Hao Ding
email: dinghao16@mails.ucas.ac.cn

Shizan Fang
email: shizan.fang@u-bourgogne.fr

Geometry on the Wasserstein space over a compact Riemannian manifold

Keywords: 58B20, 60J45 Constant vector fields, measures having divergence, Levi-Civita connection, parallel translations, Mckean-Vlasov equations

We will revisit the intrinsic differential geometry of the Wasserstein space over a Riemannian manifold, due to a series of papers by Otto, Otto-Villani, Lott, Ambrosio-Gigli-Savaré and so on.

Introduction

For the sake of simplicity, we will consider in this paper a connected compact Riemannian manifold M of dimension m. We denote by d M the Riemannian distance and dx the Riemannian measure on M such that M dx = 1. Since the diameter of M is finite, any probability measure µ on M is such that M d 2 M (x 0 , x) dµ(x) < +∞, where x 0 is a fixed point of M . As usual, we denote by P 2 (M ) the space of probability measures on M , endowed with the Wasserstein distance W 2 defined by

W 2 2 (µ 1 , µ 2 ) = inf M ×M d 2 M (x, y) π(dx, dy), π ∈ C(µ 1 , µ 2 ) ,
where C(µ 1 , µ 2 ) is the set of probability measures π on M × M , having µ 1 , µ 2 as two marginal laws. It is well known that P 2 (M ) endowed with W 2 is a Polish space. In this compact case, the weak convergence for probability measures is metrized by W 2 ; therefore (P 2 (M ), W 2 ) is a compact Polish space.

The introduction of tangent spaces of P 2 (M ) can go back to the early work [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF], as well as in [START_REF] Otto | The geometry of dissipative evolution equations: The porous medium equation[END_REF]. A more rigorous treatment was given in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]. In differential geometry, for a smooth curve {c(t); t ∈ [0, 1]} on a manifold M , the derivative c ′ (t) with respect to the time t is in the tangent space : c ′ (t) ∈ T c(t) M . A classical result says that for an absolutely continuous curve {c(t); t ∈ [0, 1]} on M , the derivative c ′ (t) ∈ T c(t) M exists for almost all t ∈ [0, 1].

Following [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], we say that a curve {c(t); t ∈ [0, 1]} on P 2 (M ) is absolutely continuous in L 2 if there exists k ∈ L 2 ([0, 1]) such that W 2 (c(t 1 ), c(t 2 )) ≤

t 2 t 1 k(s) ds, t 1 < t 2 .
The following result is our starting point:

Theorem 1.1 (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], Theorem 8.3.1). Let {c t ; t ∈ [0, 1]} be an absolutely continuous curve on P 2 (M ) in L 2 , then there exists a Borel vector field Z t on M such that

[0,1] M |Z t (x)| 2 TxM dc t (x) dt < +∞
and the following continuity equation

dc t dt + ∇ • (Z t c t ) = 0, (1.1)
holds in the sense of distribution. Uniqueness to (1.1) holds if moreover Z t is imposed to be in ∇ψ, ψ ∈ C ∞ (M )

L 2 (ct)
.

In this work, we define the tangent space Tµ of P 2 (M ) at µ by

Tµ = ∇ψ, ψ ∈ C ∞ (M ) L 2 (µ) , (1.2)
the closure of gradients of smooth functions in the space L 2 (µ). Equation (1.1) implies that for almost all t ∈ [0, 1],

d dt M f (x) dc t (x) =
M ∇f (x), Z t (x) TxM dc t (x), f ∈ C 1 (M ).

(1.3)

We will say that Z t is the intrinsic derivative of c t and use the notation

d I c t dt = Z t ∈ Tct .
In what follows, we will describe the tangent space Tµ with the least conditions as possible on the measure µ. Consider the quadratic form defined by

E(ψ) = M |∇ψ(x)| 2 dµ(x), ψ ∈ C 1 (M ).
We assume that there is a constant C µ > 0 such that

M (ψ -ψ ) 2 dµ ≤ C µ M |∇ψ| 2 dµ, (1.4)
where ψ = M ψ(x) dx. The condition (1.4) is satisfied if µ admits a positive density ρ > 0: dµ = ρ dx. In fact, let

β 1 = inf x∈M ρ(x) > 0, β 2 = sup x∈M ρ(x) < +∞.
Since M is compact, the following Poincaré inequality holds :

M (ψ -ψ ) 2 dx ≤ C M |∇ψ| 2 dx, then M (ψ -ψ ) 2 dµ ≤ Cβ 2 β 1 M |∇ψ| 2 dµ. Now let Z ∈ Tµ ; there is a sequence of functions ψ n ∈ C ∞ (M ) such that Z = lim n→+∞ ∇ψ n in L 2 (µ)
. By changing ψ n to ψ n -ψ n and by condition (1.4),

{ψ n ; n ≥ 1} is a Cauchy sequence in L 2 (µ). If the quadratic form E(ψ) is closable in L 2 (µ), then there exists a function ϕ µ in the Sobolev space D 2 1 (µ) such that Z = ∇ϕ µ , where D 2 1 (µ) is the closure of C ∞ (M ) with respect to the norm ||ϕ|| 2 D 2 1 (µ) := M |ϕ(x)| 2 dµ(x) + M |∇ϕ(x)| 2 dµ(x).
A sufficient condition to insure the closability for E is that the formula of integration by parts holds for µ; more precisely, for any C 1 vector field Z on M , there exists a function denoted by div

µ (Z) ∈ L 2 (µ) such that M ∇f (x), Z(x) TxM dµ(x) = - M f (x) div µ (Z)(x), f ∈ C 1 (M ).
(1.5) Definition 1.2. We say that the measure µ is a measure having divergence if div µ (Z) ∈ L 2 (µ) exists. We will use the notation

P div (M )
to denote the set of probability measures on M having strictly positive continuous density and satisfying conditions (1.5).

Proposition 1.3. For a measure µ ∈ P div (M ), we have

Tµ = ∇ψ; ψ ∈ D 2 1 (µ) .
The inconvenient for (1.3) is the existence of derivative for almost all t ∈ [0, 1]. In what follows, we will present two typical classes of absolutely continuous curves in P 2 (M ).

Constant vector fields on P 2 (M)

For any gradient vector field ∇ψ on M with ψ ∈ C ∞ (M ), consider the ordinary differential equation (ODE):

d dt U t (x) = ∇ψ(U t (x)), U 0 (x) = x ∈ M. Then x → U t (x) is a flow of diffeomorphisms on M . Let µ ∈ P 2 (M ), consider c t = (U t ) # µ. It is easy to see that the curve {c t ; t ∈ [0, 1]} is absolutely continuous in L 2 and for f ∈ C 1 (M ), d dt M f (x) dc t (x) = d dt M f (U t (x)) dµ(x) = M ∇f (U t (x)), ∇ψ(U t (x)) dµ(x),
which is equal to, for any t ∈ [0, 1],

M ∇f, ∇ψ dc t .

In other term, c t is a solution to the following continuity equation:

dc t dt + ∇ • (∇ψ c t ) = 0.
According to above definition, we see that for each t ∈ [0, 1],

d I c t dt = ∇ψ.
It is why we call ∇ψ a constant vector field on P 2 (M ). In order to make clearly different roles played by ∇ψ, we will use notation

V ψ
when it is seen as a constant vector field on P 2 (M ).

