PDF Duality results on gradient estimates and Wasserstein controls

Duality results on gradient estimates and Wasserstein controls

Kazumasa Kuwada



 Ochanomizu University Universit¨at Bonn





§ 1 Framework and main results

X: Polish space

• (P_x)_x∈X ⊂ P(X): Markov kernel P f (x) :=

∫

X

f dP_x, P ^∗µ(A) :=

∫

X

P_x(A)µ(dx) Assume P (C_b(X)) ⊂ C_b(X)

(e.g. P_x(dy) = P_t(x, dy): heat semigroup)

• d, d˜: lower semi-conti. pseudo-distance on X (e.g. d: distance on X, d˜ = e^−ktd)

Π(µ, ν) :=

{

π ¯¯¯ π ◦ p⁻₁ ¹ = µ, π ◦ p⁻₂ ¹ = ν }

(couplings of µ, ν ∈ P(X)) L^p-Wasserstein distance

For p ∈ [1, ∞],

d_p^W (µ, ν) := inf

π∈Π(µ,ν) kdk^Lp(π) ∈ [0, ∞] L^p-Wasserstein control

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

Gradient

|∇^df |(x) := lim

r↓0 sup

y;d(x,y)≤r

¯¯¯¯ f (x) − f (y) d(x, y)

¯¯¯¯ , k∇^df k_∞ := sup

x∈X |∇^df |(x) Subgradient

|∇⁻_d f |(x) := lim

r↓0 sup

y;d(x,y)≤r

[ f (x) − f (y) d(x, y)

]

+

, k∇⁻_d f k_∞ := sup

x∈X |∇⁻_d f |(x)

L^q-gradient estimate

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

k∇_d^˜P f k_∞ ≤ k∇^df k_∞ (G_∞) when q = ∞

(

[∇ Ã ∇⁻] ⇒ [(G_q) Ã (G⁻_q )]

)

The first duality result

For p, q ∈ [1, ∞] with 1

p + 1

q = 1, (i) (C_p) ⇒ (G_q)

(ii) Under Assumptions A1-A3, (G_q) ⇒ (C_p) Theorem A (K. ’10 JFA)

The second duality result

For p, q ∈ [1, ∞] with 1

p + 1

q = 1, (i) (C_p) ⇒ (G⁻_q )

(ii) Under Assumptions B1-B3, (G⁻_q ) ⇒ (C_p) Theorem B (K. ’10)

Remarks (without Assumptions A/B)

• For p⁰ > p,

(C_p⁰) ⇒ (C_p) and (G_q⁰) ⇒ (G_q)

• (C₁) ⇔ (G_∞) is well known

(via Kantorovich-Rubinstein formula)

• (C_∞) ⇒ (G₁) is essentially well known (Coupling method)

Interesting part: (G_q) ⇒ (C_p) for p ∈ (1, ∞]

Remarks (without Assumptions A/B)

• For p⁰ > p,

(C_p⁰) ⇒ (C_p) and (G_q⁰) ⇒ (G_q)

• (C₁) ⇔ (G_∞) is well known

(via Kantorovich-Rubinstein formula)

• (C_∞) ⇒ (G₁) is essentially well known (Coupling method)

Interesting part: (G_q) ⇒ (C_p) for p ∈ (1, ∞]

§ 2 Sketch of the proof of ( G

_q

) ⇒ ( C

_p

)

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)

• (C_p) for ∀p < ∞ ⇒ (C_∞)

ÃÃÃ We may assume p ∈ (1, ∞)

• (C_p) for µ = δ_x, ν = δ_y ⇒ (C_p) ÃÃÃ We show d_p^W (P_x, P_y)^p

p ≤ d(x, y˜ )^p p

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)

• (C_p) for ∀p < ∞ ⇒ (C_∞)

ÃÃÃ We may assume p ∈ (1, ∞)

• (C_p) for µ = δ_x, ν = δ_y ⇒ (C_p) ÃÃÃ We show d_p^W (P_x, P_y)^p

p ≤ d(x, y˜ )^p p

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)

• (C_p) for ∀p < ∞ ⇒ (C_∞)

ÃÃÃ We may assume p ∈ (1, ∞)

• (C_p) for µ = δ_x, ν = δ_y ⇒ (C_p) ÃÃÃ We show d_p^W (P_x, P_y)^p

p ≤ d(x, y˜ )^p p

Kantorovich duality d_p^W (µ, ν)^p

p = sup

f∈C_b^Lipd

[∫

X

f ^∗dµ −

∫

X

f dν ]

f ^∗(x) := inf

y∈X

[

f (y) + 1

p d(x, y)^p ]















∀x, ∀y, g(x) − f (y) ≤ 1

p d(x, y)^p

⇒ 1

p kdk^p_L^p_(π) ≥

∫

X

g dµ −

∫

X

f dν







⇒ “≥”









Kantorovich duality d_p^W (µ, ν)^p

p = sup

f∈C_b^Lipd

[∫

X

f ^∗dµ −

∫

X

f dν ]

f ^∗(x) := inf

y∈X

[

f (y) + 1

p d(x, y)^p ]















∀x, ∀y, g(x) − f (y) ≤ 1

p d(x, y)^p

⇒ 1

p kdk^p_L^p_(π) ≥

∫

X

g dµ −

∫

X

f dν







⇒ “≥”









