notations as in Setting 4.1 the following holds.
The limit curve (µ t ) given by Proposition 5.1 is unique (i.e. independent on the sequence ε n ↓ 0) and is the only W 2 -geodesic connecting µ 0 to µ 1 .
proof Inequality ≤ in (5.12a). Pick x, y ∈ X, r > 0, define and π r as the only lifting of the only W 2 -geodesic from ν r x to ν r y (recall point (i) of Theorem A.6). Since ν r x , ν r y have compact support and π r ∈ OptGeo(ν r x , ν r y ), there existx ∈ X and R > 0 sufficiently large such that Let χ be a Lipschitz cut-off function with bounded support such that χ ≡ 1 in B R (x). Then, let ε ∈ (0, 1) and 0 < t 0 < t 1 ≤ 1, putφ ε t := χϕ ε t and observe that (φ ε t ) ∈ AC loc ((0, 1], L 2 (X)) ∩ L ∞ loc ((0, 1], W 1,2 (X)) by Proposition 4.3 and the compactness of the support of χ; thus, by Lemma 5.3 applied to π r and t →φ ε (1−t)t 0 +tt 1 , we get As (5.14) implies that χ(γ t ) = 1 for all t ∈ [0, 1] for π r -a.e. γ,φ ε can be replaced by ϕ ε in the inequality above and, recalling the expression for d dt ϕ ε t and using Young's inequality, we obtain Integrating in time and recalling that π r is optimal we get Let ε ↓ 0 along the sequence (ε n ) for which (ϕ εn t ) converges to our given (ϕ t ) in the sense of Proposition 5.2 and use the uniform bound (4.10) and the fact that π r has bounded compression to deduce that and finally letting r ↓ 0 we conclude from the arbitrariness of x ∈ X that Inequality ≥ in (5.12a). To prove the opposite inequality we fixx ∈ X, r > 1, again 0 < t 0 < t 1 ≤ 1 and letR > r to be fixed later. Let χR ∈ Test ∞ (X) be given by Lemma A.2, define the vector field X ε t := χR∇ϕ ε t and apply Theorem A.4 to ((t 1 − t 0 )X ε (1−t)t 1 +tt 0 ): the inequality divX ε t ≥ χR∆ϕ ε t − |∇χR||∇ϕ ε t | and the bounds (4.9a), (4.9b) on ∇ϕ ε t , ∆ϕ ε t ensure that the theorem is applicable and we obtain existence of the regular Lagrangian flow F ε . Notice that from (4.9a) we know that for a.e. t and thus Gronwall's Lemma implies the existence of R independent ofR, ε such that for m-a.e. x it holds We now fixR := R and put π ε := m(B r (x)) −1 (F ε · ) * m | Br(x) , where F ε · : X → C([0, 1], X) is the m-a.e. defined map which sends x to t → F ε t (x), and observe that the bound (A.21) and the identity (A.22) provided by Theorem A.4 coupled with the estimates (4.9a), (4.9b) on holds in (5.15). Since both sides of (5.15) are continuous in y, we deduce that equality holds for any y ∈ B r (x) and the arbitrariness of r allows to conclude that equality actually holds for any y ∈ X.
Other properties of ϕ t . From Proposition 5.2 we already know that, for anyx ∈ X and M > 0, (ϕ t ) ∈ AC loc ((0, 1], L 1 (X, e −V m))∩L ∞ loc ((0, 1], W 1,2 (X, e −V m)), where V := M d 2 (·,x). Since ϕ t is a real-valued function for all t ∈ (0, 1], (5.12a) tells us that for all x ∈ X, t → ϕ t (x) satisfies (A.26) for a.e. t ∈ (0, 1]; taking (5.4) into account, this yields that for all δ ∈ (0, 1) and t 0 , . Up to extract a further subsequence -not relabeled -we can assume that the curves (µ εn t ) converge to a limit curve (µ t ) as in Proposition 5.1. We claim that for any t 0 , t 1 ∈ (0, 1], We now pass to the limit in ε = ε n ↓ 0: we know from Proposition 5.1 that W 2 (µ εn t , µ t ) → 0 and together with (5.3) this also grants that the left-hand side trivially converges to the lefthand side of (5.20). The contribution of the term with |∇ log ρ ε t | vanishes by (4.29b) and so does the one with ∆ϕ ε t by (4.10) and (4.17). Hence (5.20) is proved. Now notice that (5.12a) can be rewritten as Hence both (5.12b) and the fact that −(t 1 − t 0 )ϕ t 1 is a Kantorovich potential follow from Then (5.13b) and the other claims about (ψ t ) are proved in the same way.
