Geometry on the Wasserstein space over a compact Riemannian manifold.

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Geometry on the Wasserstein space over a compact Riemannian manifold.

Hao Ding, Shizan Fang

To cite this version:

Hao Ding, Shizan Fang. Geometry on the Wasserstein space over a compact Riemannian manifold..

2021. �hal-03187501�

Geometry on the Wasserstein space over a compact Riemannian manifold

Hao DING

Shizan FANG

1Institut de Math´ematiques de Bourgogne, UMR 5584 CNRS, Universit´e de Bourgogne Franche-Comt´e, F-21000 Dijon, France

2Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

March 29, 2021

Abstract

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-Savar´e and so on.

MSC 2010: 58B20, 60J45

Keywords: Constant vector fields, measures having divergence, Levi-Civita connection, parallel translations, Mckean-Vlasov equations.

1 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

the Riemannian distance and dx the Rieman- nian measure on M such that R

M

dx = 1. Since the diameter of M is finite, any probability measure µ on M is such that R

M

d

(x

, x) dµ(x) < + ∞ , where x

is a fixed point of M . As usual, we denote by P

(M ) the space of probability measures on M , endowed with the Wasserstein distance W

defined by

W

(µ

, µ

) = inf n Z

M×M

d

(x, y) π(dx, dy), π ∈ C (µ

, µ

) o ,

where C (µ

, µ

) is the set of probability measures π on M × M, having µ

, µ

as two marginal laws. It is well known that P

(M) endowed with W

is a Polish space. In this compact case, the weak convergence for probability measures is metrized by W

; therefore ( P

(M ), W

) is a compact Polish space.

The introduction of tangent spaces of P

(M ) can go back to the early work [19], as well as in [18]. A more rigorous treatment was given in [2]. 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

M . A classical result says that for an absolutely continuous curve { c(t); t ∈ [0, 1] } on M, the derivative c

(t) ∈ T

M exists for almost all t ∈ [0, 1].

∗Email: dinghao16@mails.ucas.ac.cn

†Email:Shizan.Fang@u-bourgogne.fr

Following [2], we say that a curve { c(t); t ∈ [0, 1] } on P

(M ) is absolutely continuous in L

if there exists k ∈ L

([0, 1]) such that

W

(c(t

), c(t

)) ≤ Z

t1

k(s) ds, t

< t

. The following result is our starting point:

Theorem 1.1 (see [2], Theorem 8.3.1). Let { c

; t ∈ [0, 1] } be an absolutely continuous curve on P

(M ) in L

, then there exists a Borel vector field Z

on M such that

Z

[0,1]

h Z

M

| Z

(x) |

dc

(x) i

dt < + ∞ and the following continuity equation

dc

dt + ∇ · (Z

c

) = 0, (1.1)

holds in the sense of distribution. Uniqueness to (1.1) holds if moreover Z

is imposed to be in

∇ ψ, ψ ∈ C

(M)

2(ct)

. In this work, we define the tangent space ¯ T

of P

(M ) at µ by

T ¯

=

∇ ψ, ψ ∈ C

(M )

2(µ)

, (1.2)

the closure of gradients of smooth functions in the space L

(µ). Equation (1.1) implies that for almost all t ∈ [0, 1],

d dt

Z

M

f(x) dc

(x) = Z

M

h∇ f (x), Z

(x) i

dc

(x), f ∈ C

(M ). (1.3) We will say that Z

is the intrinsic derivative of c

and use the notation

d

c

dt = Z

∈ T ¯

.

In what follows, we will describe the tangent space ¯ T

with the least conditions as possible on the measure µ. Consider the quadratic form defined by

E (ψ) = Z

M

|∇ ψ(x) |

dµ(x), ψ ∈ C

(M ).

We assume that there is a constant C

> 0 such that Z

M

(ψ − h ψ i )

dµ ≤ C

Z

M

|∇ ψ |

dµ, (1.4)

where h ψ i = Z

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

= inf

x∈M

ρ(x) > 0, β

= sup

x∈M

ρ(x) < + ∞ .

Since M is compact, the following Poincar´e inequality holds : Z

M

(ψ − h ψ i )

dx ≤ C Z

M

|∇ ψ |

dx,

then Z

M

(ψ − h ψ i )

dµ ≤ Cβ

β

Z

M

|∇ ψ |

dµ.

Now let Z ∈ T ¯

; there is a sequence of functions ψ

∈ C

(M) such that Z = lim

n→+∞

∇ ψ

in L

(µ). By changing ψ

to ψ

−h ψ

i and by condition (1.4), { ψ

; n ≥ 1 } is a Cauchy sequence in L

(µ). If the quadratic form E (ψ) is closable in L

(µ), then there exists a function ϕ

in the Sobolev space D

(µ) such that Z = ∇ ϕ

, where D

(µ) is the closure of C

(M) with respect to the norm

|| ϕ ||

:=

Z

M

| ϕ(x) |

dµ(x) + Z

M

|∇ ϕ(x) |

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

vector field Z on M , there exists a function denoted by div

(Z) ∈ L

(µ) such that

Z

M

h∇ f(x), Z (x) i

dµ(x) = − Z

M

f (x) div

(Z )(x), f ∈ C

(M). (1.5)

Definition 1.2. We say that the measure µ is a measure having divergence if div

(Z) ∈ L

(µ) exists. We will use the notation

P

(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

(M ), we have T ¯

=

∇ ψ; ψ ∈ D

(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

(M ).