Remark 1.4. In section 3 below, we will compute Lie brackets of two constant vector fields on P 2 (M ) without explicitly using the existence of density of measure, the Lie bracket of two constant vector fields is NOT a constant vector field.

Geodesics with constant speed

It is easy to introduce geodesics with constant speed when the base space is a flat space R m . A probability measure µ on R m is in

P 2 (R m ) if R m |x| 2 dµ(x) < +∞. Let c 0 , c 1 ∈ P 2 (R m ),
there is an optimal coupling plan γ ∈ C(c 0 , c 1 ) such that

W 2 2 (c 0 , c 1 ) = R m ×R m
|x -y| 2 dγ(x, y).

For each t ∈ [0, 1], define c t ∈ P 2 (R m ) by R m f (x) dc t (x) = R m ×R m f (u t (x, y)) dγ(x, y), where u t (x, y) = (1 -t)x + ty. For 0 ≤ s < t ≤ 1, define π s,t ∈ C(c s , c t ) by R m ×R m g(x, y) dπ s,t (x, y) = R m ×R m g(u s (x, y), u t (x, y)) dγ(x, y). Then W 2 2 (c s , c t ) ≤ R m ×R m |u t (x, y) -u s (x, y| 2 dγ(x, y) = (t -s) 2 W 2 (c 0 , c 1 ) 2 .
It follows that W 2 (c s , c t ) ≤ (t -s)W 2 (c 0 , c 1 ). Combing with triangulaire inequality,

W 2 (c 0 , c 1 ) ≤ W 2 (c 0 , c s ) + W 2 (c s , c t ) + W 2 (c t , c 1 ) ≤ sW 2 (c 0 , c 1 ) + (t -s)W 2 (c 0 , c 1 ) + (1 -t)W 2 (c 0 , c 1 ) = W 2 (c 0 , c 1 ),
we get the property of geodesic with constant speed:

W 2 (c s , c t ) = |t -s| W 2 (c 0 , c 1 ).
According to Theorem 1.1, there is

Z t ∈ Tct such that, for f ∈ C 1 c (R d ), d dt R m f (x)dc t (x) = R m ∇f (u t (x, y)), y -x R m dγ(x, y) = R d ∇f (x), Z t (x) R m dc t (x)
where , R m is the canonical inner product of R m . We heuristically look for Z t such that

Z t (u t (x, y)) = y -x.
Taking the derivative with respect to t yields

( d dt Z t )(u t (x, y)) + ∇Z t (u t (x, y)), y -x = 0. It follows that ( d dt Z t ) + ∇Z t (Z t ) = 0.
In the case where Z t = ∇ψ t , we have

( d dt ∇ψ t ) + ∇ 2 ψ t (∇ψ t ) = 0.
We remark that {∇ψ t , t ∈]0, 1[} satisfies heuristically the equation of Riemannian geodesic obtained in [START_REF] Lott | Some geometric calculation on Wasserstein space[END_REF] or heuristically obtained in [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF], in which the authors showed that the convexity of entropy functional along these geodesics is equivalent to Bakry-Emery's curvature condition [START_REF] Bakry | Diffusion hypercontractivities, Sém. de Probab[END_REF] (see also [START_REF] Li | W -entropy formulas and Langevin deformation of flows on the Wasserstein space over Riemannian manifolds[END_REF], [START_REF] Sturm | Transport inequalities, gradient estimates, entropy and Ricci curvature[END_REF][START_REF] Sturm | On the geometry of metric measure spaces[END_REF]).

In the case of Riemannian manifold M , it is a bit complicated. We follow the exposition of [START_REF] Gigli | On the inverse implication of Brenier-McCann theorems and the structure of (P 2 (M ), W 2 )[END_REF]. Let T M be the tangent bundle of M and π : T M → M the natural projection. For each µ ∈ P 2 (M ), we consider the set

Γ µ = γ probability measure on T M ; π # γ = µ, T M |v| 2 TxM dγ(x, v) < +∞ .
The set Γ µ is obviously non empty. Let γ ∈ Γ µ , we consider ν = exp # γ, that is,

M f (x)dν(x) = T M f (exp x (v)) dγ(x, v),
where exp x : T x M → M is the exponential map induced by geodesics on M . The map

T M → M × M, (x, v) → (x, exp x (v))
sends γ to a coupling plan γ ∈ C(µ, ν). We have

W 2 2 (µ, ν) ≤ T M d 2 M (x, exp x (v)) dγ(x, v) ≤ T M |v| 2 TxM dγ(x, v).
In order to construct geodesics {c t ; t ∈ [0, 1]} connecting µ and ν, we need to find

γ 0 ∈ Γ µ such that W 2 2 (µ, ν) = T M |v| 2 TxM dγ 0 (x, v). (1.6)
As M is connected, let x ∈ M , for each y, there is a minimizing geodesic {ξ(t), t ∈ [0, 1]} connecting x and y. Let v x,y = ξ ′ (0) ∈ T x M , then

y = exp x (v x,y ) and d M (x, y) = |v x,y | TxM .
Take a Borel version Ξ of such a map (x, y) → (x, v x,y ) from M × M to T M . Let γ0 ∈ C(µ, ν) be an optimal coupling plan; define γ 0 ∈ Γ µ by

T M g(x, v) dγ 0 (x, v) = M ×M g x, Ξ(x, y) dγ 0 (x, y). Therefore T M |v| 2 TxM dγ 0 (x, v) = M ×M |Ξ(x, y)| 2 dγ 0 (x, y) = M ×M d M (x, y) 2 dγ 0 (x, y) = W 2 2 (µ, ν).

Now we define the curve {c

t ; t ∈ [0, 1]} on P 2 (M ) by M f (x)dc t (x) = T M f (exp x (tv)) dγ 0 (x, v).
Similarly we check that

W 2 (c s , c t ) = |t -s| W 2 (c 0 , c 1 ).
The organization of the paper is as follows. In Section 2, we consider ordinary equations on P 2 (M ), a Cauchy-Peano's type theorem is established, also Mckean-Vlasov equation involved.

In Section 3, we emphasize that the suitable class of probability measures for developing the differential geometry is one having divergence and the strictly positive density with certain regularity. The Levi-Civita connection is introduced and the formula for the covariant derivative of a general but smooth enough vector field is obtained. In section 4, we precise results on the derivability of the Wasserstein distance on P 2 (M ), which enable us to obtain the extension of a vector field along a quite good curve on P 2 (M ) in Section 5 as in differentiable geometry; the parallel translation along such a good curve on P 2 (M ) is naturally and rigorously introduced. The existence for parallel translations is established for a curve whose intrinsic derivative gives rise a good enough vector field on P 2 (M ).