Kantorovich potentials from µ t 1 to µ t 0 and from µ t 1 to µ t ′ 0 respectively and thus by point (ii) of Theorem A.6 we deduce Swapping the roles of t 0 , t 1 and using the ψ's in place of the ϕ's we then get This grants that the restriction of (µ t ) to any interval [t 0 , t 1 ] ⊂ (0, 1) is a constant speed geodesic. Since (µ t ) is continuous on the whole [0, 1], this gives the conclusion. Since in this situation the W 2 -geodesic connecting µ 0 to µ 1 is unique (recall point (i) of Theorem A.6), by the arbitrariness of the subsequences chosen we also proved the uniqueness of the limit curve (µ t ).
Remark 5.5 (The vanishing viscosity limit). The part of this last proposition concerning the properties of the ϕ ε t 's is valid in a context wider than the one provided by Schrödinger problem: we could restate the result by saying that if (ϕ ε t ) solves and ϕ ε 0 uniformly converges to some ϕ 0 , then ϕ ε t uniformly converges to ϕ t := −Q t (−ϕ 0 ). In this direction, it is worth recalling that in [2] and [21] it has been developed a theory of viscosity solutions for some first-order Hamilton-Jacobi equations on metric spaces. This theory applies in particular to the equation whose only viscosity solution is given by the formula ϕ t := −Q t (−ϕ 0 ). Therefore, we have just proved that if one works not only on a metric space, but on a metric measure space which is an RCD * (K, N ) space, then the solutions of the viscous approximation (5.22) converge to the unique viscosity solution of (5.23), in accordance with the classical case.
Remark 5.6. It is not clear whether the 'full' families ϕ ε t , ψ ε t converge as ε ↓ 0 to a unique limit. This is related to the non-uniqueness of the Kantorovich potentials in the classical optimal transport problem.
We shall now make use of the following lemma. It could be directly deduced from the results obtained by Cheeger in [14], however, the additional regularity assumptions on both the space and the function allow for a 'softer' argument based on the metric Brenier's theorem, which we propose.
proof Lemma 3.3 in [32] grants that φ is locally Lipschitz on Ω and that ∂ c φ(x) = ∅ for every x ∈ Ω. The same lemma also grants that for K ⊂ Ω compact, the set ∪ x∈K ∂ c φ(x) is bounded. Recalling that ∂ c φ is the set of (x, y) ∈ Y 2 such that and that φ, φ c are upper semicontinuous, we see that ∂ c φ is closed. Hence for K ⊂ Ω compact the set ∪ x∈K ∂ c φ(x) is compact and not empty and thus by the Kuratowski-Ryll-Nardzewski Borel selection theorem we deduce the existence of a Borel map T : Ω → Y such that T (x) ∈ ∂ c φ(x) for every x ∈ Ω. Pick µ ∈ P 2 (Y) with supp(µ) ⊂⊂ Ω and µ ≤ Cm for some C > 0 and set ν := T * µ. By construction, µ, ν have both bounded support, T is an optimal map and φ is a Kantorovich potential from µ to ν.
Hence point (iii) of Theorem A.6 applies and since lip φ = max{|D + φ|, |D − φ|}, by the arbitrariness of µ to conclude it is sufficient to show that |D + φ| = |D − φ| m-a.e. This easily follows from the fact that m is doubling and φ Lipschitz, see Proposition 2.7 in [5].
With this said, we can now show that the weighted energies of the Schrödinger potentials converge to the weighted energy of the limit ones: Proposition 5.8. With the same assumptions and notations as in Setting 4.1 the following holds.
Let ε n ↓ 0 be a sequence such that (ϕ εn t ), (ψ εn t ) converge to limit curves (ϕ t ), (ψ t ) as in Proposition 5.2 and let V := M d 2 (·,x) withx ∈ X and M > 0 arbitrary. Then for every δ ∈ (0, 1) we have proof Fix δ ∈ (0, 1) and argue as in the proof of Proposition 5.2 to obtain that t →´e −V ϕ ε t dm is absolutely continuous in [δ, 1] (see in particular (5.7)) and that Choosing ε := ε n , letting n → ∞ and using the uniform bounds (4.10), (5.3) and the volume growth estimate (A.20) we obtain Combining (A.26) and (5.12a) we see that for any x ∈ X it holds By Fubini's theorem, the same identity holds for L 1 ⊗ m-a.e. (t, x) ∈ [δ, 1] × X. The identity (5.12a) also grants that ϕ t is a multiple of a c-concave function, thus the thesis of Lemma 5.7 is valid for ϕ t and recalling that (ϕ t ) ∈ AC loc ((0, 1], L 1 (X, e −V m)) by Proposition 5.2 we deduce thatˆe which together with (5.25) gives the first in (5.24). The proof of the second is analogous.
