ZETA FUNCTION REGULARIZED LAPLACIAN ON THE SMOOTH WASSERSTEIN SPACE ABOVE THE UNIT CIRCLE CHRISTIAN SELINGER Abstract. Via elements of second order differential geometry on smooth Wasserstein spaces of probability measures we give an explicit formula for a Laplacian in the case that the Wasserstein space is based on the unit circle. The Laplacian on this infinite dimensional manifold is calculated as trace of the Hessian in the sense of Zeta function regularization. Its square field operator is the square norm of the Wasserstein gradient.

1. Differential geometry of smooth probability densities Definition 1.1. Let (M, h., .ix ) denote a complete simply connected Riemannian manifold without boundary and T1 denote R mod Z equipped with the flat metric. R • P (M ) := P = {µ Borel probability measure on M and dM (x, y)2 µ(dx) < ∞} • Pac (M ) := Pac = {µ ∈ P : µ ≪ volM } dµ • P ∞ (M ) := P ∞ = {µ ∈ Pac : m(x) := dvol (x) > 0; for a.e. x ∈ M } M ∞ • Cc (M ) is the space of smooth compactly supported funtions on M . • Given µ, ν ∈ P : Π(µ, ν) = Π := {π ∈ P (M ×M ) : π(A×M ) = µ(A); π(M ×B) = ν(B) for all Borel setsR A, B.}. dW (µ, ν)2 := inf π∈Π dM (x, y)2 π(dx, dy) is called quadratic Wasserstein distance. • The metric space (P, dW ) is called Wasserstein space. Convergence in Wasserstein distance is equivalent to weak convergence plus convergence of second moments. By Prokhorov’s theorem the Wasserstein space is Polish if the underlying space is so, which is the case for M . The subspace (P ∞ , dW ) ⊂ (P, dW ) is not complete, e.g. convolution of a positive density with rescaled Gaussians converges in Wasserstein distance to a Dirac measure. Theorem 1.1 (Brenier-McCann [2][5]). Given µ, ν ∈ Pac (M ), then the optimal transport plan π realizing the Wasserstein distance between µ and ν is given by π = (id, exp(∇ϕ))#µ, i.e. π(dx, dy) = δ{y=expx (∇ϕ(x))} µ(dx), where ϕ is a µ-a.s. unique (up to constants) function on M which satisfies (ϕc )c = ϕ for ϕc (y) := min{x ∈ M : d2 (x, y)/2 − ϕ(x)}

Theorem 1.2 (Benamou-Brenier [1]). Given µ, ν ∈ Pac (Rd ). We denote by Cµ,ν the set of all curves c : [0, 1] → Pac (Rd ) satisfying c(0) = µ and c(1) = ν such that there exists a time-dependent L2 (c(t))-integrable compactly supported vector field vt for which the continuity equation c(t) ˙ = −∇.(c(t)vt ) ∞ holds, i.e. for all ϕ ∈ C0 ([0, 1] × M ) it holds that Z Z c(t)vt .∇ϕvolM dt. c(t)ϕvol ˙ M dt = − [0,1]×M

[0,1]×M

Then

dW (µ, ν)2 = inf

Cµ,ν

Z

[0,1]×M

|vt |2 c(t)volM dt.

1 2000 Mathematics Subject Classification. Primary: 46E27, 46G05, Secondary: 58E10, 28A33. Key words and phrases. Wasserstein distance, smooth Wasserstein space, smooth Lie bracket, optimal transport, entropy, Riemann zeta-function.

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CHRISTIAN SELINGER

Definition 1.2 (Tangent space). We define the tangent space at a measure µ ∈ P ∞ (M ) by: L2 (µ)

Tµ P ∞ (M ) := {∇ϕ ; ϕ ∈ Cc∞ (M )} R and denote the inner product on Tµ P ∞ (M ) by h∇ϕ, ∇ϕ′ iµ := M h∇ϕ, ∇ϕ′ ix µ(dx) for any µ ∈ P ∞ (M ).