1.1 Constant vector fields on P

( M )

For any gradient vector field ∇ ψ on M with ψ ∈ C

(M), consider the ordinary differential equation (ODE):

d

dt U

(x) = ∇ ψ(U

(x)), U

(x) = x ∈ M.

Then x → U

(x) is a flow of diffeomorphisms on M . Let µ ∈ P

(M ), consider c

= (U

)

µ. It is easy to see that the curve { c

; t ∈ [0, 1] } is absolutely continuous in L

and for f ∈ C

(M),

d dt

Z

M

f (x) dc

(x) = d dt

Z

M

f (U

(x)) dµ(x) = Z

M

h∇ f (U

(x)), ∇ ψ(U

(x)) i dµ(x),

which is equal to, for any t ∈ [0, 1], Z

M

h∇ f, ∇ ψ i dc

.

In other term, c

is a solution to the following continuity equation:

dc

dt + ∇ · ( ∇ ψ c

) = 0.

According to above definition, we see that for each t ∈ [0, 1], d

c

dt = ∇ ψ.

It is why we call ∇ ψ a constant vector field on P

(M). In order to make clearly different roles played by ∇ ψ, we will use notation

V

when it is seen as a constant vector field on P

(M).

Remark 1.4. In section 3 below, we will compute Lie brackets of two constant vector fields on P

(M ) without explicitly using the existence of density of measure, the Lie bracket of two constant vector fields is NOT a constant vector field.

1.2 Geodesics with constant speed

It is easy to introduce geodesics with constant speed when the base space is a flat space R

. A probability measure µ on R

is in P

( R

) if R

R^m

| x |

dµ(x) < + ∞ . Let c

, c

∈ P

( R

), there is an optimal coupling plan γ ∈ C (c

, c

) such that

W

(c

, c

) = Z

Rm×Rm

| x − y |

dγ(x, y).

For each t ∈ [0, 1], define c

∈ P

( R

) by Z

Rm

f (x) dc

(x) = Z

Rm×Rm

f (u

(x, y)) dγ(x, y), where u

(x, y) = (1 − t)x + ty. For 0 ≤ s < t ≤ 1, define π

∈ C (c

, c

) by

Z

Rm×Rm

g(x, y) dπ

(x, y) = Z

Rm×Rm

g(u

(x, y), u

(x, y)) dγ(x, y).

Then

W

(c

, c

) ≤ Z

R^m×R^m

| u

(x, y) − u

(x, y |

dγ(x, y) = (t − s)

W

(c

, c

)

. It follows that W

(c

, c

) ≤ (t − s)W

(c

, c

). Combing with triangulaire inequality,

W

(c

, c

) ≤ W

(c

, c

) + W

(c

, c

) + W

(c

, c

)

≤ sW

(c

, c

) + (t − s)W

(c

, c

) + (1 − t)W

(c

, c

) = W

(c

, c

),

we get the property of geodesic with constant speed:

W

(c

, c

) = | t − s | W

(c

, c

).

According to Theorem 1.1, there is Z

∈ T ¯

such that, for f ∈ C

( R

), d

dt Z

Rm

f(x)dc

(x) = Z

Rm

h∇ f (u

(x, y)), y − x i

dγ(x, y)

= Z

Rd

h∇ f (x), Z

(x) i

dc

(x)

where h , i

is the canonical inner product of R

. We heuristically look for Z

such that Z

(u

(x, y)) = y − x.

Taking the derivative with respect to t yields ( d

dt Z

)(u

(x, y)) + h∇ Z

(u

(x, y)), y − x i = 0.

It follows that

( d

dt Z

) + ∇ Z

(Z

) = 0.

In the case where Z

= ∇ ψ

, we have ( d

dt ∇ ψ

) + ∇

ψ

( ∇ ψ

) = 0.

We remark that {∇ ψ

, t ∈ ]0, 1[ } satisfies heuristically the equation of Riemannian geodesic obtained in [14] or heuristically obtained in [19], in which the authors showed that the con- vexity of entropy functional along these geodesics is equivalent to Bakry-Emery’s curvature condition [3] (see also [12], [21, 20]).

In the case of Riemannian manifold M , it is a bit complicated. We follow the exposition of [10]. Let T M be the tangent bundle of M and π : T M → M the natural projection. For each µ ∈ P

(M ), we consider the set

Γ

= n

γ probability measure on T M; π

γ = µ, Z

T M

| v |

dγ(x, v) < + ∞ o . The set Γ

is obviously non empty. Let γ ∈ Γ

, we consider ν = exp

γ , that is,

Z

M

f (x)dν(x) = Z

T M

f (exp

(v)) dγ(x, v),

where exp

: T

M → M is the exponential map induced by geodesics on M. The map T M → M × M, (x, v) → (x, exp

(v))

sends γ to a coupling plan ˜ γ ∈ C (µ, ν). We have W

(µ, ν) ≤

Z

T M

d

(x, exp

(v)) dγ(x, v ) ≤ Z

T M

| v |

dγ(x, v ).