As a direct consequence of the limit (5.24) and the local equi-Lipschitz bounds (4.9a) we obtain the following result. In order to state it, let us introduce the module L 2 (T * X, e −V m) as {ω ∈ L 0 (T * X) : |ω| ∈ L 2 (X, e −V m)}; an analogous definition can be given for L 2 ((T * ) ⊗2 X). Corollary 5.9. With the same assumptions and notations as in Setting 4.1 the following holds.
The other claims follow by analogous arguments.
The estimates that we have for the functions ϕ's tell nothing about their regularity as t ↓ 0 and similarly little we know so far about the ψ's for t ↑ 1. We now see in which sense limit functions ϕ 0 , ψ 1 exist. This is not needed for the proof of our main result, but we believe it is relevant on its own. Thus let us fix ε n ↓ 0 so that ϕ εn t → ϕ t for t ∈ (0, 1] and ψ εn t → ψ t for t ∈ [0, 1) as in Notice that the fact that the inf are equal to the stated limits is a consequence of formulas (5.12a), (5.13a), which directly imply that for every x ∈ X the maps t → ϕ t (x) and t → ψ 1−t (x) are non-decreasing. The main properties of ϕ 0 , ψ 1 are collected in the following proposition: i) The functions −ϕ t (resp. −ψ t ) Γ-converge to −ϕ 0 (resp. −ψ 1 ) as t ↓ 0 (resp. t ↑ 1).
With the analogous definition of ρ 1 ψ εn 1 we have that these converge to ρ 1 ψ 1 in L ∞ (X) as n → ∞.
proof We shall prove the claims for ϕ 0 only, as those for ψ 1 follow along similar lines.
(i) For the Γ − lim inequality we simply observe that by definition −ϕ 0 (x) = lim t↓0 −ϕ t (x). To prove the Γ − lim inequality, use the fact that −ϕ t ≥ −ϕ s for 0 < t ≤ s and the continuity of ϕ s : for given (x t ) converging to x we have ∀s > 0.
The conclusion follows letting s ↓ 0.
Therefore passing to the limit in (5.28) as ε = ε n ↓ 0 and recalling the local uniform convergence of ϕ εn t to ϕ t give −ϕ t ≥ −C t + C 5 d 2 (·,x) t for every t ∈ (0, 1], whereC ≥ 0 depends on K, N, ρ 0 , ρ 1 ,x only. It follows that Now fix x ∈ X and a sequence t n ↓ 0: the bound (5.31) grants that there are y n ∈ X such that and that these y n range in a bounded set. Thus up to pass to a subsequence we can assume that y n → y for some y ∈ X, so that taking into account the Γ − lim inequality previously proved we get so that letting t ↓ 0 and using the continuity of [0, 1) ∋ t → ψ t (x) for all x ∈ X we deduce that ϕ 0 ≤ −ψ 0 on X.
Now notice that the fact that −ϕ 0 ≤ Γ − lim(−ϕ t ) implies that Let π be the lifting of the W 2 -geodesic (µ t ) (recall point (i) of Theorem A.6); taking into account that the evaluation maps e t : C([0, 1], X) → X are continuous and that supp(π) is a compact subset of C([0, 1], X), because given by constant speed geodesics running from the compact set supp(ρ 0 ) to the compact supp(ρ 1 ), it is easy to see that for every γ ∈ supp(π) and t ∈ [0, 1] we have γ t ∈ supp(µ t ) and viceversa for every x ∈ supp(µ t ) there is γ ∈ supp(π) with γ t = x.