2 Ordinary differential equations on P 2 (M)

Let ϕ ∈ C 1 (M ), consider the function F ϕ on P 2 (M ) defined by F ϕ (µ) = M ϕ(x) dµ(x). (2.1)
A function F on P 2 (M ) is said to be a polynomial if there exists a finite number of functions

ϕ 1 , . . . , ϕ k in C 1 (M ) such that F = F ϕ 1 • • • F ϕ k . Let Z = V ψ be a constant vector field on P 2 (M ) with ψ ∈ C ∞ (M )
, and U t the flow on M associated to ∇ψ. For µ 0 ∈ P 2 (M ), we set µ t = (U t ) # µ 0 . Then we have seen in section 1.1,

d dt F ϕ (µ t ) | t=0 = M ∇ϕ(x), ∇ψ(x) dµ 0 (x) = V ϕ , V ψ Tµ 0 .
The left hand side of above equality is the derivative of F ϕ along V ψ . More generally, for a function F on P 2 (M ), we say that F is derivable at µ 0 along V ψ , if

( DV ψ F )(µ 0 ) = d dt F (µ t ) | t=0
exists.

We say that the gradient ∇F (µ

0 ) ∈ Tµ 0 exists if for each ψ ∈ C ∞ (M ), ( DV ψ F )(µ 0 ) exists and DV ψ F (µ 0 ) = ∇F, V ψ Tµ 0 . (2.2) Note that for ϕ ∈ C 1 (M ), there is a sequence of ψ n ∈ C ∞ (M ) such that ∇ψ n converge uniformly to ∇ϕ so that V ϕ ∈ Tµ for any µ ∈ P 2 (M ). It is obvious that ∇F ϕ = V ϕ . For the polynomial F = k i=1 F ϕ i , we have ∇F = k i=1 j =i F ϕ j V ϕ i .
Note that the family {F ϕ , ϕ ∈ C 1 (M )} separates the point of P 2 (M ). By Stone-Weierstrauss theorem, the space of polynomials is dense in the space of continuous functions on P 2 (M ).

Convention of notations:

We will use ∇ to denote the gradient operator on the base space M , and ∇ to denote the gradient operator on the Wasserstein space (P 2 (M ), W 2 ). For example, if (µ, x) → Φ(µ, x) is a function on P 2 (M ) × M , then ∇Φ(µ, x) is the gradient with respect to x, while ∇Φ(µ, x) is the gradient with respect to µ.

Definition 2.1. We will say that Z is a vector field on P 2 (M ) if there exists a Borel map

Φ : P 2 (M ) × M → R such that for any µ ∈ P 2 (M ), x → Φ(µ, x) is C 1 and Z(µ) = V Φ(µ,•) .
A class of test vector fields on P 2 (M ) is

χ(P) = f inite α i V ψ i , α i polynomial, ψ i ∈ C ∞ (M ) . (2.3)
Let Z be a vector field on P 2 (M ), how to construct a solution µ t ∈ P 2 (M ) to the following

ODE d I µ t dt = Z(µ t )?
Theorem 2.2. Let Z be a vector field on P 2 (M ) given by Φ. Assume that (µ, x) → ∇Φ(µ, x) is continuous, then for any µ 0 ∈ P 2 (M ), there is an absolutely curve

{µ t ; t ∈ [0, 1]} on P 2 (M ) such that d I µ t dt = Z(µ t ), µ | t=0 = µ 0 . (2.4) If moreover, for any µ ∈ P 2 (M ), x → Φ(µ, x) is C 2 and C 2 := sup µ∈P 2 (M ) sup x∈M ||∇ 2 Φ(µ, x)|| < +∞, (2.5)
then there is a flow of continuous maps (t, x) → U t (x) on M , solution to the following Mckean-Vlasov equation

d dt U t (x) = ∇Φ(µ t , U t (x)), µ t = (U t ) # µ 0 . (2.6)
Proof. We use the Euler approximation to construct a solution. We first note that

C 1 := sup (µ,x)∈P 2 (M )×M |∇Φ(µ, x)| < +∞.
(2.7)

Let P t = e t∆ M be the heat semi-group associated to the Laplace operator ∆ M on functions, and T t = e -t the heat semigroup on differential forms, with de Rham-Hodge operator .

It is well-known that

|T t (∇ϕ)| ≤ e -tκ/2 P t |∇ϕ|, ϕ ∈ C 1 (M )
where κ is lower bound of Ricci tensor on M . As t → 0, T t (∇ϕ) converges to ∇ϕ uniformly.

For n ≥ 1, let

Z n (µ, x) = T 1/n ∇Φ(µ, •) (x).
According to (2.7) and above estimate, for n big enough, sup

(µ,x)∈P 2 (M )×M |Z n (µ, x)| ≤ 2C 1 . (2.8) Now let t k = k2 -n for k = 1, . . . , 2 n and [t] = t k if t ∈ [t k , t k+1 [.
On the intervall [t 0 , t 1 ], consider the ODE on M :

dU (n) t dt = Z n µ 0 , U (n) t , U (n) 0 (x) = x, (2.9)
and µ

(n) t = (U (n) t ) # µ 0 for t ∈ [t 0 , t 1 ]; inductively, on [t k , t k+1 ], we consider dU (n) t dt = Z n µ (n) t k , U (n) t , U (n) |t=t k (x) = U (n) t k (x), (2.10
)

and for t ∈ [t k , t k+1 ], µ (n) t = (U (n) t ) # µ (n) t k (2.11)
and so on, we get a curve {µ

(n) t ; t ∈ [0, 1]} on P 2 (M ). We now prove that this family is equicontinuous in C([0, 1], P 2 (M )). Let 0 ≤ s < t ≤ 1, define γ(θ) = U (n) (1-θ)s+θt , then dγ(θ) dθ = (t -s)Z n µ (n) [(1-θ)s+θt] , U (n)
(1-θ)s+θt .

We have, according to (2.8),

d M U (n) t (x), U (n) s (x) ≤ 1 0 dγ(θ) dθ dθ ≤ 2C 1 (t -s).
Define a probability measure π on M × M by

M ×M g(x, y)π(dx, dy) = M g U (n) t (x), U (n) s (x) dµ 0 (x).
Then π ∈ C(µ

(n) t , µ (n)
s ), we have

W 2 2 µ (n) t , µ (n) s ≤ M d 2 M U (n) t (x), U (n) s (x) dµ 0 (x) ≤ 4C 2 1 (t -s) 2 .
By Ascoli theorem, up to a subsequence, µ

(n) • converges in C([0, 1], P 2 (M )) to a continuous curve {µ t ; t ∈ [0, 1]} such that W 2 (µ t , µ s ) ≤ 2C 1 (t -s).
For proving that {µ t ; t ∈ [0, 1]} is a solution to ODE (2.4), we need the following preparation:

Lemma 2.3. Set Φ µ (x) = Φ(µ, x), then sup (µ,x)∈P 2 (M )×M |(T t ∇Φ µ )(x) -∇Φ(x)| TxM → 0, as t → 0.
(2.12)

Proof. We use || • || ∞ to denote the uniform norm on M . Let ε > 0, for µ ∈ P 2 (M ), there is tµ > 0 such that sup

t≤ tµ ||T t ∇Φ µ -∇Φ µ || ∞ < ε. Since (µ, t) → ||T t ∇Φ µ -∇Φ µ || ∞ is continuous, there is δ µ > 0 such that for t ≤ tµ , W 2 (µ, ν) < δ µ ⇒ ||T t ∇Φ ν -∇Φ ν || ∞ < ε.
Let B(µ, δ) be the open ball in (P 2 (M ), W 2 ) centered at µ, of radius δ. We have

P 2 (M ) = ∪ µ∈P 2 (M ) B(µ, δ µ );
so there is a finite number of {µ 1 , . . . , µ K } such that

P 2 (M ) = ∪ K i=1 B(µ i , δ µ i ).
Let t = min tµ i , i = 1, . . . , K > 0. Then for 0 < t < t, sup

µ∈P 2 (M ) ||T t ∇Φ µ -∇Φ µ || ∞ ≤ ε.
So we get (2.12).