Hamilton-Jacobi semigroup Q_tf (x) := inf

y∈X

[

f (y) + t · 1 p

( d(x, y) t

)^p ]

⇒ f ^∗ = Q₁f

We expect (under our assumptions):

• Q_·f (x): Lipschitz, Q_tf (·): d-Lipschitz

• Hamilton-Jacobi equation

∂_tQ_tf = − 1

q |∇^dQ_tf |^q









˜

γ : [0, 1] → X d˜-minimal geodesic,

˜

γ₀ = y, γ˜₁ = 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

[∫ 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.

upp. 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

) ¥

§ 3 Assumptions

(1) Assumptions A1-A3

v: Radon measure on X with supp(v) = X Assumption A1

• d: compatible with the topology on X

• d: length metric

• ∀r > 0, ∀x, {y | d(x, y) ≤ r}: compact

• local (uniform) volume doubling condition

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

Assumptions A1 ⇒ HJ eq. for Q_tf v-a.e.



 Lott & Villani ’07 JMPA

Balogh, Engoulatov, Hunziker & Maasalo ’09





◦ ◦

Assumption A2

d˜: conti. length metric Assumption A3

P_x ¿ v, x 7→ dP_x

dv (y): conti

(2) Assumptions B1-B3

“Assumption B1”

• d: length pseudo-metric

• Q_tf is measurable for ∀f ∈ C_b(X)

• d(x, ·)² is locally semiconcave

• d-geodesic is locally uniformly extendable

◦ ◦

Assumption B1 ⇒ HJ eq. for Q_tf w.r.t. ∇⁻

& local uniform bound of 1

s (Q_t+sf − Q_tf ) (Extension of an argument in Villani’s book)

Assumption B2

d˜: length pseudo-metric Assumption B3

∀f ∈ C_b(X), P Q_tf : d˜-upper semi-conti.

◦ ◦

When [ ˜d(x_n, x) → 0] ⇒ [x_n → x in X],

• P : strong Feller, or

• ∀f ∈ C_b(X), Q_tf : u.s.c.

(true if X: vect sp., d˜: seminorm)

⇒ Assumption B3

Examples

Examples satisfying A1-A2

• Complete Riemannian manifold with Ric ≥ k₀

• Carnot groups (see below)

(dim X < ∞, P_x ¿ v for a base measure v) Examples satisfying B1-B2

• Cpl. Riem. mfds (no curvature assumption)

• Vecter space, d: Hilbertian seminorm (P_x ¿ v is not necessary)

§ 4 Related results and Applications

(1) Connection with geometry,

curvature bound

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

X: cpl. Riem. mfd

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, ∞], ∀t > 0

(iii) |∇P_tf |(x) ≤ e^−ktP_t(|∇f |^q)(x)^1/q for some q ∈ [1, ∞], ∀t > 0

Remark

“(a lower Ricci bound) ⇒ (C_p)” in the literature No direct way “(G_q) ⇒ (C_p)” was known

E.g. in von Renesse & Sturm ’05, Ric ≥ k

⇓ coupling method

(C_∞) ⇒ (C_p) ⇒ (C₁)

⇓ ⇓

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

(2) H¨ ormander-type operators

on a Lie group

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 A := 1

2

(X₁² + X₂²)

, P = P_t := e^tA

Diffusion BBB_t = (B_t¹, B_t², B_t³) associated with A starting from (x, y, z), i.e.,

dBBB_t = X₁(BBB_t)dW_t¹ + X₂(BBB_t)dW_t², BBB₀ = (x, y, z)

More explicitly,

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 starting from (x, y)

|Γf | := |X₁f |² + |X₂f |²: carr´e du champ L^q-gradient estimate

∃K_q > 1,

|ΓP_tf |(x) ≤ K_qP_t(|Γf |^q/2)(x)^2/q (G^∗_q)

◦ q > 1: Driver & Melcher ’05 JFA

◦ q = 1: H.-Q. Li ’06 JFA/

Bakry, Baudoin, Bonnefont & Chafa¨ı ’08 JFA

Carnot-Caratheodory distance For V ∈ T_xX,

|V | :=





√

a²₁ + a²₂ if V = a₁X₁ + a₂X₂,

∞ otherwise.

d(x, y) := inf





∫ 1 0

|γ˙ _s|ds

¯¯¯¯

¯¯ γ₀ = x, γ₁ = y





(i) (X, d, v; P ) satisfies Assumption A1-A3 (ii) (G^∗_q) ⇒ (G_q)

Proposition

(G^∗_q) ⇒ (C_p) for p ∈ [1, ∞] Corollary

(C_∞): For each t > 0,

∃ a coupling (BBB_t, BBB˜ _t) of (B_t¹, B_t², B_t³) s.t.

d(BBB_t, BBB˜ _t) ≤ K₁d(BBB₀, BBB₀) PPP-a.s.

◦ ◦

Remark ∃C₁, C₂ > 0 s.t.

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

(x² + y²)² + z²)1/4

Extension of (G^∗_q)

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

• X: group of type H, q = 1, K_q(t) ≡ K_q: Eldredge ’10 JFA

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

Proposition (and Corollary) is still valid

⇒ Theorem A implies (C_p)

Extension of (G^∗_q)

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

• X: group of type H, q = 1, K_q(t) ≡ K_q: Eldredge ’10 JFA

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

Proposition (and Corollary) is still valid

⇒ Theorem A implies (C_p)