Thus it remains to prove that ϕ 0 = −∞ outside supp(ρ 0 ). To this aim, we notice again that the supports of f ε , g ε coincide with those of ρ 0 , ρ 1 and use the second in (A.5) to get for every t ∈ (0, 1) and constants c i > 0 depending on K, N, ρ 0 , ρ 1 ,x only. From these bounds, the identity ρ ε t = f ε t g ε t and the estimates (5.30) and (5.29) we deduce that Now let ε n ↓ 0 be the sequence such that ϕ εn t , ψ εn t converge to ϕ t , ψ t as in Proposition 5.2 and put S(x) := sup ε∈(0,1), (iv) By the point (iii) just proven we havê ϕ 0 ρ 0 dm +ˆψ 1 ρ 1 dm = −ˆψ 0 ρ 0 dm −ˆϕ 1 ρ 1 dm so that taking into account the weak continuity of t → µ t , the fact that the measures µ t have equibounded supports and the continuity of t → ϕ t (resp. t → ψ t ) for t close to 1 (resp. close to 0) in the topology of local uniform convergence (direct consequence of the continuity in L 1 (X, e −V m) and the uniform local Lipschitz estimates provided by Proposition 5.2), we get (v) Since ρ 0 ∈ L ∞ (X), we also have ρ 0 log(ρ 0 ) ∈ L ∞ (X). The claim then follows from the identity ρ 0 ϕ ε 0 = ερ 0 log ρ 0 − ρ 0 ψ ε 0 , the compactness of supp(ρ 0 ), the local uniform convergence of ψ εn 0 to ψ 0 as n → ∞ and the fact that ψ 0 = −ϕ 0 on supp(ρ 0 ).

Proof of the main theorem
We start with the following simple continuity statement: Lemma 5.11. With the same assumptions and notation as in Setting 4.1, let t → µ t = ρ t m be the W 2 -geodesic from µ 0 to µ 1 and (ϕ t ) t∈(0,1] and (ψ t ) t∈[0,1) any couple of limit functions given by Proposition 5.2. Then the maps are all continuous w.r.t. the strong topologies.
proof By Lemma A.8 we know that for any p < ∞ we have ρ s → ρ t in L p (X) as s → t and thus in particular √ ρ s → √ ρ t as s → t. Moreover, the compactness of the supports of ρ 0 and ρ 1 implies that there existx ∈ X and R > 0 such that supp(ρ t ) ⊂ B R (x) for all t ∈ [0, 1]. Consider a Lipschitz cut-off function χ with support in B R+1 (x) such that χ ≡ 1 in B R (x). The closure of the differential and the fact that ϕ s → ϕ t weakly in W 1,2 (X, e −V m) as s → t > 0 (as a consequence of (ϕ t ) ∈ C((0, 1], C(X, e −V )) ∩ L ∞ loc ((0, 1), W 1,2 (X, e −V m)), see Proposition 5.4 and the notation therein) grant that dϕ s → dϕ t weakly in L 2 (T * X, e −V m) and thus χdϕ s → χdϕ t in L 2 (T * X). Together with the previous claim about the densities, the fact that the latter are uniformly bounded in L ∞ (X) and how χ is constructed, this is sufficient to conclude that t → √ ρ t dϕ t ∈ L 2 (T * X) is weakly continuous.
We now claim that t → √ ρ t dϕ t ∈ L 2 (T * X) is strongly continuous and to this aim we show that their L 2 (T * X)-norms are constant. To see this, recall that by Proposition 5.4 we know that for t ∈ (0, 1] the function −(1 − t)ψ t is a Kantorovich potential from µ t to µ 1 while from (5.5) and the locality of the differential we get that |dϕ t | = |dψ t | µ t -a.e., thus by point (iii) in Theorem A.6 we have that Multiplying the √ ρ t dϕ t 's by √ ρ t and using again the L 2 (X)-strong continuity of √ ρ t and the uniform L ∞ (X)-bound we conclude that t → ρ t dϕ t ∈ L 2 (T * X) is strongly continuous, as desired.
To prove the strong continuity of t → ρ t dϕ t ⊗dϕ t ∈ L 2 ((T * ) ⊗2 X) we argue as in Corollary 5.9: the strong continuity of t → √ ρ t dϕ t ∈ L 2 (T * X) and the fact that these are, locally in t ∈ (0, 1], uniformly bounded (thanks again to supp(ρ t ) ⊂ B R (x) for all t ∈ [0, 1]), grant both that t → ρ t dϕ t ⊗ dϕ t L 2 ((T * ) ⊗2 X) is continuous and that t → ρ t dϕ t ⊗ dϕ t ∈ L 2 ((T * ) ⊗2 X) is weakly continuous. The claims about the ψ t 's follow in the same way.
We now have all the tools needed to prove our main result. Notice that we shall not make explicit use of Theorem 1.4 but rather reprove it for (the restriction to [δ, 1 − δ] of) entropic interpolations.