End of the proof of theorem : {µ

(n t ; t ∈ [0, 1]} satisfies the following continuity equation [0,1]×M α ′ (t)f (x)dµ (n) t (x)dt = α(0) M f (x)dµ 0 (x) + [0,1]×M α(t) ∇f (x), Z n µ (n) [t] , x dµ (n) t (x)dt, (2.13)
for all α ∈ C 1 c ([0, 1)) and f ∈ C 1 (M ). We have

[0,1]×M α(t) ∇f (x), Z n µ (n) [t] , x dµ (n) t dt - [0,1]×M α(t) ∇f (x), ∇Φ µ t , x dµ t dt = [0,1]×M α(t) ∇f (x), Z n µ (n) [t] , x -∇Φ(µ t , x) dµ (n) t dt + [0,1]×M α(t) ∇f (x), ∇Φ µ t , x dµ (n) t dt - [0,1]×M α(t) ∇f (x), ∇Φ µ t , x dµ t dt.
It is obvious that the sum of two last terms converge to 0 as n → +∞. Let I n be the first term on the right side, then

|I n | ≤ ||∇f || ∞ 1 0 |α(t)| ||T 1/n ∇Φ µ (n) [t] -∇Φ µt || ∞ dt Note that ||T 1/n ∇Φ µ (n) [t] -∇Φ µt || ∞ ≤ ||T 1/n ∇Φ µ (n) [t] -∇Φ µ (n) [t] || ∞ + ||∇Φ µ (n) [t] -∇Φ µt || ∞ . The term ||T 1/n ∇Φ µ (n) [t] -∇Φ µ (n) [t]
|| ∞ → 0 is due to above lemma. As n → +∞, µ

[t] converges to µ t . By continuity of (µ, x) → ∇Φ(µ, x), the last term tends to 0. Letting n → +∞ in (2.13) yields

[0,1]×M α ′ (t)f (x)dµ t (x)dt = α(0) M f (x)dµ 0 (x) + [0,1]×M α(t) ∇f (x), ∇Φ µ t , x dµ t (x)dt,
which is the meaning of Equation (2.4) in distribution sense.

For the proof of second part, since x → Φ(µ, x) is C 2 , we can directly use ∇Φ(µ, •) instead of Z n in (2.9), (2.10), (2.11). On the intervall [t 0 , t 1 ], consider the ODE on M :

dU (n) t dt = ∇Φ µ 0 , U (n) t , U (n) 0 (x) = x, (2.14)
and µ

(n) t = (U (n) t ) # µ 0 for t ∈ [t 0 , t 1 ]; inductively, on [t k , t k+1 ], we consider dU (n) t dt = ∇Φ µ (n) t k , U (n) t , U (n) |t=t k (x) = U (n) t k (x), (2.15)
and for t ∈ [t k , t k+1 ], µ

(n) t = (U (n) t ) # µ (n) t k . (2.16)
By above result, up to a subsequence, {µ

(n) t , t ∈ [0, 1]} converges to {µ t , t ∈ [0, 1]} in C([0, 1], P 2 (M )
). We use this subsequence to prove the convergence of {U

(n) t (x), t ∈ [0, 1]}. Now we prove that, under Condition (2.7), d M U (n) t (x), U (n) t (y) ≤ e C 2 t d M (x, y), x, y ∈ M.
(2.17)

For x, y ∈ M given, there is a minimizing geodesic {ξ s , s ∈ [0, 1]} connecting x and y such that d M (x, y) =

1 0 |ξ ′ s | ds. Set σ(t, s) = U (n) t (ξ s ).
Since the torsion is free, we have the relation:

D ds d dt σ(t, s) = D dt d ds σ(t, s), (2.18)
where D ds denotes the covariant derivative. We have

d dt U (n) t (ξ s ) = ∇Φ µ (n) [t] , U (n) t (ξ s ) .
Taking the derivative with respect to s, we get

D ds d dt U (n) t (ξ s ) = ∇ 2 Φ µ (n) [t] , U (n) t (ξ s ) • d ds U (n) t (ξ s ).
Combining with (2.18) yields

D dt d ds U (n) t (ξ s ) = ∇ 2 Φ µ (n) [t] , U (n) t (ξ s ) • d ds U (n) t (ξ s ). Now, d dt d ds U (n) t (ξ s ) 2 = 2 ∇ 2 Φ µ (n) [t] , U (n) t (ξ s ) • d ds U (n) t (ξ s ), d ds U (n) t (ξ s ) ,
which is, by Condition (2.7), less than

2C 2 d ds U (n) t (ξ s ) 2 .
By Gronwall lemma,

d ds U (n) t (ξ s ) ≤ e C 2 t |ξ ′ s |, which implies that d M U (n) t (x), U (n) t (y) ≤ e C 2 t d M (x, y).
Therefore the family (t,

x) → U (n) t (x); n ≥ 1 is equicontinuous in C([0, 1] × M ). By Ascoli theorem, up to a subsequence, U (n) t (x) converges to U t (x) uniformly in (t, x) ∈ [0, 1] × M . It is obvious to see that (U t , µ t ) solves Mckean-Vlasov equation (2.6).
Remark 2.4. Comparing to [START_REF] Buckdahn | Mean-field stochastic differential equations and associated PDEs[END_REF], as well to [START_REF] Wang | Diffusions and PDEs on Wasserstein Space[END_REF], we did not suppose the Lipschitz continuity with respect to µ; in counterpart, we have no uniqueness of solutions of (2.6).

Remark 2.5. Many interesting PDE can be interpreted as gradient flows on the Wasserstein space P 2 (M ) (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], [START_REF] Villani | Optimal transport, Old and New[END_REF], [START_REF] Villani | Topics in optimal transportation[END_REF], [START_REF] Fang | Fokker-Planck equation with respect to heat measures on loop groups Bull[END_REF]). The interpolation between geodesic flows and gradient flows were realized using Langevin's deformation in [START_REF] Li | W -entropy formulas and Langevin deformation of flows on the Wasserstein space over Riemannian manifolds[END_REF][START_REF] Li | W -entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds[END_REF].

3 Levi-Civita connection on P 2 (M)

In this section, we will revisit the paper by J. Lott [START_REF] Lott | Some geometric calculation on Wasserstein space[END_REF]: we try to reformulate conditions given there as weak as possible, also to expose some of them in an intrinsic way, avoiding the use of density. In order to obtain good pictures on the geometry of P 2 (M ), the suitable class of probability measures should be the class P div (M ) of probability measures on M having divergence (see Definition 1.2).

