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.

§ 2 Framework and main result

(X, d): Polish metric space.

• (P_x)_x∈X ⊂ P(X): Markov kernel.

P : B^b(X) → B^b(X) P f (x) :=

∫

X

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)

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

d_p^W (µ, ν) := inf

π∈Π(µ,ν) kdk^Lp(π) ∈ [0, ∞].

( Π(µ, ν): couplings of µ and ν ) 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).

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),

k∇_d^˜P fk_∞ ≤ 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.

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.

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, Ric ≥ k

⇓ coupling method

(C_∞) ⇒ (C_p) ⇒ (C₁)

⇓ ⇓

(G₁) ⇒ (G_q) ⇒ (G_∞) ⇒ Ric ≥ k. Bochner

§ 3 H¨ ormander-type operators

on a Lie group

X: Lie group with a right-Haar measure v. {X_i}ⁿ_i=1: left-invariant vector fields

satisfying the 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 & Baudoin &

Bonnefont & Chafa¨ı ’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 ’10.

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

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) with d˜ = K_q(t)d.

Corollary

(G^∗_q) ⇒ (C_p) for q ∈ [ 1, ∞].

§ 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 < ∞.

• For (C_p), it suffices to show

d_p^W (P_x, P_y) ≤ d(x, y˜ ).

General theory of the Hamilton-Jacobi semigroup (Lott & Villani ’07 / Balogh & Engoulatov &

Hunziker & Maasalo ’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 = − 1

q |∇^dQ_tf |^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 (P_x, P_y)^p

p = sup

f

[P Q₁f (x) −P f (y)] .

∃γ : [0, 1] → X : d˜-min. geod. of const. speed, γ₀ = y, γ₁ = x. (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))

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

HJ eq. ≤ d(x, y˜ ) ¯¯∇_d^˜P Q_tf ¯¯ (γ_t)

− 1

q P (|∇^dQ_tf |^q)(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 )

Hence

d_p^W (P_x, P_y)^p

p = sup

f

[∫ ₁

0

∂_t(P Q_tf (γ_t))dt ]

≤ sup

f

∫ 1 0

d(x, y˜ )^p

p dt

=

d(x, y˜ )^p p .

¥

§ 5 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_∞) ⇒ “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”...