34)
where φ t is any function such that for some s = t, s ∈ [0, 1], the function −(s − t)φ t is a Kantorovich potential from µ t to µ s . and recalling the formula for d

Related differentiation formulas
In this last part we collect some direct consequences of Theorem 5.12. For the notion of covariant derivative and the related calculus rules we refer to [25].
for every η of the form η = χ Γ with Γ ⊂ C([0, 1], X) Borel, where here and below the integral are intended in the Bochner sense. Then the fact that the linear span of such η's is dense in L 2 (π) forces the equalities Hess(h)(∇φ r , ∇φ r ) • e r dr which is the claim.
(ii) By (i) and the Leibniz rule for the covariant derivative (see [25]) we see that the claim holds for W = n i=1 f i ∇g i , with n ∈ N and (f i ), (g i ) ∈ Test ∞ (X). These vector fields are dense in H 1,2 C (T X), hence the claim follows noticing that if W n → W in H 1,2 C (T X) and the φ t 's are chosen uniformly Lipschitz (which as discussed in the proof of Theorem 5.12 is always admissible) then W n , ∇φ t → W, ∇φ t and ∇W n : (∇φ t ⊗ ∇φ t ) → ∇W : (∇φ t ⊗ ∇φ t ) in L 2 (X) as n → ∞. Therefore, since (e t ) * π ≤ Cm for every t ∈ [0, 1] and some C > 0, we have that in L 2 (π) uniformly in t ∈ [0, 1]. The conclusion follows. (iii) Direct consequence of (ii) and an integration w.r.t. π.

A Reminders about analysis on RCD spaces
In this appendix we recall the basic definitions and properties of the various objects that we used in the body of the paper. We also provide detailed bibliographical references.
By P(X) we denote the space of Borel probability measures on (X, d) and by P 2 (X) ⊂ P(X) the subclass of those with finite second moment.
Let (X, d, m) be a complete and separable metric measure space endowed with a Borel non-negative measure which is finite on bounded sets.
For the definition of test plans, of the Sobolev class S 2 (X) and of minimal weak upper gradient |Df | see [5] (and the previous works [14], [52] for alternative -but equivalent -definitions of Sobolev functions). The local counterpart of S 2 (X) is introduced as follows: L 2 loc (X) is defined as the space of functions f ∈ L 0 (X) such that for all compact set Ω ⊂ X there exists a function g ∈ L 2 (X) such that f = g m-a.e. in Ω and the local Sobolev class S 2 loc (X) is then defined as The local minimal weak upper gradient of a function f ∈ S 2 loc (X) is denoted by |Df |, omitting the locality feature, and defined for all Ω ⊂⊂ X as |Df | := |Dg| m-a.e. in Ω, where g is as in (A.1). The definition does depend neither on Ω nor on the choice of g associated to it by locality of the minimal weak upper gradient.
The Sobolev space W 1,2 (X) (resp. W 1,2 loc (X)) is defined as L 2 (X) ∩ S 2 (X) (resp. L 2 loc ∩ S 2 loc (X)). When endowed with the norm f 2 W 1,2 := f 2 L 2 + |Df | 2 L 2 , W 1,2 (X) is a Banach space. The Cheeger energy is the convex and lower-semicontinuous functional E : L 2 (X) → [0, ∞] given by is infinitesimally Hilbertian (see [27]) if W 1,2 (X) is Hilbert. In this case E is a Dirichlet form and its infinitesimal generator ∆, which is a closed self-adjoint operator on L 2 (X), is called Laplacian on (X, d, m) and its domain denoted by D(∆) ⊂ W 1,2 (X). The flow (h t ) associated to E is called heat flow (see [5]), and for any f ∈ L 2 (X) the curve t → h t f ∈ L 2 (X) is continuous on [0, ∞), locally absolutely continuous on (0, ∞) and the only solution of d dt If moreover (X, d, m) is an RCD(K, ∞) space (see [6]), the following a priori estimates hold true for every f ∈ L 2 (X) and t > 0: Still within the RCD framework, there exists the heat kernel, namely a function for every f ∈ L 2 (X). For every x ∈ X and t > 0, r t [x] is a probability density and thus (A.4) can be used to extend the heat flow to L 1 (X) and shows that the flow is mass preserving and satisfies the maximum principle, i.e.