Remark 1.1. P ∞ (M ) is an infinite dimensional manifold in the sense of convenient calculus [3], i.e. the modelling space is not a Banach but a locally convex one, for instance if M is compact then P ∞ (M ) ⊂ C ∞ (M ), where the latter is a locally convex vector space which is complete (in the sense that each Mackey-Cauchy sequence this space is also converging in this space). As topological space P ∞ (M ) is endowed with the smooth topology, which is the final topology with respect to smooth curves into it. The tangent space Tµ P ∞ corresponds to notion of kinematic tangent space in [3] p.284. See also [4]. Vector fields are defined as smooth (in the sense of c∞ -topology, see [3]) sections of the tangent bundle and Otto’s formalism suggests to write V ∈ Γ(P ∞ ← T P ∞ ) as a distribution: V (µ) = −div(µ ∇v), dµ i.e. for all ϕ ∈ Cc∞ (M ), a function v ∈ C ∞ (M ) and m0 = dvol : Z h∇v, ∇ϕix m0 (x)vol(dx). (V (µ)|ϕ) = M

We emphasize that the smooth function v : M ∋ x 7→ v¯(m0 (x)) ∈ R for v¯ : R → R is a 0 (x) possible choice, i.e. take v¯(x) = log(x) + 1, then ∇ v(x) = ∇ (log(m0 (x)) + 1) = ∇m m0 (x) gives the vector field associated to the entropy via Wasserstein gradient [6]: Z P∞ ∇ m0 log(m0 ) vol(dx) = −div(m0 ∇ v) M

Tangent map. For a smooth mapping F : P ∞ (M ) → R, an interval I = (−a, a) and any smooth curve c : I → P ∞ (M ) such that c(0) = µ and c(0) ˙ = −div(µ ∇ u) the tangent map T(F): TP → R × R d |0 (F ◦ c)(t)) (µ, c(0)) ˙ 7→ (F (µ), T (F )(µ).c(0)) ˙ := (F (µ), dt Proposition 1.1 (smooth Lie bracket [7]). Given U, V ∈ Γ(T P∞ ← P∞ ). Since vector fields are not complete we have to construct their respective flows explicitely: For 0 < V ∞ a ≪ ǫ we define FlU → P∞ by t , Flt : (−a, a) × P ∂ U Fl (µ) ∂t t

=

U (FlU t (µ))

=

−div(FlU u(FlU t (µ)∇¯ t (µ)))

resp. ∂ V Fl (µ) = ∂t t =

V (FlVt (µ)) −div(FlU v (FlVt (µ))) t (µ)∇¯

Then the Lie bracket reads as follows: [U, V ](µ)

= div(V (µ) ∇u) − div(U (µ) ∇v) +

+div(µ ∇T (u)(µ).V (µ)) − div(µ ∇T (v)(µ).U (µ))

Here T (u)(µ) is the tangent map of u at µ, since u is a real-valued function on P it is the differential of u at µ.

WASSERSTEIN LAPLACIAN

3

Proof. As a prerequisite we calculate ∂  U  T Fl−t (µ), ∂t

i.e. the expression we differentiate is the tangent map of FlU −t at µ. By the product rule applied to the flow equation:   ∂ ∂ U U T (Fl−t )(µ) = T Fl (µ) ∂t ∂t −t o  n U = −T −div(FlU ∇ u ◦ Fl −t −t ) (µ) h n o i U = div T (FlU −t )(µ)∇ u ◦ Fl−t (µ) + h n o i U +div (FlU )(µ)∇ T (u ◦ Fl ) (µ) −t −t o i n h U U = div T (Fl−t )(µ)∇ u ◦ Fl−t (µ) + h n oi U U +div (FlU )(µ)∇ T (u)(Fl (µ))T (Fl )(µ) −t −t −t By definition [U, V ](µ)