In order to construct geodesics { c

; t ∈ [0, 1] } connecting µ and ν, we need to find γ

∈ Γ

such that

W

(µ, ν) = Z

T M

| v |

dγ

(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

= ξ

(0) ∈ T

M, then

y = exp

(v

) and d

(x, y) = | v

|

.

Take a Borel version Ξ of such a map (x, y) → (x, v

) from M × M to T M. Let ˜ γ

∈ C (µ, ν) be an optimal coupling plan; define γ

∈ Γ

by

Z

T M

g(x, v) dγ

(x, v) = Z

M×M

g x, Ξ(x, y)

d˜ γ

(x, y).

Therefore Z

T M

| v |

dγ

(x, v) = Z

M×M

| Ξ(x, y) |

d˜ γ

(x, y)

= Z

M×M

d

(x, y)

d˜ γ

(x, y) = W

(µ, ν).

Now we define the curve { c

; t ∈ [0, 1] } on P

(M ) by Z

M

f (x)dc

(x) = Z

T M

f (exp

(tv)) dγ

(x, v).

Similarly we check that

W

(c

, c

) = | t − s | W

(c

, c

).

The organization of the paper is as follows. In Section 2, we consider ordinary equations on P

(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 deriva- tive 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

(M), which enable us to obtain the ex- tension of a vector field along a quite good curve on P

(M) in Section 5 as in differentiable geometry; the parallel translation along such a good curve on P

(M) is naturally and rig- orously introduced. The existence for parallel translations is established for a curve whose intrinsic derivative gives rise a good enough vector field on P

(M).

2 Ordinary differential equations on P ₂ ( M )

Let ϕ ∈ C

(M), consider the function F

on P

(M) defined by F

(µ) =

Z

M

ϕ(x) dµ(x). (2.1)

A function F on P

(M) is said to be a polynomial if there exists a finite number of functions ϕ

, . . . , ϕ

in C

(M ) such that F = F

· · · F

. Let Z = V

be a constant vector field on P

(M) with ψ ∈ C

(M), and U

the flow on M associated to ∇ ψ. For µ

∈ P

(M), we set µ

= (U

)

µ

. Then we have seen in section 1.1,

n d

dt F

(µ

) o

|^t=0

= Z

M

h∇ ϕ(x), ∇ ψ(x) i dµ

(x) = h V

, V

i

.

The left hand side of above equality is the derivative of F

along V

. More generally, for a function F on P

(M), we say that F is derivable at µ

along V

, if

( ¯ D

F)(µ

) = n d

dt F (µ

) o

|t=0

exists.

We say that the gradient ¯ ∇ F (µ

) ∈ T ¯

exists if for each ψ ∈ C

(M ), ( ¯ D

F )(µ

) exists and

D ¯

F (µ

) = h ∇ ¯ F, V

i

. (2.2) Note that for ϕ ∈ C

(M ), there is a sequence of ψ

∈ C

(M) such that ∇ ψ

converge uniformly to ∇ ϕ so that V

∈ T ¯

for any µ ∈ P

(M ). It is obvious that ¯ ∇ F

= V

. For the polynomial F = Q

i=1

F

, we have

∇ ¯ F =

k

X

i=1

Y

j6=i

F

V

.

Note that the family { F

, ϕ ∈ C

(M ) } separates the point of P

(M ). By Stone-Weierstrauss theorem, the space of polynomials is dense in the space of continuous functions on P

(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

(M ), W

). For example, if (µ, x) → Φ(µ, x) is a function on P

(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

(M ) if there exists a Borel map Φ : P

(M ) × M → R such that for any µ ∈ P

(M), x → Φ(µ, x) is C

and Z (µ) = V

. A class of test vector fields on P

(M ) is

χ( P ) = n X

f inite

α

V

, α

polynomial, ψ

∈ C

(M ) o

. (2.3)

Let Z be a vector field on P

(M ), how to construct a solution µ

∈ P

(M ) to the following

ODE d

µ

dt = Z(µ

)?

Theorem 2.2. Let Z be a vector field on P

(M ) given by Φ. Assume that (µ, x) → ∇ Φ(µ, x) is continuous, then for any µ

∈ P

(M), there is an absolutely curve { µ

; t ∈ [0, 1] } on P

(M ) such that

d

µ

dt = Z(µ

), µ

= µ

. (2.4)

If moreover, for any µ ∈ P

(M ), x → Φ(µ, x) is C

and C

:= sup

µ∈P₂(M)

sup

x∈M

||∇

Φ(µ, x) || < + ∞ , (2.5) then there is a flow of continuous maps (t, x) → U

(x) on M, solution to the following Mckean-Vlasov equation

d

dt U

(x) = ∇ Φ(µ

, U

(x)), µ

= (U

)

µ

. (2.6)

Proof. We use the Euler approximation to construct a solution. We first note that C

:= sup

(µ,x)∈P₂(M)×M

|∇ Φ(µ, x) | < + ∞ . (2.7) Let P

= e

be the heat semi-group associated to the Laplace operator ∆

on functions, and T

= e

the heat semigroup on differential forms, with de Rham-Hodge operator . It is well-known that

| T

( ∇ ϕ) | ≤ e

P

|∇ ϕ | , ϕ ∈ C

(M )

where κ is lower bound of Ricci tensor on M. As t → 0, T

( ∇ ϕ) converges to ∇ ϕ uniformly.