For general metric measure spaces, the differential is a well defined linear map d from S 2 (X) with values in the cotangent module L 2 (T * X) (see [25]) which is a closed operator when seen as unbounded operator on L 2 (X). If (X, d, m) is infinitesimally Hilbertian, which from now on we shall always assume, the cotangent module is canonically isomorphic to its dual, the tangent module L 2 (T X), and the isomorphism sends the differential df to the gradient ∇f . Replacing the language of L 2 -normed modules with the L 0 's one (see [25]), the differential can be extended to d : S 2 loc (X) → L 0 (T * X), where L 0 (T * X) denotes the family of (measurable) 1-forms. The dual of L 0 (T * X) as L 0 -normed module is denoted by L 0 (T X), it is canonically isomorphic to L 0 (T * X) and its elements are called vector fields. With this said, L 2 loc (T * X) ⊂ L 0 (T * X) (resp. L 2 loc (T X) ⊂ L 0 (T X)) is defined as the collection of the 1-forms ω such that |ω| ∈ L 2 loc (X) (resp. the vector fields v such that |v| ∈ L 2 loc (X)). The divergence of a vector field is defined as (minus) the adjoint of the differential, i.e. we say that v ∈ L 2 (T X) (resp. v ∈ L 2 loc (T X)) has a divergence in L 2 (X) (resp. in L 2 loc (X)), and write v ∈ D(div) (resp. v ∈ D(div loc )), provided there is a function g ∈ L 2 (X) (resp. g ∈ L 2 loc (X)) such thatˆf (resp. for all Lipschitz functions f with bounded support). In this case g is unique and is denoted div(v). A function f ∈ W 1,2 loc (X) has Laplacian in L 2 loc (X), and we shall write f ∈ D(∆ loc ), if there exists g ∈ L 2 loc (X) such that φgdm = −ˆ ∇φ, ∇f dm, ∀φ Lipschitz with bounded support and in this case, since g is unique, we set ∆f := g. It can be verified that f ∈ D(∆ loc ) if and only if ∇f ∈ D(div loc ) and in this case ∆f = div(∇f ), in accordance with the smooth case.
As regards the properties of d, div, ∆, the differential satisfies the following calculus rules which we shall use extensively without further notice: where it is part of the properties the fact that ϕ • f, f g ∈ S 2 (X) for ϕ, f, g as above. For the divergence, the formula holds, where it is intended in particular that f v ∈ D(div) for f, v as above, and for the Laplacian where in the first equality we assume that f ∈ D(∆), ϕ ∈ C 2 (R) are such that f, |df | ∈ L ∞ (X) and ϕ ′ , ϕ ′′ ∈ L ∞ (R) and in the second that f, g ∈ D(∆) ∩ L ∞ (X) and |df |, |dg| ∈ L ∞ (X) and it is part of the claims that ϕ • f, f g are in D(∆). On S 2 loc (X) as well as on D(div loc ) and D(∆ loc ) the same calculus rules hold with slight adaptations. For sake of information, we present the chain rule for differetial and Laplacian, as they will be widely exploited without further mention.
Lemma A.1 (Calculus rules). Let (X, d, m) be an RCD * (K, N ) with K ∈ R and N ∈ [1, ∞). Then: (i) for all f ∈ S 2 loc (X) and ϕ : R → R such that for all K ⊂⊂ X there exists I K ⊂⊂ R in such a way that L 1 (f (K) \ I K ) = 0 and ϕ | I K is Lipschitz it holds where it is part of the statement the fact that ϕ•f ∈ S 2 loc (X) for ϕ, f as above; analogous statements hold for the gradient; (ii) for all f ∈ D(∆ loc ) and ϕ : The proof is based on the locality of such differentiation operators and the analogous properties of their global counterparts defined on S 2 (X), D(∆).
Beside this notion of L 2 -valued Laplacian, we shall also need that of measure-valued Laplacian ( [27]). A function f ∈ W 1,2 (X) is said to have measure-valued Laplacian, and in this case we write f ∈ D(∆), provided there exists a Borel (signed) measure µ whose total variation is finite on bounded sets and such that g dµ = −ˆ ∇g, ∇f dm, ∀g Lipschitz with bounded support.
In this case µ is unique and denoted ∆f . This notion is compatible with the previous one in the sense that f ∈ D(∆), ∆f ≪ m and d∆f dm ∈ L 2 (X) ⇔ f ∈ D(∆) and in this case ∆f = d∆f dm .