For convenience of readers, we will briefly prepare materials needed for our exposition. For a measure µ ∈ P 2 (M ), for any

C 1 vector field A on M , the divergence div µ (A) ∈ L 2 (M, µ) is such that M ∇φ(x), A(x) TxM dµ(x) = - M φ(x) div µ (A)(x) dµ(x) for any φ ∈ C 1 (M ). It is easy to see that div µ (f A) = f div µ (A) + ∇f, A for f ∈ C 1 (M ). If dµ = ρ dx has a density ρ > 0 in the space C 1 (M ), we have M ∇φ, A dµ = M ∇φ, ρA dx = - M φ div(ρA) dx = - M φ div(ρA) ρ -1 dµ, It follows that div µ (A) = ρ -1 div(ρA) = div(A) + ∇(log ρ), A . (3.1)
For µ ∈ P div (M ) and φ ∈ C 2 (M ), we denote

L µ (φ) ∈ L 2 (µ) such that M ∇f, ∇φ dµ = - M f L µ φ dµ, for any f ∈ C 1 (M ), (3.2)
where L µ φ = div µ (∇φ) is a negative operator.

Let ψ ∈ C 3 (M ), consider the ODE

dU t dt = ∇ψ(U t ), U 0 (x) = x.
Proposition 3.1. Let dµ = ρ dx be a probability measure in P div (M ) with a strictly positive density ρ in C 1 (M ) and ψ ∈ C 3 (M ). Then for each t ∈ [0, 1], µ t := (U t ) # µ ∈ P div (M ).

Proof. By Kunita [START_REF] Kunita | Stochastic Flows and Stochastic Differentail Equations[END_REF] (see also [START_REF] Cruzeiro | Equations différentielles sur l'espace de Wiener et formules de Cameron-Martin non linéaires[END_REF], [START_REF] Malliavin | Stochastic analysis[END_REF]), the push-forward measure (U -1 t ) # µ by inverse map of U t admits a density Kt with respect to µ, having the following explicit expression

Kt = exp - t 0 div µ (∇ψ)(U s (x))ds .
It follows that the density K t of µ t with respect to µ has the expression

K t = exp t 0 div µ (∇ψ)(U -s (x))ds .
According to (3.1), x → div µ (∇ψ(x)) is C 1 . Therefore the condition in [START_REF] Cruzeiro | Equations différentielles sur l'espace de Wiener et formules de Cameron-Martin non linéaires[END_REF] M exp(λdiv µ (∇ψ(x)) dµ(x) < +∞, for all λ > 0 is automatically satisfied. Again by (3.1), x → K t (x) is in C 1 . Now let A be a C 1 vector field on M and f ∈ C 1 (M ), we have

M ∇f (x), A(x) TxM dµ t (x) = M ∇f, A TxM K t (x)dµ(x) = - M f div µ (K t Z) dµ. It follows that div µt (A) = div µ (K t A) K -1 t .
For ψ 1 , ψ 2 ∈ C 2 (M ), we denote by V ψ 1 , V ψ 2 the associated constant vector fields on P 2 (M ).

In what follows, we will compute the Lie bracket

[V ψ 1 , V ψ 2 ]. For f ∈ C 1 (M ), we set F f (µ) = M f dµ.
According to preparations given at the beginning of Section 2,

( DV ψ 2 F f )(µ) = M ∇ψ 2 , ∇f dµ = F ∇ψ 2 ,∇f (µ).
Using again above formula, we have

( DV ψ 1 DV ψ 2 F f )(µ) = M ∇ψ 1 , ∇ ∇ψ 2 , ∇f dµ = - M L µ ψ 1 ∇ψ 2 , ∇f dµ. Therefore [V ψ 2 , V ψ 1 ]F f = DV ψ 2 DV ψ 1 F f -DV ψ 1 DV ψ 2 F f = M (L µ ψ 1 ∇ψ 2 -L µ ψ 2 ∇ψ 1 ), ∇f dµ. Let C ψ 1 ,ψ 2 (µ) = L µ ψ 1 ∇ψ 2 -L µ ψ 2 ∇ψ 1 . (3.3) Note that C ψ 1 ,ψ 2 (µ) is in L 2 (M, T M ; µ)
, not in Tµ . Consider the orthogonal projection:

Π µ : L 2 (M, T M ; µ) → Tµ .
As µ ∈ P div (M ) and by Proposition 1.3, there exists Φµ ∈ D 2 1 (µ) such that

Π µ (C ψ 1 ,ψ 2 (µ)) = ∇ Φµ . (3.4) Then we have [V ψ 2 , V ψ 1 ]F f = M ∇ Φµ , ∇f dµ = ( DV Φµ F f )(µ). (3.5)
Above equality can be extended to the class of polynomials on P 2 (M ), that is to say that

[V ψ 2 , V ψ 1 ] µ = V Φµ on polynomials, (3.6)
We emphasize that Lie bracket of two constant vector fields is no more a constant vector field.

Proposition 3.2. Let ψ 1 , ψ 2 ∈ C 3 (M ), for dµ = ρ dx with ρ > 0 and ρ ∈ C 2 (M ), the function Φµ obtained in (3.4) has the following expression :

Φµ = (L µ ) -1 div µ C ψ 1 ,ψ 2 (µ) . (3.7)
Proof. By (3.1),

L µ ψ = ∆ M ψ + ∇ log ρ, ∇ψ ,
where ∆ M denotes the Laplace operator on M . It is well-known that L µ has a spectral gap if log ρ ∈ C 2 (M ). In [START_REF] Lott | Some geometric calculation on Wasserstein space[END_REF], the Lie bracket [V ψ 2 , V ψ 1 ] was expressed using Hodge decomposition for vector fields in L 2 (µ). For

ψ 1 , ψ 2 ∈ C 3 (M ), we have div µ C ψ 1 ,ψ 2 (µ) = ∇L µ ψ 1 , ∇ψ 2 -∇L µ ψ 2 , ∇ψ 1 .
By Hodge decomposition, C ψ 1 ,ψ 2 (µ) admits the decomposition

C ψ 1 ,ψ 2 (µ) = d µ * ω + ∇f + h,
where ω is a differential 2-form on M , d µ * is adjoint operator of exterior derivative in L 2 (µ), h is harmonic form : (d µ * d + dd µ * )h = 0. Taking the divergence div µ on the two sides of above equality, we see that f is a solution the following equation

L µ f = div µ C ψ 1 ,ψ 2 (µ) .

It follows that Φµ has the expression (3.7).

Proof. It is enough to prove (3.9). We have

V ψ 3 , [V ψ 1 , V ψ 2 ] Tµ = M -L µ ψ 1 ∇ψ 2 + L µ ψ 2 ∇ψ 1 , ∇ψ 3 dµ = M ∇ψ 1 , ∇ ∇ψ 2 , ∇ψ 3 dµ - M ∇ψ 2 , ∇ ∇ψ 1 , ∇ψ 3 dµ = M ∇ 2 ψ 2 , ∇ψ 1 ⊗ ∇ψ 3 + ∇ 2 ψ 3 , ∇ψ 1 ⊗ ∇ψ 2 dµ - M ∇ 2 ψ 1 , ∇ψ 2 ⊗ ∇ψ 3 + ∇ 2 ψ 3 , ∇ψ 2 ⊗ ∇ψ 1 dµ = M ∇ 2 ψ 2 , ∇ψ 1 ⊗ ∇ψ 3 -∇ 2 ψ 1 , ∇ψ 2 ⊗ ∇ψ 3 dµ,
due to the symmetry of the Hessian ∇ 2 ψ 3 . On the other hand,