For n ≥ 1, let

Z

(µ, x) = T

∇ Φ(µ, · ) (x).

According to (2.7) and above estimate, for n big enough, sup

(µ,x)∈P₂(M)×M

| Z

(µ, x) | ≤ 2C

. (2.8) Now let t

= k2

for k = 1, . . . , 2

and

[t] = t

if t ∈ [t

, t

[.

On the intervall [t

, t

], consider the ODE on M : dU

dt = Z

µ

, U

, U

(x) = x, (2.9)

and µ

= (U

)

µ

for t ∈ [t

, t

]; inductively, on [t

, t

], we consider dU

dt = Z

µ

k

, U

, U

t=tk

(x) = U

k

(x), (2.10)

and for t ∈ [t

, t

],

µ

= (U

)

µ

(2.11) and so on, we get a curve { µ

; t ∈ [0, 1] } on P

(M ). We now prove that this family is equicontinuous in C([0, 1], P

(M )). Let 0 ≤ s < t ≤ 1, define γ(θ) = U

−θ)s+θt

, then dγ(θ)

dθ = (t − s)Z

µ

, U

.

We have, according to (2.8),

d

U

(x), U

(x)

≤ Z

0

dγ(θ) dθ

dθ ≤ 2C

(t − s).

Define a probability measure π on M × M by Z

M×M

g(x, y)π(dx, dy) = Z

M

g U

(x), U

(x)

dµ

(x).

Then π ∈ C (µ

, µ

), we have W

µ

, µ

≤ Z

M

d

U

(x), U

(x)

dµ

(x) ≤ 4C

(t − s)

.

By Ascoli theorem, up to a subsequence, µ

converges in C([0, 1], P

(M)) to a continuous curve { µ

; t ∈ [0, 1] } such that W

(µ

, µ

) ≤ 2C

(t − s).

For proving that { µ

; t ∈ [0, 1] } is a solution to ODE (2.4), we need the following preparation:

Lemma 2.3. Set Φ

(x) = Φ(µ, x), then sup

(µ,x)∈P₂(M)×M

| (T

∇ Φ

)(x) − ∇ Φ(x) |

→ 0, as t → 0. (2.12) Proof. We use || · ||

to denote the uniform norm on M . Let ε > 0, for µ ∈ P

(M ), there is t ˆ

> 0 such that

sup

t≤ˆtµ

|| T

∇ Φ

− ∇ Φ

||

< ε.

Since (µ, t) → || T

∇ Φ

− ∇ Φ

||

is continuous, there is δ

> 0 such that for t ≤ ˆ t

, W

(µ, ν) < δ

⇒ || T

∇ Φ

− ∇ Φ

||

< ε.

Let B(µ, δ) be the open ball in ( P

(M), W

) centered at µ, of radius δ. We have P

(M) = ∪

B (µ, δ

);

so there is a finite number of { µ

, . . . , µ

} such that P

(M ) = ∪

B(µ

, δ

).

Let ˆ t = min ˆ t

, i = 1, . . . , K > 0. Then for 0 < t < ˆ t, sup

µ∈P₂(M)

|| T

∇ Φ

− ∇ Φ

||

≤ ε.

So we get (2.12).

End of the proof of theorem : { µ

; t ∈ [0, 1] } satisfies the following continuity equation Z

[0,1]×M

α

(t)f (x)dµ

(x)dt

= α(0) Z

M

f (x)dµ

(x) + Z

[0,1]×M

α(t) h∇ f(x), Z

µ

, x

i dµ

(x)dt,

(2.13)

for all α ∈ C

([0, 1)) and f ∈ C

(M ). We have Z

[0,1]×M

α(t) h∇ f (x), Z

µ

, x

i dµ

dt − Z

[0,1]×M

α(t) h∇ f (x), ∇ Φ µ

, x i dµ

dt

= Z

[0,1]×M

α(t) h∇ f (x), Z

µ

, x

− ∇ Φ(µ

, x) i dµ

dt +

Z

[0,1]×M

α(t) h∇ f (x), ∇ Φ µ

, x

i dµ

dt − Z

[0,1]×M

α(t) h∇ f (x), ∇ Φ µ

, x

i dµ

dt.

It is obvious that the sum of two last terms converge to 0 as n → + ∞ . Let I

be the first term on the right side, then

| I

| ≤ ||∇ f ||

Z

0

| α(t) | || T

∇ Φ

µ⁽ⁿ⁾_[t]

− ∇ Φ

||

dt Note that

|| T

∇ Φ

µ⁽ⁿ⁾_[t]

− ∇ Φ

||

≤ || T

∇ Φ

µ⁽ⁿ⁾_[t]

− ∇ Φ

µ⁽ⁿ⁾_[t]

||

+ ||∇ Φ

µ⁽ⁿ⁾_[t]

− ∇ Φ

||

. The term || T

∇ Φ

µ⁽ⁿ⁾_[t]

− ∇ Φ

µ⁽ⁿ⁾_[t]

||

→ 0 is due to above lemma. As n → + ∞ , µ

converges to µ

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

Z

[0,1]×M

α

(t)f (x)dµ

(x)dt

= α(0) Z

M

f (x)dµ

(x) + Z

[0,1]×M

α(t) h∇ f(x), ∇ Φ µ

, x

i dµ

(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

, we can directly use ∇ Φ(µ, · ) instead of Z

in (2.9), (2.10), (2.11).