On RCD(K, ∞) spaces, the vector space of 'test functions' (see [51]) is an algebra dense in W 1,2 (X) for which it holds Combining the Gaussian estimates on RCD * (K, N ) spaces, N < ∞, with the results in [51] we see that The fact that Test ∞ (X) is an algebra is based on the property and actually a further regularity property of test functions is that so that it is possible to introduce the measure-valued Γ 2 operator ([51]) as By construction, the assignment f → Γ 2 (f ) is a quadratic form. An important property of the heat flow on RCD(K, ∞) spaces is the Bakry-Émery contraction estimate (see [6]): We also recall that RCD(K, ∞) spaces have the Sobolev-to-Lipschitz property ( [6], [24]), i.e.
f ∈ W 1,2 (X), |df | ∈ L ∞ (X) ⇒ ∃f = f m − a.e. with Lip(f ) ≤ |df | L ∞ , (A.10) and thus we shall typically identify Sobolev functions with bounded differentials with their Lipschitz representative; in particular this will be the case for functions in Test ∞ (X).
A well-known consequence of lower Ricci curvature bounds (see e.g. [15], [16], [17]) is the existence of 'good cut-off functions', typically intended as cut-offs with bounded Laplacian; for our purposes the following result will be sufficient: Lemma A.2. Let (X, d, m) be an RCD * (K, N ) space with K ∈ R and N ∈ [1, ∞). Then for all R > 0 and x ∈ X there exists a function χ R : X → R satisfying: Moreover, there exist constants C, C ′ > 0 depending on K, N only such that The proof can be obtained following verbatim the arguments given in Lemma 3.1 of [47] (inspired by [8], see also [31] for an alternative approach): there the authors are interested in cut-off functions such that χ ≡ 1 on B R (x) and supp(χ) ⊂ B 2R (x): for this reason they fix R > 0 and then claim that for all x ∈ X and 0 < r < R there exists a cut-off function χ satisfying (i), (ii) and (A.11) with C, C ′ also depending on R. However, as far as one is concerned with cut-off functions χ where the distance between {χ = 0} and {χ = 1} is always equal to 1, the proof of [47] in the case R = 1 applies and does not affect (A.11).
A direct consequence of the existence of such cut-off functions is that {f ∈ L 2 loc (X) : ∀Ω ⊂⊂ X ∃g ∈ Test ∞ (X) s.t. f = g m-a.e. in Ω} = f ∈ D(∆ loc ) ∩ L ∞ loc (X) : |∇f | ∈ L ∞ loc (X), ∆f ∈ W 1,2 loc (X) . (A.12) Indeed the '⊂' inclusion is obvious, while for the opposite one if f belongs to the second set and Ω ⊂ X is a bounded open set, consider a cut-off function χ ∈ Test ∞ (X) with compact support and χ ≡ 1 on Ω: it is clear that χf ∈ Test ∞ (X) and χf ≡ f on Ω. We shall call the set in (A.12) the space of local test functions and denote it Test ∞ loc (X). The existence of the space of test functions and the language of L 2 -normed L ∞ -modules allow to introduce the spaces W 2,2 (X) and W 1,2 C (T X) as follows (see [25]). We first consider the tensor product L 2 ((T * ) ⊗2 X) of L 2 (T * X) with itself. The pointwise norm on such module is denoted | · | HS (as in the smooth case it coincides with the Hilbert-Schmidt one) and : is the scalar product inducing it. Then we say that a function f ∈ W 1,2 (X) belongs to W 2,2 (X) provided there exists A ∈ L 2 ((T * ) ⊗2 X) symmetric, i.e. such that A(v 1 , v 2 ) = A(v 2 , v 1 ) m-a.e. for every v 1 , v 2 ∈ L 2 (T X), for which it holdŝ hA(∇g, ∇g) dm =ˆ− ∇f, ∇g div(h∇g) − h ∇f, ∇ |∇g| 2 2 dm ∀g, h ∈ Test ∞ (X).
The Bochner inequality on RCD(K, ∞) spaces takes the form of an inequality between measures ( [25] -see also the previous contributions [51], [57]): 16) and if the space is RCD * (K, N ) for some finite N it also holds ( [20], [9]): Notice that since the Laplacian is in general not the trace of the Hessian, the former does not trivially imply the latter (in connection to this, see [34]).