V ψ 3 , V ∇ψ 1 ,∇ψ 2 Tµ = M ∇ 2 ψ 2 , ∇ψ 3 ⊗ ∇ψ 1 + ∇ 2 ψ 1 , ∇ψ 3 ⊗ ∇ψ 2 dµ.
Summing these last two equalities yields (3.9). Remark 3.4. By (3.8), for two constant vector fields

V ψ 1 , V ψ 2 , the covariant derivative ∇V ψ 1 V ψ 2 is not a constant vector field on P 2 (M ) if ψ 1 = ψ 2 . Let α : P 2 (M ) → R be a differentiable function, we define ∇V ψ 1 α V ψ 2 = DV ψ 1 α • V ψ 2 + α ∇V ψ 1 V ψ 2 . (3.10)
Proposition 3.5. Let Z be a vector field on P 2 (M ) in the test space χ(P), that is, Z = k i=1 α i V ψ i with α i polynomials. Then ∇Z Z still is in the test space; moreover

∇Z Z = V Φ 1 + 1 2 V |∇Φ 2 | 2 ,
where

Φ 1 = k j=1 k i=1 α i DV ψ i α j ψ j , Φ 2 = k i=1 α i ψ i .
Proof. Using the rule concerning covariant derivatives, ∇Z Z is equal to

k i,j=1 α i DV ψ i α j V ψ j + 1 2 k i,j=1 α i α j V ∇ψ i ,∇ψ j + 1 2 k i,j=1 α i α j [V ψ i , V ψ j ].
The last sum is equal to 0 due to the skew-symmetry of [V ψ i , V ψ j ], the first one gives rise to Φ 1 and the second one gives rise to Φ 2 .

In what follows, we will extend the definition of covariant derivative (3.10) for a general vector field Z on P 2 (M ). Let ∆ be the Laplace operator on M , let {ϕ n , n ≥ 0} be the eigenfunctions of ∆:

-∆ϕ n = λ n ϕ n .

We have λ 0 = 0 and ϕ 0 = 1. It is well-known, by Weyl's result, that

λ n ∼ n 2/m , n → +∞
where m is the dimension of M . The functions {ϕ n ; n ∈ N} are smooth, chosen to form an orthonormal basis of L 2 (M, dx). A function f on M is said to be in

H k (M ) for k ∈ N, if ||f || 2 H k = M |(I -∆) k/2 f | 2 dx < +∞.
By Sobolev embedding inequality, for k > m 2 + q,

||f || C q ≤ C ||f || H k . For f ∈ H k (M ), put f = n≥0 a n ϕ n which holds in L 2 (M, dx) with a n = M f (x)ϕ n (x) dx.
We have :

||f || 2 H k = n≥0 a 2 n (1 + λ n ) k .
The system

∇ϕ n √ λ n ; n ≥ 1 is orthonormal. Let V n = V ϕn/ √ λn , then {V n ; n ≥ 1} is an orthonormal basis of Tdx .
Let Z be a vector field on P 2 (M ) given by Z(µ) = V Φ(µ,•) or Z(µ) = ∇Φ(µ, •). In the sequel, we denote: Φ µ (x) = Φ(µ, x), Φ x (µ) = Φ(µ, x). Then, if x → ∇Φ µ (x) is continuous,

∇Φ µ = n≥1 M ∇Φ µ , ∇ϕ n √ λ n dx ∇ϕ n √ λ n = n≥1 M Φ µ ϕ n dx ∇ϕ n ,
which converges in L 2 (M, dx). Let µ ∈ P div (M ), the above series converges also in Tµ . Let

a n (µ) = M Φ µ (x)ϕ n (x) dx. (3.11)
Let V ψ be a constant vector field on P 2 (M ) with ψ ∈ C ∞ (M ). For q ≥ p ≥ 1, set

S p,q = q n=p DV ψ a n V ϕn + a n ∇V ψ V ϕn = S 1 p,q + S 2 p,q (3.12)
respectively. Let φ ∈ C ∞ (M ), according to (3.9), we have

S 2 p,q , V φ Tµ = M q n=p a n (µ)∇ 2 ϕ n (∇ψ(x), ∇φ(x)) dµ(x).
It follows that

| S 2 p,q , V φ Tµ | ≤ q n=p a n (µ)∇ 2 ϕ n ∞ |V ψ | Tµ |V φ | Tµ , therefore |S 2 p,q | Tµ ≤ q n=p a n (µ)∇ 2 ϕ n ∞ |V ψ | Tµ .
We have

|| q n=p a n (µ)(I -∆) k/2 ϕ n || 2 L 2 (dx) = q n=p a n (µ) 2 (1 + λ n ) k = q n=p M (I -∆) k/2 Φ µ ϕ n dx 2 → 0 as p, q → +∞ if Φ µ ∈ H k (M ).
On the other hand, we have

( DV ψ a n )(µ) = M ( DV ψ Φ x )(µ)ϕ n (x) dx = M ∇ DV ψ Φ x , ∇ϕ n √ λ n dx √ λ n , then S 1 p,q = q n=p M ∇ DV ψ Φ x , ∇ϕ n √ λ n dx ∇ϕ n √ λ n and M |S 1 p,q | 2 dx = q n=p M ∇ DV ψ Φ x , ∇ϕ n √ λ n dx 2 → 0 as p, q → +∞ if M |∇ DV ψ Φ x | 2 dx < +∞.
Therefore for dµ = ρ dx with µ ∈ P div (M ), as p, q → ∞,

|S 1 p,q | 2 Tµ ≤ ||ρ|| ∞ M |S 1 p,q | 2 dx → 0.
We get the following result Theorem 3.6. Let Z be a vector field on P 2 (M ) given by Φ :

P 2 (M ) × M → R. Assume that (i) for any µ ∈ P 2 (M ), Φ µ ∈ H k (M ) with k > m 2 + 2,
(ii) for any x ∈ M, DV ψ Φ x exists and ∇ DV ψ Φ • ∈ L 2 (M, dx).

Then the covariant derivative ∇V ψ Z is well defined at µ ∈ P div (M ) and for φ ∈ C ∞ (M ),

∇V ψ Z, V φ Tµ = M (∇ DV ψ Φ • ), ∇φ dµ + M ∇ 2 Φ µ ∇ψ, ∇φ dµ. (3.13) Proof. Let Z q = q n=1
a n V ϕn . Then ∇V ψ Z q = S 1,q .

Letting q → +∞ yields the result.

4 Derivability of the square of the Wasserstein distance Let {c t ; t ∈ [0, 1]} be an absolutely continuous curve on P 2 (M ), for σ ∈ P 2 (M ) given, the derivability of t → W 2 2 (σ, c t ) was established in chapter 8 of [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] , as well as in [START_REF] Villani | Optimal transport, Old and New[END_REF] (see pages 636-649); however they hold true only for almost all t ∈ [0, 1]. The derivability at t = 0 was proved in Theorem 8.13 of [START_REF] Villani | Topics in optimal transportation[END_REF] if σ and c 0 have a density with respect to dx. When {c t } is a geodesic of constant speed, the derivability at t = 0 was given in theorem 4.2 of [START_REF] Gigli | On the inverse implication of Brenier-McCann theorems and the structure of (P 2 (M ), W 2 )[END_REF] where the property of semi concavity was used. In what follows, we will use constant vector fields on P 2 (M ).

