Duality on gradient estimates and Wasserstein controls

Duality on gradient estimates and Wasserstein controls

Kazumasa Kuwada



 Ochanomizu University Universit¨at Bonn





§ 0 Motivation

Equivalent conditions for a lower Ricci curvature bound (von Renesse & Sturm ’05, etc...)

X: complete Riemannian manifold

P_t: heat semigroup associated with ∆ (i) Ric ≥ k,

(ii) d_p^W (P_t^∗µ, P_t^∗ν) ≤ e^−ktd_p^W (µ, ν) for some p ∈ [1, ∞].

(iii) |∇P_tf |(x) ≤ e^−ktP_t(|∇f |^q)(x)^1/q for some q ∈ [1, ∞],

Our goal:

Generalization of (ii) ⇔ (iii), to obtain a (ii)/(iii)-type estimate from the other one.

(ii) d_p^W (P_t^∗µ, P_t^∗ν) ≤ e^−ktd_p^W (µ, ν) (L^p-Wasserstein control)

(iii) |∇P_tf |(x) ≤ e^−ktP_t(|∇f |^q)(x)^1/q (L^q-gradient estimate)

§ 1 Framework and Main Result

Framework

• (X, d): Polish, proper length metric space.

⇒



 ∀x, y ∈ X,

∃a minimal geodesic joining x and y





• P(X): probability measures on X.

For µ, ν ∈ P(X), Π(µ, ν) ⊂ P(X × X) by Π(µ, ν) :=



π

¯¯¯¯

¯¯ π(A × X) = µ(A), π(X × A) = ν(A)



 .

L^p-Wasserstein distance For p ∈ [1, ∞],

d_p^W (µ, ν) := inf

π∈Π(µ,ν) kdk^Lp(π) ∈ [0, ∞].

Gradient

|∇^df |(x) := lim

r↓0 sup

y∈B_r(x)

¯¯¯¯ f (y) − f (x) d(y, x)

¯¯¯¯ , k∇^df k_∞ := sup

x∈X |∇^df |(x).

? |∇^df | is an upper gradient of f ,

i.e. for any curve γ joining x and y, f (y) − f (x) ≤

Z d(x,y) 0

|∇^df |(γ(s))ds.

• (P_x)_x∈X ⊂ P(X) s.t. x 7→ P_x continuous, P : C_b(X) → C_b(X),

P f(x) :=

Z

M

f dP_x. (e.g. P = P_t: heat semigroup)

• d˜: continuous geodesic distance on X. (e.g. d˜ = e^−ktd)

We also use notations |∇_d^˜f | and d˜_p^W (µ, ν).

L^p-Wasserstein control

d_p^W (P ^∗µ, P ^∗ν) ≤ d˜_p^W (µ, ν) (C_p) for p ∈ [1, ∞] and µ, ν ∈ P(X).

L^q-gradient estimate

|∇_d^˜P f|(x) ≤ P (|∇^df |^q)(x)^1/q (G_q) for q ∈ [1, ∞) and f ∈ C_b^Lip(X),

|∇_d^˜P f|(x) ≤ k∇^df k_∞ (G_∞) for q = ∞.

Assumptions

∃ Radon measure v on X with supp(v) = X s.t.

(i) (X, d, v) satisfies

• the local volume doubling condition,

• (1, ρ)-local Poincar´e inequality (∃ρ ≥ 1).

(ii) P_x ¿ v and x 7→ P_x(y): continuous.

Assumptions

∃ Radon measure v on X with supp(v) = X s.t.

(i) (X, d, v) satisfies

• the local volume doubling condition,

• (1, ρ)-local Poincar´e inequality (∃ρ ≥ 1).



 for employing a general theory of the Hamilton-Jacobi semigroup



 (ii) P_x ¿ v and x 7→ P_x(y): continuous.

(technical)

Local volume doubling condition

∃D > 0, ∃R₁ > 0 s.t. ∀x ∈ X, ∀r < R₁ v(B_2r(x)) ≤ Dv(B_r(x)).

(1, ρ)-local Poincar´e inequality

∀R > 0, ∃λ ≥ 1, ∃C_P > 0 s.t. ∀r < R,

− Z

B_r(x)

|f −f_x,r|dv ≤ C_P r

Ã

− Z

B_λr(x)

g^ρ dv

!1/ρ

for ∀f and ∀g: upper gradient of f , where f_x,r := 1

v(B_r(x)) Z

B_r(x)

f dv =: − Z

B_r(x)

f dv.

Theorem (K.)