On the intervall [t

, t

], consider the ODE on M : dU

dt = ∇ Φ µ

, U

, U

(x) = x, (2.14)

and µ

= (U

)

µ

for t ∈ [t

, t

]; inductively, on [t

, t

], we consider dU

dt = ∇ Φ µ

k

, U

, U

t=tk

(x) = U

k

(x), (2.15)

and for t ∈ [t

, t

],

µ

= (U

)

µ

k

. (2.16)

By above result, up to a subsequence, { µ

, t ∈ [0, 1] } converges to { µ

, t ∈ [0, 1] } in C([0, 1], P

(M )). We use this subsequence to prove the convergence of { U

(x), t ∈ [0, 1] } . Now we prove that, under Condition (2.7),

d

U

(x), U

(y)

≤ e

d

(x, y), x, y ∈ M. (2.17) For x, y ∈ M given, there is a minimizing geodesic { ξ

, s ∈ [0, 1] } connecting x and y such that d

(x, y) = R

0

| ξ

| ds. Set

σ(t, s) = U

(ξ

).

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

denotes the covariant derivative. We have d

dt U

(ξ

) = ∇ Φ

µ

, U

(ξ

) . Taking the derivative with respect to s, we get

D ds

d

dt U

(ξ

) = ∇

Φ

µ

, U

(ξ

)

· d

ds U

(ξ

).

Combining with (2.18) yields D dt

d

ds U

(ξ

) = ∇

Φ

µ

, U

(ξ

)

· d

ds U

(ξ

).

Now,

d dt

d

ds U

(ξ

)

2

= 2 D

∇

Φ

µ

, U

(ξ

)

· d

ds U

(ξ

), d

ds U

(ξ

) E , which is, by Condition (2.7), less than

2C

d

ds U

(ξ

)

2

.

By Gronwall lemma,

d

ds U

(ξ

)

≤ e

| ξ

| , which implies that

d

U

(x), U

(y)

≤ e

d

(x, y).

Therefore the family

(t, x) → U

(x); n ≥ 1 is equicontinuous in C([0, 1] × M ). By Ascoli theorem, up to a subsequence, U

(x) converges to U

(x) uniformly in (t, x) ∈ [0, 1] × M. It is obvious to see that (U

, µ

) solves Mckean-Vlasov equation (2.6).

Remark 2.4. Comparing to [5], as well to [24], 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

(M ) (see [2], [22],[23], [9]). The interpolation between geodesic flows and gradient flows were realized using Langevin’s deformation in [12, 13].

3 Levi-Civita connection on P ₂ ( M )

In this section, we will revisit the paper by J. Lott [14]: 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

(M ), the suitable class of probability measures should be the class P

(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

(M ), for any C

vector field A on M , the divergence div

(A) ∈ L

(M, µ) is such that

Z

M

h∇ φ(x), A(x) i

dµ(x) = − Z

M

φ(x) div

(A)(x) dµ(x)

for any φ ∈ C

(M). It is easy to see that div

(f A) = f div

(A) + h∇ f, A i for f ∈ C

(M). If dµ = ρ dx has a density ρ > 0 in the space C

(M), we have

Z

M

h∇ φ, A i dµ = Z

M

h∇ φ, ρA i dx = − Z

M

φ div(ρA) dx = − Z

M

φ div(ρA) ρ

dµ, It follows that

div

(A) = ρ

div(ρA) = div(A) + h∇ (log ρ), A i . (3.1) For µ ∈ P

(M) and φ ∈ C

(M ), we denote L

(φ) ∈ L

(µ) such that

Z

M

h∇ f, ∇ φ i dµ = − Z

M

f L

φ dµ, for any f ∈ C

(M), (3.2) where L

φ = div

( ∇ φ) is a negative operator.

Let ψ ∈ C

(M ), consider the ODE dU

dt = ∇ ψ(U

), U

(x) = x.

Proposition 3.1. Let dµ = ρ dx be a probability measure in P

(M ) with a strictly positive density ρ in C

(M ) and ψ ∈ C

(M ). Then for each t ∈ [0, 1], µ

:= (U

)

µ ∈ P

(M ).

Proof. By Kunita [11] (see also [7], [17]), the push-forward measure (U

)

µ by inverse map of U

admits a density ˜ K

with respect to µ, having the following explicit expression

K ˜

= exp

− Z

0

div

( ∇ ψ)(U

(x))ds .

It follows that the density K

of µ

with respect to µ has the expression K

= exp Z

0

div

( ∇ ψ)(U

(x))ds .