= = = = =

=

∂ ∗ |0 (FlU t ) V (µ) ∂t ∂ U |0 (T (FlU −t ) ◦ V ◦ Flt )(µ) ∂t     ∂ ∂ U U U |0 T (FlU )(V ◦ Fl | ) (µ) + T (Fl )| ◦ | V ◦ Fl (µ) 0 −t t 0 −t 0 t ∂t ∂t ∂ ∂ |0 T (FlU |0 V (FlU −t )(V (µ)) + t (µ)) ∂t ∂t div [V (µ)∇ u(µ)] + div [µ ∇ {T (u)(µ).V (µ)}] + h i ∂  U + |0 −div FlU t (µ)∇ v(Fl t (µ)) ∂t div [V (µ)∇ u(µ)] + div [µ ∇ {T (u)(µ).V (µ)}] + −div [U (µ)∇ v(µ)] − div [µ ∇ {T (v)(µ).U (µ)}]

 In view of the Lie bracket we define the covariant derivative (compare to [4]): Proposition 1.2 (smooth Levi-Civit` a connection). e U V (µ) := ∇ =

−div [U (µ)∇ v(µ)] − div [µ ∇( T (v)(µ).U (µ))]

div [div(µ ∇ u(µ))∇ v(µ)] − div [µ ∇( T (v)(µ).U (µ))] ,

i.e. for all ϕ ∈ Cc∞ (M ) : Z Z e U V (µ)|ϕ) := (∇ h∇h∇ϕ, ∇vix , ∇uix µ(dx) + h∇(T (v)(µ).U (µ)), ∇ϕix µ(dx) M

M

e U V (µ) ∈ Tµ P ∞ , for details Note that if v does not depend on the density µ, then ∇ see Lemma 4.14 in [4]. Proof. We have to show that

e U V, W iµ + hV, ∇ e U W iµ , U hV, W iµ = h∇

i.e. for I(µ) = hV (µ), W (µ)iµ

4

CHRISTIAN SELINGER

U hV, W iµ = T (I).U Z d |0 {h∇ v((id + t∇ u(µ))#µ), ∇w((id + t∇ u(µ))#µ)ix × = dt M × (id + t∇ u(µ))#µ)}  Z  d = |0 ∇ v((id + t∇ u(µ))#µ), ∇ w(µ) µ(dx) dt M x  Z  d ∇ v(µ), |0 ∇ w((id + t∇ u(µ))#µ) + µ(dx) dt M x Z + h∇(h∇ v(µ), ∇ w(µ)ix ), ∇u(µ)ix Z Z M h∇ v(µ), ∇ T (w).U ix µ(dx) h∇ T (v).U, ∇ w(µ)ix µ(dx) + = M ZM Z hh∇ v, ∇∇ wix , ∇ uix µ h∇∇ v, ∇ wix , ∇ uix µ + + M

M

= −hdiv(µ ∇ T (v).U ), wiµ − hdiv(µ ∇ T (w).U ), viµ

+hdiv(div µ∇ u)∇ v), wiµ + hdiv(div µ∇ v)∇ u), viµ e U V, W iµ + hV, ∇ e U W iµ . = h∇

Taking some Riemannian connection ∇ definied in terms of the Koszul formula 2h∇U V, W iµ

= U hV, W iµ + V hW, U iµ − W hU, V iµ + hW, [U, V ]iµ −hV, [U, W ]iµ − hU, [V, W ]iµ

and substituting the Lie bracket and the calculations of U hV, W iµ into this formula shows e It is the Levi-Civit` e is torsion-free by definition. a connection since ∇  that ∇ = ∇.

2. Second order calculus R In [6] the Hessian of the entropy functional Ent(µ) = Rn µlog(µ) vol(dx) with respect to Kantorovich-Rubinstein metric was calculated by second order variation of the entropy functional along constant speed geodesics. This is possible for any smooth functional E : Pac → R of the type Z dµ (x) = m(x), e : R+ → R C 2 . e(m(x)) vol(dx), E(µ) = dvol M The calculation of the Hessian will be done in normal coordinates, i.e. covariant derivatives are calculated in directions U ∈ Γ(T P∞ ) giving rise to geodesics: U (µ) = −div(µ ∇ u) for some u ∈ Cc∞ (M ) depending not on µ: Proposition 2.1 (The Hessian: a variational approach). Let M have Ricci curvature bounded from below. Given a functional E : Pac (M ) → R of the type Z e(µ(x))vol(dx) E(µ) = M

where e : R+ → R is twice differentiable we define p(µ) = p2 (µ) =

µe′ (µ) − e(µ) µp′ (µ) − p(µ).