As regards the geometric features of finite-dimensional RCD * (K, N ) spaces, we recall the Bishop-Gromov inequality in the form we shall need (see [55], [56]): for any x ∈ supp(m) and for any 0 < r ≤ R < ∞ it holds and there exists a constant C > 0 such that the following volume growth condition is satisfied m(B r (x)) ≤ Ce Cr , ∀x ∈ X, r > 0. (A.20) We conclude the section recalling the notion of Regular Lagrangian Flow, introduced by Ambrosio-Trevisan in [10] as the generalization to RCD spaces of the analogous concept existing on R d as proposed by Ambrosio in [1]: Definition A.3 (Regular Lagrangian Flow). Given (v t ) ∈ L 1 ([0, 1], L 2 (T X)), the function F : [0, 1] × X → X is a Regular Lagrangian Flow for (v t ) provided: ii) for every f ∈ Test ∞ (X) and m-a.e. x the map t → f (F t (x)) belongs to W 1,1 ([0, 1]) and iii) it holds (F t ) * m ≤ Cm ∀t ∈ [0, 1] for some constant C > 0.
Then there exists a unique, up to m-a.e. equality, Regular Lagrangian Flow F for (v t ).
For such flow, the quantitative bound holds for every t ∈ [0, 1] and for m-a.e. x the curve t → F t (x) is absolutely continuous and its metric speed ms t (F · (x)) at time t satisfies ms t (F · (x)) = |v t |(F t (x)) a.e. t ∈ [0, 1]. (A.22) To be precise, (A.22) is not explicitly stated in [10]; its proof is anyway not hard and can be obtained, for instance, following the arguments in [25].

A.2 Optimal transport on RCD spaces
It is well known that on R d , curves of measures which are W 2 -absolutely continuous are in correspondence with appropriate solutions of the continuity equation ( [4]). It has been proved in [28] that the same connection holds on arbitrary metric measure spaces (X, d, m), provided the measures are controlled by Cm for some C > 0, the formulation of such result which we shall need is: Theorem A.5 (Continuity equation and W 2 -AC curves). Let (X, d, m) be infinitesimally Hilbertian, (µ t ) ⊂ P(X) be weakly continuous and t → X t ∈ L 0 (T X) be a family of vector fields, possibly defined only for a.e. t ∈ [0, 1]. Assume that the map t →´|X t | 2 dµ t is Borel and: and that the continuity equation is satisfied in the following sense: for any f ∈ W 1,2 (X) the map [0, 1] ∋ t →´f dµ t is absolutely continuous and it holds d dtˆf dµ t =ˆdf (X t ) dµ t a.e. t.
A property related to the above is the fact that although the Kantorovich potentials are not uniquely determined by the initial and final measures, their gradients are. This is expressed by the following result, which also says that if we sit in the intermediate point of a geodesic and move to one extreme or the other, then the two corresponding velocities are one the opposite of the other (see Lemma 5.8 and Lemma 5.9 in [24] for the proof): Lemma A.7. Let (X, d, m) be an RCD(K, ∞) space with K ∈ R and (µ t ) ⊂ P 2 (X) a W 2geodesic such that µ t ≤ Cm for every t ∈ [0, 1] for some C > 0. For t ∈ [0, 1] let φ t , φ ′ t : X → R be locally Lipschitz functions such that for some s, s ′ = t the functions −(s−t)φ t and −(s ′ −t)φ ′ t are Kantorovich potentials from µ t to µ s and from µ t to µ s ′ respectively.
On RCD spaces, W 2 -geodesics made of measures with bounded density also have the weak continuity property of the densities expressed by the following lemma. The proof follows by a simple argument involving Young's measures and the continuity of the entropy along a geodesic (see Corollary 5.7 in [24]): Lemma A.8. Let (X, d, m) be an RCD(K, ∞) space with K ∈ R and (µ t ) ⊂ P 2 (X) a W 2geodesic such that µ t ≤ Cm for every t ∈ [0, 1] for some C > 0. Let ρ t be the density of µ t .
Then for any t ∈ [0, 1] and any sequence (t n ) n∈N ⊂ [0, 1] converging to t there exists a subsequence (t n k ) k∈N such that ρ tn k → ρ t , m-a.e.
We conclude recalling some properties of the Hopf-Lax semigroup in metric spaces, also in connection with optimal transport. For f : X → R ∪ {+∞} and t > 0 we define the function Q t f : X → R ∪ {−∞} as and set t * := sup{t > 0 : Q t f (x) > −∞ for some x ∈ X}; it is worth saying that t * does not actually depend on x, since if Q t f (x) > −∞, then Q s f (y) > −∞ for all s ∈ (0, t) and all y ∈ X. With this premise we have the following result (see [5]):