According to (3.1), x → div

( ∇ ψ(x)) is C

. Therefore the condition in [7]

Z

M

exp(λdiv

( ∇ ψ(x)) dµ(x) < + ∞ , for all λ > 0

is automatically satisfied. Again by (3.1), x → K

(x) is in C

. Now let A be a C

vector field on M and f ∈ C

(M ), we have

Z

M

h∇ f (x), A(x) i

dµ

(x) = Z

M

h∇ f, A i

K

(x)dµ(x) = − Z

M

f div

(K

Z ) dµ.

It follows that

div

(A) = div

(K

A) K

.

For ψ

, ψ

∈ C

(M), we denote by V

, V

the associated constant vector fields on P

(M).

In what follows, we will compute the Lie bracket [V

, V

].

For f ∈ C

(M ), we set F

(µ) = R

M

f dµ. According to preparations given at the beginning of Section 2,

( ¯ D

2

F

)(µ) = Z

M

h∇ ψ

, ∇ f i dµ = F

(µ).

Using again above formula, we have ( ¯ D

1

D ¯

2

F

)(µ) = Z

M

h∇ ψ

, ∇h∇ ψ

, ∇ f ii dµ = − Z

M

L

ψ

h∇ ψ

, ∇ f i dµ.

Therefore

[V

, V

]F

= ¯ D

2

D ¯

1

F

− D ¯

1

D ¯

2

F

= Z

M

h ( L

ψ

∇ ψ

− L

ψ

∇ ψ

), ∇ f i dµ.

Let

C

(µ) = L

ψ

∇ ψ

− L

ψ

∇ ψ

. (3.3) Note that C

(µ) is in L

(M, T M ; µ), not in ¯ T

. Consider the orthogonal projection:

Π

: L

(M, T M ; µ) → T ¯

. As µ ∈ P

(M ) and by Proposition 1.3, there exists ˜ Φ

∈ D

1

(µ) such that

Π

( C

(µ)) = ∇ Φ ˜

. (3.4) Then we have

[V

, V

]F

= Z

M

h∇ Φ ˜

, ∇ f i dµ = ( ¯ D

F

)(µ). (3.5) Above equality can be extended to the class of polynomials on P

(M ), that is to say that

[V

, V

]

= V

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

, ψ

∈ C

(M), for dµ = ρ dx with ρ > 0 and ρ ∈ C

(M ), the function Φ ˜

obtained in (3.4) has the following expression :

Φ ˜

= ( L

)

div

C

(µ)

. (3.7)

Proof. By (3.1),

L

ψ = ∆

ψ + h∇ log ρ, ∇ ψ i ,

where ∆

denotes the Laplace operator on M . It is well-known that L

has a spectral gap if log ρ ∈ C

(M ). In [14], the Lie bracket [V

, V

] was expressed using Hodge decomposition for vector fields in L

(µ). For ψ

, ψ

∈ C

(M), we have

div

C

(µ)

= h∇L

ψ

, ∇ ψ

i − h∇L

ψ

, ∇ ψ

i . By Hodge decomposition, C

(µ) admits the decomposition

C

(µ) = d

ω + ∇ f + h,

where ω is a differential 2-form on M, d

is adjoint operator of exterior derivative in L

(µ), h is harmonic form : (d

d + dd

)h = 0. Taking the divergence div

on the two sides of above equality, we see that f is a solution the following equation

L

f = div

C

(µ)

.

It follows that ˜ Φ

has the expression (3.7).

Now we introduce the covariant derivative ¯ ∇

V

associated to the Levi-Civita connection on P

(M ) by

2 h ∇ ¯

V

, V

i = ¯ D

1

h V

, V

i + ¯ D

2

h V

, V

i − D ¯

3

h V

, V

i + h V

, [V

, V

] i − h V

, [V

, V

] i − h V

, [V

, V

] i . We have h V

, V

i =

Z

M

h∇ ψ

, ∇ ψ

i dµ = F

. Then D ¯

1

h V

, V

i = Z

M

h∇ ψ

, ∇ h∇ ψ

, ∇ ψ

ii dµ = − Z

M

hL

ψ

∇ ψ

, ∇ ψ

i dµ.

Replacing ψ

by ψ

, ψ

by ψ

and ψ

by ψ

, we get D ¯

2

h V

, V

i = − Z

M

hL

ψ

∇ ψ

, ∇ ψ

i dµ.

We have, in the same way D ¯

3

h V

, V

i = − Z

M

hL

ψ

∇ ψ

, ∇ ψ

i dµ.

Now using expression of [V

, V

], we have h V

, [V

, V

] i =

Z

M

h−L

ψ

∇ ψ

+ L

ψ

∇ ψ

, ∇ ψ

i dµ.

In the same way, we get

h V

, [V

, V

] i = Z

M

h−L

ψ

∇ ψ

+ L

ψ

∇ ψ

, ∇ ψ

i dµ and

h V

, [V

, V

] i = Z

M

h−L

ψ

∇ ψ

+ L

ψ

∇ ψ

, ∇ ψ

i dµ.

Combining all these terms, we finally get 2 h ∇ ¯

V

, V

i =

Z

M

h∇h∇ ψ

, ∇ ψ

i , ∇ ψ

i dµ + Z

M

hL

ψ

∇ ψ

− L

ψ

∇ ψ

, ∇ ψ

i dµ.