Before stating our result, we recall some well-known facts concerning optimal transport maps (see [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF][START_REF] Brenier | Polar factorization and monotone rearrangement of vector valued functions[END_REF][START_REF] Mccann | Polar factorization of maps on Riemannian manifolds[END_REF][START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Villani | Optimal transport, Old and New[END_REF]). Let σ ∈ P 2,ac (M ) be absolutely continuous with respect to dx and µ ∈ P 2 (M ), then there is an unique Borel map φ ∈ D 2 1 (σ) such that

M |∇φ(x)| 2 dσ(x) = W 2 2 (σ, µ)
and x → T (x) = exp x (∇φ(x)) pushes σ forward to µ. If µ is also in P 2,ac (M ), the map T : M → M is invertible and its inverse map T -1 is given by y → exp y (∇ φ(y)) with some function φ such that M |∇ φ| 2 dµ < +∞. We need also the following result Lemma 4.1. Let x, y ∈ M and {ξ(t); t ∈ [0, 1]} be a minimizing geodesic connecting x and y, given by ξ(t) = exp x (tu) with some u ∈ T x M . Then

d 2 M (exp y (v), x) -d 2 M (y, x) ≤ 2 v, ξ ′ (1) Ty M + o(|v|) as |v| → 0. ( 4.1)
Proof. See [START_REF] Mccann | Polar factorization of maps on Riemannian manifolds[END_REF], page 10.

Theorem 4.2. Assume that σ ∈ P 2,ac (M ) is absolutely continuous with respect to dx, then µ → χ(µ) := W 2 2 (σ, µ) is derivable along each constant vector field V ψ at any µ ∈ P 2 (M ). If µ ∈ P 2,ac (M ), the gradient ∇χ exists and admits the expression :

∇χ(µ) = ∇ φ. ( 4

.2)

Proof. Let ψ ∈ C ∞ (M ) and (U t ) t∈R be the associated flow of diffeomorphisms of M :

dU t (x) dt = ∇ψ(U t (x)), x ∈ M. (4.3)
The inverse map U -1

t of U t satisfies the ODE dU -1 t (x) dt = -∇ψ(U -1 t (x)), x ∈ M. (4.4) Set µ t = (U t ) # µ, then µ = (U -1 t ) # µ t . Let γ ∈ C o (σ, µ) be the optimal coupling plan such that W 2 2 (σ, µ) = M ×M d 2 M (x, y) dγ(x, y).
The map (x, y) → (x, U t (y)) pushes γ forword to a coupling plan γ t ∈ C(σ, µ t ). Then for t > 0,

1 t W 2 2 (σ, µ t ) -W 2 2 (σ, µ) ≤ 1 t M ×M d 2 M (x, U t (y)) -d 2 M (x, y) dγ(x, y) = 1 t M ×M d 2 M (x, U t (y)) -d 2 M (x, exp y (t∇ψ(y)) dγ(x, y) + 1 t M ×M d 2 M (x, exp y (t∇ψ(y)) -d 2 M (x, y) dγ(x, y) = I 1 (t) + I 2 (t)
respectively. Let ξ(t) = exp x (t∇φ(x)), by [START_REF] Mccann | Polar factorization of maps on Riemannian manifolds[END_REF], ξ is a minimizing geodesic connecting x and y = T (x). By Lemma 4.1, we have To justifier the passage of limit throught the integral, we note that for t > 0, For obtaining the minoration, we use as in [START_REF] Villani | Topics in optimal transportation[END_REF] the fact that lim t↓0 (-a t ) = -lim t↓0 a t .

Let γt ∈ C o (σ, µ t ) be the optimal coupling plan: Let T t : M → M be the optimal transport map which pushes σ forword to µ t , with T t (x) = exp x (∇φ t (x)). As t ↓ 0, the map T t converges in measure to T (see for example [START_REF] Villani | Topics in optimal transportation[END_REF], page

W 2 2 (σ, µ t ) = M ×M d 2 M (x,
Proof. Following Section 5 of [START_REF] Ambrosio | Construction of the parallel transport in the Wasserstein space[END_REF], for s ≤ t, we define P t,s : Tcs → Tct , u s → Π ct τ t-s u s • U -1 s,t .

For a subdivision D = {0 = t 0 < t 1 < . . . < t n = 1} of [0, 1], we define

P D : Tc 0 → Tc 1 , u 0 → (P 1,t n-1 • • • • • P t 1 ,0 )(u 0 ).
Under the assumption of Theorem, we have the uniform bound sup (t,x)∈[0,1]×M ||∇ 2 Φ t (x)|| < +∞, which allows us to mimic the construction of section 5 in [START_REF] Ambrosio | Construction of the parallel transport in the Wasserstein space[END_REF], so that we get that P D converges as D becomes finer and finer, with |D| = max i |t i -t i-1 | → 0.

As a result of (5.10), we have as in [START_REF] Ambrosio | Construction of the parallel transport in the Wasserstein space[END_REF] the following property: In what follows, we will relaxe a bit conditions in Proposition 5.5. We return to the situation in Theorem 5.1. Let {c t ; t ∈ [0, 1]} be an absolutely curve in P div (M ) satisfying conditions in Theorem 5.1, set d I c t dt = V Φt .

If furthermore (t, x) → ∇ 2 Φ t (x) is continuous, according to the the construction, the extension Φ(µ, x) of (t, x) → ∇ 2 Φ t (x) obtained in (5.3) satisfies (µ, x) → ∇ 2 Φ(µ, x) is continuous.

In particular, the condition (2.5)

d 2 M 2 M

 22 x, exp y (t∇ψ(y) -d 2 M (y, x) ≤ 2t ∇ψ(y), ξ ′ (1) Ty M + o(|t|) as t → 0. On other hand, ξ ′ (1) = d exp x (∇φ(x)) • ∇φ(x) = // ξ 1 ∇φ(x), where // ξ t denotes the parallel translation along the geodesic ξ. Hence |ξ ′ (1)| = |∇φ(x)|. Therefore I 2 (t) ≤ ∇ψ(T (x)), d exp x (∇φ(x)) • ∇φ(x) dσ(x) + o(1)

1 t d 2 MI 2 (t) ≤ 2 M

 1222 x, exp y (t∇ψ(y)) -d 2 M (x, y) ≤ 2 t diam(M ) d M y, exp y (t∇ψ(y)) ≤ 2 diam(M ) |∇ψ(y)|.Then lim t↓0 ∇ψ(T (x)), d exp x (∇φ(x)) • ∇φ(x) dσ(x).

For estimating I 1 2 M

 12 (t), it is obvious that lim t↓0 1 t sup y∈M d M U t (y), exp y (t∇ψ(y)) , µ t ) -W 2 2 (σ, µ) ≤ ∇ψ(T (x)), d exp x (∇φ(x)) • ∇φ(x) dσ(x).(4.6)

  y) dγ t (x, y).Let η t ∈ C(σ, µ t ) be defined by

	Then for t > 0,
	1 t	W 2 2 (σ, µ) -W 2 2 (σ, µ t ) ≤	1 t M ×M	d 2 M (x, U -1 t (y)) -d 2 M (x, y) dγ t (x, y).

M ×M f (x, y)dη t (x, y) = M ×M f x, U -1 t (y) dγ t (x, y).