WASSERSTEIN LAPLACIAN

5

By Hessvar E(µ, ˙ µ) ˙ we denote the second order variation of E along a geodesic path t 7→ µt in Pac of the form ( ∂t µ + ∇.(µ∇ϕ) = 0 2 = 0. ∂t ϕ + |∇ϕ| 2 for ϕ a time-dependent compactly supported function on M . By Γ resp. Γ2 we denote the square field resp. the iterated square field operator with respect to the Laplace-Beltrami opererator ∆: Γ(f, g) := 12 {∆(f g) − g∆f − f ∆g} and Γ2 (f ) ≡ Γ2 (f, f ) := 21 ∆Γ(f, f ) − Γ(f, ∆f ). Then Z Z (∆ϕ0 )2 p2 (µ) vol. Γ2 (ϕ0 ) p(µ) vol + (1) Hessvar E(µ, ˙ µ) ˙ = M

M

Proof. See [9], p441f.  3. A regularized Laplacian on P∞ (T1 ) Proposition 3.1 (Renormalized Laplacian on P∞ (T1 )). Given a functional E : P∞ (T1 ) → R of the type Z e(µ(x))vol(dx), E(µ) = T1

3

where e : R+ → R is C . For an orthonormal system {ek (µ)}k∈N of Tµ P∞ (T1 ) := C ∞ (T1 )/R

L2 (µ)

we define an operator A on Tµ P∞ (T1 ) by diagonalization in the basis {ek (µ)}k∈N : 3 A : ek (µ) 7→ ⌊k/2⌋−a ek (µ); k ∈ {2, 3, . . . }, a > 2 √ For the first mode we define A : e1 (µ) 7→ 2π 2e1 (µ). g Let HessE be the Hessian operator associated to the the (variational) Hessian var Hess E(., .)(µ). The renormalized Wasserstein Laplacian in an open neighbourhood of µ as defined below is finite: ∆aP∞ (T1 ) E(µ) :=

∞ X

k=1

g hHessEAe k (µ), Aek (µ)iµ < ∞

∞ 1 ′ ′ Proof. For the inner product hek , ek ivol ≡ hek , ek iH 1 (vol) := (2π) (T1 ), 2 hek , ek iL2 on Tvol P ∞ 1 we are given a complete orthonormal system on Tvol P (T ) by  √ k∈N  e2k (x) = 2√k −1 sin 2πkx, e2k+1 (x) = 2 k −1 cos 2πkx, k ∈ N  e1 (x) = 1. Likewise by

 d d √ 1 dx e2k (x), e (µ)(x) such that  dx e2k (µ)(x) =  2k µ(x) d d 1 e2k+1 (µ)(x) such that dx e2k+1 (µ)(x) = √ e2k+1 (x), µ(x) dx   e1 (µ)(x) = 1. with inital data  e2k (µ)(0) = 0, k∈N e2k+1 (µ)(0) = 0, k ∈ N

k∈N k∈N

we are given a complete orthonormal system of Tµ P∞ (T1 ): On the torus we can solve the defining differential equation by integration and orthonormality of {ek (µ)}k∈N is given by definition. To show that {ek (µ)}k∈N ⊂ Tµ P∞ (T1 ) we consider a vector field

6

CHRISTIAN SELINGER

u such that divµ u = 0. We have to show that ek (µ) ⊥ u with respect to h., .iµ for all k ∈ N: Z

T1

e′k (µ).uµ

Z

=

T1

=

−

√ e′k .u µ

Z

Z

µ′ (ek uµ) p 2 µ3 T1

√ ( µ/µ)ek .(uµ)′ + 1 T | {z }

ϕ1 ∈ Cc∞ (T1 ) ! Z µ′ ek p 0+ uµ = 0 2 µ3 T1 {z } |

=

ϕ′2 for ϕ2 ∈ Cc∞ (T1 )