Theorem 3.3. For two constant vector fields V

, V

, we have

∇ ¯

V

= 1

2 V

+ 1

2 [V

, V

]. (3.8) Moreover, for any constant vector field V

,

h ∇ ¯

V

, V

i

= Z

M

h∇

ψ

, ∇ ψ

⊗ ∇ ψ

i dµ. (3.9)

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

, [V

, V

] i

=

Z

M

h−L

ψ

∇ ψ

+ L

ψ

∇ ψ

, ∇ ψ

i dµ

= Z

M

h∇ ψ

, ∇h∇ ψ

, ∇ ψ

ii dµ − Z

M

h∇ ψ

, ∇h∇ ψ

, ∇ ψ

ii dµ

= Z

M

h∇

ψ

, ∇ ψ

⊗ ∇ ψ

i + h∇

ψ

, ∇ ψ

⊗ ∇ ψ

i dµ

− Z

M

h∇

ψ

, ∇ ψ

⊗ ∇ ψ

i + h∇

ψ

, ∇ ψ

⊗ ∇ ψ

i dµ

= Z

M

h∇

ψ

, ∇ ψ

⊗ ∇ ψ

i − h∇

ψ

, ∇ ψ

⊗ ∇ ψ

ii dµ,

due to the symmetry of the Hessian ∇

ψ

. On the other hand, h V

, V

i

=

Z

M

h∇

ψ

, ∇ ψ

⊗ ∇ ψ

i + h∇

ψ

, ∇ ψ

⊗ ∇ ψ

ii dµ.

Summing these last two equalities yields (3.9).

Remark 3.4. By (3.8), for two constant vector fields V

, V

, the covariant derivative

∇ ¯

V

is not a constant vector field on P

(M) if ψ

6 = ψ

. Let α : P

(M ) → R be a differentiable function, we define

∇ ¯

α V

= ¯ D

1

α · V

+ α ∇ ¯

V

. (3.10) Proposition 3.5. Let Z be a vector field on P

(M ) in the test space χ( P ), that is, Z =

k

X

i=1

α

V

with α

polynomials. Then ∇ ¯

Z still is in the test space; moreover

∇ ¯

Z = V

+ 1

2 V

, where

Φ

=

k

X

j=1

X

i=1

α

D ¯

α

ψ

, Φ

=

k

X

i=1

α

ψ

.

Proof. Using the rule concerning covariant derivatives, ¯ ∇

Z is equal to

k

X

i,j=1

α

D ¯

α

V

+ 1 2

k

X

i,j=1

α

α

V

+ 1 2

k

X

i,j=1

α

α

[V

, V

].

The last sum is equal to 0 due to the skew-symmetry of [V

, V

], the first one gives rise to Φ

and the second one gives rise to Φ

.

In what follows, we will extend the definition of covariant derivative (3.10) for a general vector field Z on P

(M ). Let ∆ be the Laplace operator on M , let { ϕ

, n ≥ 0 } be the eigenfunctions of ∆:

− ∆ϕ

= λ

ϕ

.

We have λ

= 0 and ϕ

= 1. It is well-known, by Weyl’s result, that λ

∼ n

, n → + ∞

where m is the dimension of M . The functions { ϕ

; n ∈ N } are smooth, chosen to form an orthonormal basis of L

(M, dx). A function f on M is said to be in H

(M ) for k ∈ N , if

|| f ||

= Z

M

| (I − ∆)

f |

dx < + ∞ . By Sobolev embedding inequality, for k > m

2 + q,

|| f ||

≤ C || f ||

. For f ∈ H

(M ), put f = X

n≥0

a

ϕ

which holds in L

(M, dx) with

a

= Z

M

f (x)ϕ

(x) dx.

We have :

|| f ||

= X

n≥0

a

(1 + λ

)

.

The system n ∇ ϕ

√ λ

; n ≥ 1 o

is orthonormal. Let V

= V

, then { V

; n ≥ 1 } is an orthonormal basis of ¯ T

.

Let Z be a vector field on P

(M) given by Z (µ) = V

or Z(µ) = ∇ Φ(µ, · ). In the sequel, we denote: Φ

(x) = Φ(µ, x), Φ

(µ) = Φ(µ, x). Then, if x → ∇ Φ

(x) is continuous,

∇ Φ

= X

n≥1

Z

M

h∇ Φ

, ∇ ϕ

√ λ

i dx ∇ ϕ

√ λ

= X

n≥1

Z

M

Φ

ϕ

dx

∇ ϕ

,

which converges in L

(M, dx). Let µ ∈ P

(M), the above series converges also in ¯ T

. Let a

(µ) =

Z

M

Φ

(x)ϕ

(x) dx. (3.11)

Let V

be a constant vector field on P

(M) with ψ ∈ C

(M ). For q ≥ p ≥ 1, set S

=

q

X

n=p

D ¯

a

V

+ a

∇ ¯

V

= S

+ S

(3.12) respectively. Let φ ∈ C

(M ), according to (3.9), we have

h S

, V

i

= Z

M

X

n=p

a

(µ) ∇

ϕ

( ∇ ψ(x), ∇ φ(x)) dµ(x).