  Proposition 5.6. Let {∇Ψ t ; t ∈ [0, 1]} be given in Proposition 5.5, then Proof. We have c t+ε = (U t,t+ε ) # c t , and

			d dt	||∇Ψ t || 2 ct = 0.	(5.11)
	Therefore
	||u t+ε || 2 Tt+ε -||u t || 2 Tc t	=
	It follows that
	d dt	||∇Φ t || 2 ct = 2

M |∇Ψ t+ε (x)| 2 dc t+ε (x) = M |∇Ψ t+ε (U t,t+ε (x))| 2 dc t (x). M |τ -1 ε ∇Ψ t+ε (U t,t+ε (x))| 2 -|∇Ψ t (x)| 2 dc t (x) = M τ -1 ε ∇Ψ t+ε (U t,t+ε (x)) -∇Ψ t (x), τ -1 ε ∇Ψ t+ε (U t,t+ε (x)) dc t (x) + M ∇Ψ t (x), τ -1 ε ∇Ψ t+ε (U t,t+ε (x)) -∇Ψ t (x) dc t (x). M lim ε→0 τ -1 ε ∇Ψ t+ε (U t,t+ε (x)) -∇Ψ t (x) ε , ∇Ψ t (x) dc t (x) = 0.

Acknowledgement: This work has been prepared in a joint PhD program between the Institute of Applied Mathematics, Academy of Mathematics and Systems Science (Beijing, China) and the Institute of Mathematics of Burgundy, University of Burgundy (Dijon, France), the first named author is grateful to the hospitality of these two institutions, the financial support of China Scholarship Council is particularly acknowledged.

Now we introduce the covariant derivative ∇V ψ 1 V ψ 2 associated to the Levi-Civita connection on P 2 (M ) by

Replacing ψ 1 by ψ 2 , ψ 2 by ψ 3 and ψ 3 by ψ 1 , we get

We have, in the same way

In the same way, we get

Combining all these terms, we finally get

Moreover, for any constant vector field V ψ 3 ,

265). We have

respectively. According to (4.5), lim t↓0 J 1 (t) = 0. Concerning J 2 (t), we note as above,

Then by Lemma 4.1,

Combining (4.6) and (4.7), we finally get lim

Now if µ ∈ P 2,ac (M ) and the map y → exp y (∇ φ(y)) is the optimal transport map which pushes µ to σ. Consider the minimizing geodesic

which connects x and y. We have ξ ′ (1) = ∇ φ(y). In this case, replacing

from which we get (4.2). The proof is complete.

Parallel translations

Before introducing parallel translations on the space P div (M ), let's give a brief review on the definition of parallel translations on the manifold M , endowed with an affine connection. Let {γ(t); t ∈ [0, 1]} be a smooth curve on M , and {Y t ; t ∈ [0, 1]} a family vector fields along γ: Y t ∈ T γ(t) M . Then there exist vector fields X and Y on M such that

} be an absolutely curve on P div (M ) such that

(5.1)

then there are functions

moreover for x ∈ M , µ → Φ(µ, x) and µ → Ψ(µ, x) are derivable on P 2 (M ) along any constant vector fields V ψ , their gradients exist on P 2,ac (M ).

which is > 0 at t = t 0 . Therefore there is an open interval I(t 0 ) of t 0 such that t → α(t) is a C 1 diffeomorphism from I(t 0 ) onto an interval J(t 0 ) containing α(t 0 ). Let β : J(t 0 ) → I(t 0 ) be the inverse map of α. We have

which is an open set in P 2 (M ). Let r > 0 and ν ∈ P 2 (M ), we denote by B(ν, r) the open ball in P 2 (M ) centered at ν of radius r. Take r 0 > 0 small enough such that B(c t 0 , r 0 ) ⊂ U (t 0 ).

We define, for µ ∈ B(c t 0 , r 0 ),

(5.4)

We remark that for t ∈ [0, 1] such that c t ∈ U (t 0 ), we have: β(F Φt 0 (c t )) = t. Note that {c t ; t ∈ [0, 1]} is a compact set of P 2 (M ) and

There exists a finite number of t 1 , . . . , t k ∈ [0, 1] such that

Let µ ∈ U , then µ ∈ B(c t i , r i ); according to (5.4), we define,

Then for t ∈ [0, 1] such that c t ∈ B(c t i , r i ), Φt i (c t ) = Φ t and Ψt i (c t ) = Ψ t . Now for r > 0 and ν ∈ P 2 (M ), we define

and g r,ν (µ) = 0 otherwise. Then g r,ν (µ) > 0 if and only if µ ∈ B(ν, r). By Theorem 4.2, if

We have

It is the same for Ψ. The proof is completed.

Notice that for such a curve {c t ; t ∈ [0, 1]} given in Theorem 5.1, and {Y t ; t ∈ [0, 1]} a vector field along {c t ; t ∈ [0, 1]} given by Ψ t . If furthermore for any t ∈ [0, 1], Ψ t ∈ H k (M ) with k > m 2 + 2, then the extension obtained Ψ obtained in Theorem 5.1 satisfies conditions in Theorem 3.6. Definition 5.2. We say that {Y

Proof. Note that

Then (5.6) follows from (3.13).

When

, it is more convenient to put Equation (5.6) in the following form :

where Π ct the orthogonal projection from L 2 (M, T M, c t ) onto Tct . By arguments in the proof of Proposition 3.2, when dc t = ρ t dx with ρ t ∈ C 2 (M ) and ρ t > 0, Π ct admits the expression

The price for this pointwise formulation of (5.7) as well as of (5.8) is the involement of second order derivative of Ψ.

Remark 5.4. Let s → ξ(s) is a smooth curve on M such that ξ(0) = x and ξ ′ (0) = ∇Φ t (x), then

where τ s is the parallel translation along s → ξ(s). We refind the similar expression of parallel translations given in [START_REF] Ambrosio | Construction of the parallel transport in the Wasserstein space[END_REF].

Proposition 5.5. Assume that the curve {c t ; t ∈ [0, 1]} is induced by a flow of diffeomorphisms Φ t , that is, there is a C

Then for any u 0 = ∇Ψ 0 ∈ Tc 0 , there is a unique vector field

holds. By theorem 2.2, there exists a solution (µ t , U t ) to the following Mckean-Vlasov equation

with µ t = (U t ) # c 0 which solves the ODE on P 2 (M ):

(5.12)

Theorem 5.7. If the ODE (5.12) has the unique solution, then for each V Ψ 0 ∈ Tc 0 , there is a vector field {V Ψt ∈ Tct ; t ∈ [0, 1]} along {c t ; t ∈ [0, 1]} such that

holds in L 2 (c t ).

Proof. Note that ∇ Φ(c t , x) = ∇Φ t (x), then V Φ(ct,•) = V Φt . The curve {c t ; t ∈ [0, 1]} is therefore a solution to

Under the assumption of uniqueness of solution to (5.12), we get that c t = µ t for t ∈ [0, 1]. Now by arguments in the proof of Propositions 5.5 and 5.6, we obtain the result.

Remark 5.8. The parallel translations along diffusion paths on the Wasserstein space are discussed in a forthcoming paper [START_REF] Hao Ding | Stochastic parallel translations on the Wasserstein space[END_REF].