R since µu.ϕ′ = 0 for any ϕ ∈ Cc∞ (S 1 ). Note that at this place it is crucial to deal with differentiable densities with full support. The function ϕ2 is obtained by integration. a functional E : P∞ → R and a distribution U ∈ T P∞ such that (U (µ)|ϕ) = R Given ′ ′ T1 u ϕ µ for smooth, compactly supported functions u and ϕ. According to ([9]): Hessvar E(U, U )(µ) =

Z

′ Γ∆ 2 (u)(µe (µ) − e(µ))vol +

T1

with

Z

T1

(∆u)2 (µp′ (µ) − p(µ))vol,

p(x) = xe′ (x) − e(x) and p′ (x) = xe′′ (x) + e′ (x) − e′ (x) = xe′′ (x) and Γ∆ e du champ operator with respect to the flat Laplacian ∆ = 2 the iterated carr´ d2 ∆T1 = dx . Then 2 Z (u′′ )2 (µe′ (µ) − e(µ))vol Hessvar E(U, U )(µ) = T1 Z + (u′′ )2 (µ2 e′′ (µ) − µe′ (µ) + e(µ))vol T1 Z = (u′′ )2 µ2 e′′ (µ)vol T1

∆aP∞ (T1 ) E(µ)

=

Z

2(2π)2 ((e1 (µ))′′ )2 µ2 e′′ (µ)vol(dx)

T1 ∞ Z X

+

k=2

= 2(2π)2 +

T1

Z

∞ Z X k=1

⌊k/2⌋−2a((ek (µ))′′ )2 µ2 e′′ (µ)vol(dx)

T1

T1

((log µ)′ )2 2 ′′ µ e (µ)vol(dx) 4µ

k −2a (((µ−1/2 e′2k )′ )2 + ((µ−1/2 e′2k+1 )′ )2 )µ2 e′′ (µ)vol(dx)

Since e′′2k = 2πke′2k+1 resp. e′′2k+1 = −2πke′2k and (e′2k )2 + (e′2k+1 )2 = 2(2π)2 it follows that ((µ−1/2 e′2k )′ )2 + ((µ−1/2 e′2k+1 )′ )2

= ((e′2k )2 + (e′2k+1 )2 )(1/4µ−3 (µ′ )2 ) + ((e′2k )2 + (e′2k+1 )2 )(µ−1 (2πk)2 ) + e′2k e′2k+1 (−µ−3/2 µ′ µ−1/2 2πk + µ−3/2 µ′ µ−1/2 2πk) = 2(2π)2 {1/4µ−3 (µ′ )2 + µ−1 (2πk)2 }

WASSERSTEIN LAPLACIAN

7

Consequently ∆aP∞ (T1 ) E(µ)

=

∞ Z X k=1

T1

 k −2a 2(2π)2 1/4µ−3 (µ′ )2 + µ−1 (2πk)2 µ2 e′′ (µ)vol(dx)

=

Z

((log µ)′ )2 2 ′′ µ e (µ)vol(dx) 4µ T1 Z ∞ X  2(2π)2 k −2a 1/4((log µ)′ )2 + (2πk)2 µe′′ (µ)vol(dx)

+2(2π)2

T1

k=1

+2(2π)2

Z

T1

< since

∞

((log µ)′ )2 ′′ µe (µ)vol(dx) 4

||((log µ)′ )2 µe′′ (µ)||∞ < +∞

||µe′′ (µ)||∞ < +∞,

which is guaranteed since the densities are supposed to have full support and to be sufficiently regular.  P ∞ For the Riemann zeta function ζ R (s) = k=1 k1s , ℜ(s) > 1 there exists a meromorphic continuation (see [10]) to the complex plane with single pole at s = 1 which was proved by Riemann in 1859 by the following functional equation: sπ ζ R (s) = 2s π s−1 sin Γ(1 − s)ζ R (1 − s); s ∈ C \ {1} 2 which enables us to calculate a specific value: sπ 1 lim sin Γ(1 − s)ζ R (1 − s) ζ R (0) = π s→0 2    1 sπ s3 π 3 1 1 = lim − + ... − + ... = − . π s→0 2 48 s 2 We usedP that Res(ζ, 1) = lims→1 (s − 1)ζ R (s) = 1 = a−1 and the Laurent series reads 1 R n R ζ (s) = ∞ n=−1 an (s − 1) i.e. ζ (1 − s) = − s + . . . .