It follows that

|h S

, V

i

| ≤

q

X

n=p

a

(µ) ∇

ϕ

| V

|

| V

|

,

therefore

| S

|

≤

q

X

n=p

a

(µ) ∇

ϕ

| V

|

. We have

||

q

X

n=p

a

(µ)(I − ∆)

ϕ

||

=

q

X

n=p

a

(µ)

(1 + λ

)

=

q

X

n=p

Z

M

(I − ∆)

Φ

ϕ

dx

→ 0

as p, q → + ∞ if Φ

∈ H

(M ). On the other hand, we have ( ¯ D

a

)(µ) =

Z

M

( ¯ D

Φ

)(µ)ϕ

(x) dx = Z

M

h∇ D ¯

Φ

, ∇ ϕ

√ λ

i dx

√ λ

,

then

S

=

q

X

n=p

Z

M

h∇ D ¯

Φ

, ∇ ϕ

√ λ

i dx ∇ ϕ

√ λ

and

Z

M

| S

|

dx =

q

X

n=p

Z

M

h∇ D ¯

Φ

, ∇ ϕ

√ λ

i dx

→ 0 as p, q → + ∞ if

Z

M

|∇ D ¯

Φ

|

dx < + ∞ . Therefore for dµ = ρ dx with µ ∈ P

(M), as p, q → ∞ ,

| S

|

≤ || ρ ||

Z

M

| S

|

dx → 0.

We get the following result

Theorem 3.6. Let Z be a vector field on P

(M ) given by Φ : P

(M) × M → R . Assume that (i) for any µ ∈ P

(M ), Φ

∈ H

(M ) with k > m

2 + 2, (ii) for any x ∈ M, D ¯

Φ

exists and ∇ D ¯

Φ

∈ L

(M, dx).

Then the covariant derivative ∇ ¯

Z is well defined at µ ∈ P

(M ) and for φ ∈ C

(M), h ∇ ¯

Z, V

i

=

Z

M

h ( ∇ D ¯

Φ

), ∇ φ i dµ + Z

M

∇

Φ

∇ ψ, ∇ φ

dµ. (3.13)

Proof. Let Z

=

q

X

n=1

a

V

. Then

∇ ¯

Z

= S

.

Letting q → + ∞ yields the result.

4 Derivability of the square of the Wasserstein distance

Let { c

; t ∈ [0, 1] } be an absolutely continuous curve on P

(M ), for σ ∈ P

(M) given, the derivability of t → W

(σ, c

) was established in chapter 8 of [2] , as well as in [22] (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 [23] if σ and c

have a density with respect to dx. When { c

} is a geodesic of constant speed, the derivability at t = 0 was given in theorem 4.2 of [10] where the property of semi concavity was used. In what follows, we will use constant vector fields on P

(M ).

Before stating our result, we recall some well-known facts concerning optimal transport maps (see [4, 6, 16, 2, 22]). Let σ ∈ P

(M ) be absolutely continuous with respect to dx and µ ∈ P

(M ), then there is an unique Borel map φ ∈ D

(σ) such that

Z

M

|∇ φ(x) |

dσ(x) = W

(σ, µ)

and x → T (x) = exp

( ∇ φ(x)) pushes σ forward to µ. If µ is also in P

(M), the map T : M → M is invertible and its inverse map T

is given by y → exp

( ∇ φ(y)) with some ˜ function ˜ φ such that R

M

|∇ φ ˜ |

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

(tu) with some u ∈ T

M . Then

d

(exp

(v), x) − d

(y, x) ≤ 2 h v, ξ

(1) i

+ o( | v | ) as | v | → 0. (4.1) Proof. See [16], page 10.

Theorem 4.2. Assume that σ ∈ P

(M) is absolutely continuous with respect to dx, then µ → χ(µ) := W

(σ, µ) is derivable along each constant vector field V

at any µ ∈ P

(M ). If µ ∈ P

(M ), the gradient ∇ χ exists and admits the expression :

∇ χ(µ) = ∇ φ. ˜ (4.2)

Proof. Let ψ ∈ C

(M) and (U

)

be the associated flow of diffeomorphisms of M : dU

(x)

dt = ∇ ψ(U

(x)), x ∈ M. (4.3)

The inverse map U

of U

satisfies the ODE dU

(x)

dt = −∇ ψ(U

(x)), x ∈ M. (4.4)

Set µ

= (U

)

µ, then µ = (U

)

µ

. Let γ ∈ C

(σ, µ) be the optimal coupling plan such that

W

(σ, µ) = Z

M×M

d

(x, y) dγ(x, y).

The map (x, y) → (x, U

(y)) pushes γ forword to a coupling plan γ

∈ C (σ, µ

). Then for t > 0,

1 t h

W

(σ, µ

) − W

(σ, µ) i

≤ 1 t

Z

M×M

d

(x, U

(y)) − d

(x, y)

dγ(x, y)

= 1 t

Z

M×M

d

(x, U

(y)) − d

(x, exp

(t ∇ ψ(y))

dγ(x, y)

+ 1 t

Z

M×M

d

(x, exp

(t ∇ ψ(y)) − d

(x, y)

dγ(x, y) = I

(t) + I

(t)