For p, q ∈ [1, ∞] with 1

p + 1

q = 1, (i) (C_p) ⇒ (G_q).

(ii) Under Assumption (i)(ii), (G_q) ⇒ (C_p).

Remarks

• For p⁰ > p,





(G_p) ⇒ (G_p⁰ ), (C_p⁰) ⇒ (C_p).

(without Assumption (i)(ii))

• (G _∞) ⇔ (C₁) is well-known.

 via Kantorovich-Rubinstein formula;

without Assumption (i)(ii)





• (C_∞) ⇒ (G₁) is essentially well-known.

§ 2 Examples and Applications

In this §,

P = P_t: heat semigroup associated with ∆ (or a diffusion semigroup).

(A) How do we obtain (C_p)?

(mainly the case d˜ = e^−ktd) (B) Analytic approach to (G_q).

(the case d˜ = e^−ktd and beyond it)

(C) (C_p) for hypoelliptic diffusions on a Lie group.

(A) How do we obtain (C

) ?

(1) Coupling method

X: cpl. Riem. mfd, Ric ≥ k.

∃(B_t⁽¹⁾, B_t⁽²⁾): “infinitesimally parallel” coupling of two Brownian motions s.t.

d(B_t⁽¹⁾, B_t⁽²⁾) ≤ e^−kt/2d(B₀⁽¹⁾, B₀⁽²⁾)

∀t > 0, almost surely.

⇒ (C_∞)

Brownian motion under backward (super)Ricci flow (X, g(t))_{t∈[0,T ]}: cpl. Riem. mfds.

∂_tg(t) ≤ Ric_g(t).

(B_t)_{t∈[0,T ]}: g(t)-Brownian motion

(sol. to the martingale problem of ∂_t + 1

2 ∆_g(·)).

∃(B_t⁽¹⁾, B_t⁽²⁾): a coupling of two g(t)-BMs s.t.

d_g(t)(B_t⁽¹⁾, B_t⁽²⁾) ≤ d_g(0)(B₀⁽¹⁾, B₀⁽²⁾)



 McCann & Topping ’08 for (C₂).

Arnaudon & Coulibaly & Thalmaier ’09,

cf. K.-Philipowski ’09 for non-explosion of B_t.





(A) How do we obtain ( C

) ? (2) Gradient flow formulation

of the heat flow

• Heuristically, heat distributions (µ_t)_t≥0 is a

“gradient flow” of the relative Entropy functional E(µ) =

Z

X

dµ

dv log dµ

dv dv.

on L²-Wasserstein space P²(X) ⊂ P(X). (Otto ’98, Ambrosio & Gigli & Savar´e ’05,...)

• When X: cpl. Riem. mfd, Ric ≥ k

⇔ “Hess E ≥ k” on (P²(X), d₂^W ). (von Renesse & Sturm ’05)

Heuristically,

“Hess E ≥ k”

⇒ d₂^W (µ_t, ν_t) ≤ e^−ktd₂^W (µ₀, ν₀). If X: cpl. Riem. mfd., it can be made rigorous.

Moreover, µ_t = P_t^∗µ₀ (identification).

(Villani ’08, Erbar ’08, Ohta ’09,...)

⇒ (C₂)

More singular spaces (e.g. Alexandrov spaces):

Savar´e ’07, Ohta ’09,...

Remark on (A)

To obtain (C_p), we have used a notion of lower curvature bound which is different from (G_q). E.g. in von Renesse & Sturm ’05,

(G_∞) ⇒ Ric ≥ k

⇒ (C_∞) (coupling method)

⇒ (G₁) (easy)

⇒ (G_∞) (monotonicity)

(B) Analytic approach to (G

)

(1) Bakry & ´ Emery’s Γ

-criterion

A: (L²-)generator of P_t.

• Γ(f, f) := 1

2 (A(f ²) − 2f Af ).

• |∇f | := Γ(f, f)^1/2.

• Γ₂(f, f) := 1

2 (AΓ(f, f ) − 2Γ(f, Af ). If X: cpl. Riem. mfd. & A = ∆,

Ric ≥ k ⇔ Γ₂(f, f ) ≥ kΓ(f, f ).

Γ₂(f, f) ≥ kΓ(f, f)

⇔ |∇P_tf |(x) ≤ e^−ktP_t|∇f |(x)

⇒ if |∇f | = |∇^df |,

Γ₂(f, f ) ≥ kΓ(f, f) ⇔ (G₁)

(with d˜ = e^−ktd).

Remark

|∇f | = |∇^df | does not seem to be trivial even if there is a strong connection between d and P_t.