Definition 3.1 (Zeta function regularized Laplacian).

∆P∞ (T1 ) E(µ) := lim ∆aP∞ (T1 ) E(µ) a→0

is called (Zeta function) regularized Laplacian. Proposition 3.2. Let e : R+ → R is C 3 . Given a functional E : P∞ (T1 ) → R of the type Z e(µ(x))vol(dx). E(µ) = T1

Then

∆P∞ (T1 ) E(µ)

Z

 = 2(2π) (ζ (0) + 1) ((log µ)′ /2)2 }µe′′ (µ) vol(dx) T1 Z 2 ′ 2 = π {(log µ) )} µe′′ (µ)vol(dx). 2

R

T1

We used additionally the fact that for the analytical continuation of the Zeta function ζ R (−2) = 0 holds. R Example 3.1. For Ent(µ) = T1 µ(x) log µ(x) vol(dx) we have ∆P∞ (T1 ) Ent(µ) = π 2 ||(log µ)′ ||2L2 (vol)

8

CHRISTIAN SELINGER

R Example 3.2. For functionals E(µ) = T1 f (x)vol(dx) with f a measurable function on T1 we have ∆P∞ (T1 ) E(µ) = 0 for all µ ∈ P∞ . R Example 3.3. Set E(µ) = 21 T1 µ2 vol, then ∆P∞ E(µ) = π 2 ||∇P Ent(µ)||2µ ∞

for all µ ∈ P∞ .

In the following proposition we denote the L2 (µ) inner product by h., .iµ , ifRno measure is specified we consider the inner product on L2 (vol), furthermore hf, µi := f µ(dx).

Proposition 3.3. Given a functional F : P∞ (T1 ) → R of the type F (µ) = Φ (hf, µi) ,

1

where f ∈ Cb (T ) and Φ ∈ Cb (R). Then ∆P∞ F (µ)

√ 2(2π)2 Φ′′ (hf, µi)||f ′ µ||2L2

=

and the square-field operator with respect to ∆P∞ applied to functionals F reads: Γ(F ) = 2(2π)2 ||∇P F (µ)||2µ ∞

Proof. Following ([4]) a geodesic (µt )t∈[0,T ] in P∞ starting at µ0 = µ satisfies µ˙ t = −div(µt ∇vt )

where the smooth function vt satisfies

−|∇vt |2 . 2 The second order variation of F along (µt )t∈[0,T ] reads v˙ t =

d2 Φ(hf, µt i) = dt2 =

d ′ (Φ (hf, µt i)hf, µ˙ t i) dt Φ′′ (hf, µt i)hf, µ˙ t i2 + Φ′ (hf, µt i)

d hf, µ˙ t i dt

Remark since v˙ t′ = −vt′ vt′′ and µ˙ t = −(µt vt′ )′ Z d d hf, µ˙ t i = f ′ vt′ µt vol dt dt = hf ′ , −vt′ vt′′ µt + vt′ µ˙ t i =

=

Hess(F )(v ′ , v ′ )(µ)

= =

−hf ′ , ((vt′ )2 )′ µt + µ′t (vt′ )2 i −hf ′ , ((vt′ )2 µt )′ i

d2 Φ(hf, µt i) dt2 t=0 Φ′′ (hf, µi)hf ′ , v ′ i2µ + Φ′ (hf, µi)hf ′′ , (v ′ )2 µi

In this formula at the place of v ′ we plug in (here for s > 1/2) k −s ek (µ)′ as in the proof of Proposition 3.1 in order to calculate ∞ X ∆sP∞ (T1 ) F (µ) = Φ′′ (hf, µi)hf ′ , k −s ek (µ)′ i2µ + Φ′ (hf, µi)hf ′′ , (k −s ek (µ)′ )2 µi k=1

which equals

Φ′′ (hf, µi)

