Duality on gradient estimates and Wasserstein controls

Duality on gradient estimates and Wasserstein controls

Kazumasa Kuwada



 Ochanomizu University Universit¨at Bonn





§ 1 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)

§ 2 Framework and Main Result

(X, d): Polish metric space.

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

P f(x) :=

∫

M

f dP_x, P ^∗µ(A) :=

∫

X

P_x(A)µ(dx) (e.g. P = P_t: heat semigroup)

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

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).

f : Lipschitz ⇒ |∇^df | is an upper gradient of f . i.e. for any 1-Lipschitz curve γ : [a, b] → X

joining x and y,

f (y) − f (x) ≤

∫ b a

|∇^df |(γ(s))ds.

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 = ∞.

v: Radon measure on X with supp(v) = X. Assumption 1 (X, d): proper length space.

Assumption 2 (X, d, v) supports

• local (uniform) volume doubling condition,

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

Assumption 3 d˜: geodesic distance.

Assumption 4 P_x ¿ v, x 7→ dP_x

dv (y): continuous.

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,

−

∫

B_r(x)

|f −f_x,r|dv ≤ C_P r

(

−

∫

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))

∫

B_r(x)

f dv =: −

∫

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 1-4, (G_q) ⇒ (C_p).

Remarks

• For p⁰ > p,





(G_p) ⇒ (G_p⁰ ), (C_p⁰) ⇒ (C_p).

(without Assumption 1-4)

• (G _∞) ⇔ (C₁) is well-known.

 via Kantorovich-Rubinstein formula;

without Assumption 1-4





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

§ 3 Examples and Applications

How do we obtain (C_p)?

(A) Known derivation of ( C

)

( X: cpl. Riem. mfd., P = P_t, d˜ = e^−ktd )

• Coupling by parallel transport of B.m.’s:

Ric ≥ k ⇒ (C_∞) Ã Extension to backward Ricci flow.

? d˜ is essentially different from d!

• Gradient flow formulation of the heat flow µ_t:

∂_tµ_t = −∇E(µ_t).

◦ Ric ≥ k ⇔ “Hess E ≥ k”,

◦ Hess E ≥ k ⇒ (C₂) for µ_t (= P_t^∗µ).

à Extension to singular spaces (e.g. Alex. sp.).

Remark

To obtain (C_p), we have used some notion of lower curvature bound which is different from (G_q).

E.g. in von Renesse & Sturm ’05,

(G_∞) ⇒ Ric ≥ k (Bochner)

⇒ (C_∞) (coupling method)

⇒ (G₁)

⇒ (G_∞) (monotonicity)

(B) 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).

P_t := e^tA, A :=

∑n i=1

X_i².

|∇f |² := 1 2

(A(f ²) − 2f Af )

=

∑n i=1

|X_if |². 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.

X, v, {X_i}ⁿ_i=1, P : as before.

Carnot-Caratheodory distance For V ∈ T_xX,

|V | =









( ∑ⁿ

i=1

a_i²

)1/2

if V =

∑n

i=1

a_iX_i(x),

∞ otherwise.

d(x, y) := inf





∫ 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 1-4 (ii) “(G^∗_q) ⇒ (G_q)”.

Corollary

(G^∗_q) ⇒ (C_p) for q ∈ [ 1, ∞].

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

∫ _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}ⁿ_i=1 will be an ONB of V.

Remarks

• Any group of type H is a stratified nilpotent Lie group of step 2.

• ∀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.

§ 4 Sketch of the Proof of ( G

) ⇒ ( C

)

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 = ∞) is well-known.

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

⇒ We may assume p < ∞.

• (C_p) for µ = δ_x, ν = δ_y ⇒ (C_p).

(disintegration of measures)

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

( d(x, y) t

)^p ] .

• Under Assumption 1,

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

• Under Assumption 1-2, 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

[∫

X

f ^∗ dµ −

∫

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

[∫

X

Q₁f dµ −

∫

X

f dν ]

.









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

d(γ˜ _s, γ_t) = |t − s|d(x, y˜ ).

(Assumption 3)

◦ ◦

d_p^W (P_x, P_y)^p

p = sup

f

[P Q₁f (x) − P f(y)]

“ = ”

interpolation

sup

f

[∫ 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 . (

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) Relation between Bakry-´Emery’s Γ₂-criterion and (G_q) (in the case P = P_t, d˜ = e^−ktd) (When does |∇^df | = Γ(f, f)^1/2 hold?).

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