(B) Analytic approach to (G

) (2) H¨ ormander-type operators

on a Lie group

X: Lie group with a right-Haar measure v.

{X_i}ⁿ_i=1: left-invariant, linearly independent vector fields generating all left-invariant vector fields in the sense of Lie algebra (H¨ormander condition).

A :=

Xm i=1

X_i², P_t: semigroup associated with A.

L^q-Gradient estimate

|∇P_tf |(x) ≤ K_q(t)P_t(|∇f |^q)(x)^1/q. (G^∗_q)

Known results

• 3-dim. Heisenberg group, K_q(t) ≡ K_q > 1 – q > 1: Driver & Melcher ’05.

– q = 1: H.-Q. Li ’06 / Bakry et al. ’08.

• X: general, q > 1: Melcher ’08 (K_q(t) ≡ K_q if X: nilpotent).

• X: group of type H, q = 1, K_q(t) ≡ K_q: Eldredge ’09.

• X = SU (2), q > 1, K_q(t) = K_qe^−t: Baudoin & Bonnefont ’09.

“Lower curvature bound” on the Heisenberg group

• Formally, Ric / Γ₂ is unbounded from below.

• Curvature-dimension condition CD(k, N )

(“dim X ≤ N and Ric ≥ k”) does not hold for any N ∈ [1, ∞), k ∈ RRR (Juillet ’09).

– CD(k, ∞) ⇔ “Hess E ≥ k” .

– Weaker condition (MCP (k, N )) holds for N = 5 and k = 0 (Juillet ’09).

An appropriate notion of a lower curvature bound is not clear until now!

(C) ( C

) for hypoelliptic diffusions on a Lie group

(1) General result

X, v, {X_i}^m_i=1, P : as before.

Carnot-Caratheodory distance For V ∈ T_xX,

|V | =









³ X^m

i=1

a_i²

´1/2

if V =

Xm

i=1

a_iX_i(x),

∞ otherwise.

d(x, y) := inf





Z 1 0

|γ˙_s|ds

¯¯¯¯

¯¯ γ₀ = x, γ₁ = y



 .

Proposition

(X, d, v), P = P_t: as above.

(i) (X, d, v; P ) satisfies Assumption (i)(ii).

(ii) (G^∗_q) ⇒ (G_q).

Corollary

(G^∗_q) ⇒ (C_p) for q ∈ [ 1, ∞].

(C) ( C

) for hypoelliptic diffusions on a Lie group

(2) Examples

3-dim. Heisenberg group X = RRR³, v: Lebesgue.

(x, y, z) · (x⁰, y⁰, z⁰)

= (x + x⁰, y + y⁰, z + z⁰ + 1

2 (xy⁰ − yx⁰)), X₁ = ∂_x − y

2 ∂_z , X₂ = ∂_y + x

2 ∂_z .

Associated diffusion (B_t¹, B_t², B_t³) from (x, y, z): B_t¹ = W_t¹, B_t² = W_t²,

B_t³ = z + 1 2

Z _t

0

W_t¹dW_t² − W_t²dW_t¹, where (W_t¹, W_t²): 2-dim. BM from (x, y).

(C_∞): For each t > 0, ∃ a coupling (BBB_t, BBB˜ _t) of (B_t¹, B_t², B_t³) with initial conditions

aaa ∈ RRR³ and bbb ∈ RRR³ respectively s.t.

d(BBB_t, BBB˜ _t) ≤ K₁d(aaa, bbb) PPP-a.s..

◦ ◦

In this case, ∃C₁, C₂ > 0 s.t.

C₁kbbb⁻¹aaak ≤ d(aaa, bbb) ≤ C₂kbbb⁻¹aaak, where k(x, y, z)k = ¡

(x² + y²)² + z²¢1/4

.

Definition

X: a group of type H iff, for X : Lie alg.

associated with X with a scalar product h·, ·i,

• X = V ⊕ Z with [V, V] = Z, [V, Z] = [Z, Z] = 0.

• J : Z → End V given by

hJ(Z)V₁, V₂i = hZ, [V₁, V₂]i satisfies J(Z)² = −kZkId.

{X_i}^m_i=1 will be ONB of V.

Remarks

• ∀m, the (2m + 1)-dim. Heisenberg group is of type H.

• A free nilpotent Lie group of step 2 is of type H iff it is the 3-dim. Heisenberg group.

• Possible dimension of a group of type H is completely determined.