∞ X

k=1

hf ′ , k −s µ−1/2 e′k i2µ + 2(2π)2 Φ′ (hf, µi)hf ′′ , 1iζ(2s)

WASSERSTEIN LAPLACIAN

9

Since ζ(2s) is finite for 2s > 1 and we now that hf ′′ , 1i = 0 the second term vanishes and by the functional equation for ζ we define again ∆P∞ (T1 ) F (µ) := lim ∆sP∞ (T1 ) F (µ) s→0

which equals lim Φ′′ (hf, µi)2(2π)2 ||f ′ µ1/2 ||2H −s = Φ′′ (hf, µi)2(2π)2 ||f ′ µ1/2 ||2L2 .

s→0

Note that the limit is taken for s ∈ C. The square-field operator Γs (F ) with respect to ∆sP∞ is defined by 1 s ∆ ∞ (F 2 ) − F ∆sP∞ (F ). 2 P In a first step we remark that d d2 1 d2 2 2 (F (µ )) = ( F (µt ) F (µ )) + F (µ ) t t t 2 dt2 dt dt2 and so 1 d2 d2 d 2 (F (µ )) − F (F (µt )) = ( F (µt ))2 = (Φ′ (hf, µi)hf ′ , v ′ iµ )2 t 2 2 2 dt dt dt which entails ∞ X Γs (F ) = (Φ′ (hf, µi)hf ′ , k −s ek (µ)′ iµ )2 = 2(2π)2 (Φ′ (hf, µi))2 ||f ′ ||2H −s µ

k=1

But

lim ||f ′ ||2H −s = ||f ′ ||2L2 (µ)

s→0

µ

and consequently lim Γs (F ) = 2(2π)2 ||∇P F (µ)||2µ ∞

s→0

 Remark 3.1. By the chain rule the formulas for the regularized Wasserstein Laplacian can be extended to the set of test functions  Z = F (µ) ≡ Φ (hf, µi) ; Φ ∈ C 2 (Rd ), f = (f1 , . . . , fd ) ∈ C 2 (T1 ; Rd ); µ ∈ P∞ (T1 ) , i.e.

∆

P∞

F (µ) =

d X

∂i ∂j Φ (hf, µi)

i,j=1

Z

0

1

fi′ fj′ µ

which equals the generator of the Wasserstein diffusion (see [8]) on P∞ (T1 ) with inverse temperature β = 0. Acknowledgments. The author would like to thank Josef Teichmann and Max-Konstantin von Renesse for useful comments. References [1] J.-D. Benamou and Y. Brenier, A computational Fluid Mechanics solution to the MongeKantorovich mass transfer problem., Numer. Math. 84 (2000), 375–393. [2] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions., Comm. Pure Appl. Math. 44 (1991), 375–417. [3] A. Kriegl, and P. Michor, A convenient setting of global analysis, AMS, Providence 1997. [4] J. Lott, Some geometric calculations on Wasserstein space, Comm. Math. Phys. 277 (2008), 423– 437. [5] R. McCann, Polar factorization of maps on Riemannian manifolds., Geom. Funct. Anal. 11 (2001), 589–608. [6] F. Otto, The geometry of dissipative evolution equations: The porous medium equation., Comm. Partial Differential Equations 26 (2001), 101–174.

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CHRISTIAN SELINGER

[7] C. Selinger, Gradient Flows on the space of probability measures. On differential-geometric aspects of optimal transport., unpublished Master’s Thesis, Universit¨ at Wien, september 2006. [8] Sturm, K.-Th. and von Renesse, M.-K., Entropic Measure and Wasserstein Diffusion, Annals of Prob. 37 (2009), 1114–1191. [9] C. Villani, Optimal transport, old and new, Springer, Grundlehren der mathematischen Wissenschaften 2008. [10] F.R.S. Thitchmarsh, The theory of the Riemann zeta-function, Clarendon Press, 1951 ematiques, 6, rue Coudenhovee de Recherche en Math´ e du Luxembourg, Unit´ Universit´ Kalergi, L-1359 Luxembourg, Grand-Duchy of Luxembourg E-mail address: [email protected]