§ 3 Sketch of the Proof

Recall:

d_p^W (P ^∗µ, P ^∗ν) ≤ d˜_p^W (µ, ν), (C_p)

|∇_d^˜P f|(x) ≤ P (|∇^df |^q)(x)^1/q. (G_q)

• The case p = 1 (q = ∞) and (C_∞) ⇒ (G₁) are easy and well-known.

• d_p^W (µ, ν) ^p→∞→ d_∞^W (µ, ν) ∈ [0, ∞].

⇒ For (G_q) ⇒ (C_p), we may assume p < ∞.

• (C_p) for µ = δ_x, ν = δ_y ⇒ (C_p).

(disintegration of measures)

(A) Proof of ( C

) ⇒ ( G

)

π ∈ Π(P_x, P_y), kdk^Lp(π) = d_p^W (P_x, P_y). P f (x) − P f (y) =

Z

X

f dP_x − Z

X

f dP_y

= Z

X×X

(f (z) − f (w))π(dzdw)

= Z

{d(z,w)≤r} · · · + Z

{d(z,w)>r} · · ·

= (I) + (II).

(I) ≤ Z

{d(z,w)≤r}

f (z) − f (w)

d(z, w) d(z, w)π(dzdw)

≤

(Z

X

sup

w∈B_r(z)

¯¯¯¯ f (z) − f (w) d(z, w)

¯¯¯¯^qP_x(dz)

)^1/q

× d_p^W (P_x, P_y)

≤ (C_p)

{· · ·}^1/q

| {z }

↓^r→0

d(x, y˜ ).

P (|∇^df |^q)(x)^1/q

(II) ≤ 2kf k_∞ Z

{d(z,w)>r}

π(dzdw) Chebyshev ≤ 2kf k_∞ d_p^W (P_x, P_y)^p

r^p (C_p) ≤ 2kf k_∞d(x, y˜ )

d(x, y˜ )^p−1 r^p . Choose r = r(x, y) s.t.

ylim→x r = 0 & lim

y→x

d(x, y˜ )^p−1

r^p = 0

(e.g. r = ˜d(x, y)^1/(2q)). ¥

(B) Proof of ( G

) ⇒ ( C

)

General theory of the Hamilton-Jacobi semigroup (Lott & Villani ’07, Balogh et al. ’09)

Q_tf (x) := inf

y∈X

·

f (y) + t · 1 p

µ d(x, y) t

¶^p ¸ .

• Q_·f ∈ C_b^Lip([0, ∞) × X) if f ∈ C_b^Lip(X).

• Under Assumption (i) , for ∀t > 0, v-a.e.

∂_tQ_tf (x) = − 1

q |∇^dQ_tf |(x)^q .

¡Note: q⁻¹u^q = sup

s≥0

¡us − p⁻¹s^p¢¢

Kantorovich duality

d_p^W (µ, ν)^p = sup

f∈C_b^Lip

·Z

X

f ^∗ dµ − Z

X

f dν

¸ ,

f ^∗(x) : = inf

y∈X [ f (y) + d(x, y)^p ]

= p Q₁(p⁻¹f )(x).

⇓ d_p^W (µ, ν)^p

p = sup

f

·Z

X

Q₁f dµ − Z

X

f dν

¸ .









γ : [0, 1] → X d˜-minimal geodesic, γ₀ = y, γ₁ = x,

d(γ˜ _s, γ_t) = |t − s|d(x, y˜ ).

◦ ◦

d_p^W (P_x, P_y)^p

p = sup

f

[P Q₁f (x) − P f(y)]

“ = ”

interpolation

sup

f

·Z 1 0

∂_t(P Q_tf (γ_t))dt

¸ .

∂_t(P Q_tf (γ_t))

“=” P (∂_tQ_tf )(γ_t) + h∇P Q_tf (γ_t), γ˙ _ti HJ eq.

up. grad. ≤ − 1

q P (|∇^dQ_tf |^q)(γ_t)

+ d(x, y˜ ) ¯¯∇_d^˜P Q_tf ¯¯ (γ_t) (G_q) ≤ d(x, y˜ )σ − 1

q σ^q ≤ d(x, y˜ )^p p .

³

σ := P (|∇^dQ_tf |^q)(γ_t)^1/q

´ ¥

Questions

(i) When does (C_p) ⇒ (C_p⁰ ) / (G_p⁰) ⇒ (G_p) occur for p⁰ > p?

(OK if X: Riem., P = P_t)

(ii) When does (C_p) ⇒ “pathwise control” occur?

(in the case P = P_t)

(iii) When does |∇^df | = |∇f |(= Γ(f, f)^1/2) hold?

(OK if X: (sub-)Riem., P = P_t)

(iv) Relation with other “lower curvature bounds”...