Wasserstein contractions associated with the curvature-dimension condition

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Wasserstein contractions associated with the curvature-dimension condition

Kazumasa Kuwada (Ochanomizu University)

(Markov processes and stochastic analysis (Kyoto university) Jan. 13, 2013)

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1. Introduction

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Framework

M: cpl. Riem. mfd., dim ≥ 2, ∂M = ∅ P_t = e^t∆: heat semigroup on M, P_t1 ≡ 1

Goal

Characterize

Ric ≥ K & dimM ≤ N

in terms of behavior of Wasserstein distances between heat distributions P_t^∗µ (µ ∈ P(M), t > 0)

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lower Ricci bound on metric meas. sp.

Recent developments:

Generalization of “Ric ≥ K”

Equivalent conditions in terms of P_t

or only “metric and measure” Same equivalence beyond Riem. mfds

⇒⇒⇒ e.g. “Stable” suﬃcient cond. for Lipschitz regularity of P_tf

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lower Ricci bound on metric meas. sp.

Generalization of “Ric ≥ K” Equivalent conditions in terms of P_t

or only “metric and measure”

Same equivalence beyond Riem. mfds

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lower Ricci bound on metric meas. sp.

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lower Ricci bound on metric meas. sp.

⇒⇒⇒ e.g. “Stable” suﬃcient cond.

for Lipschitz regularity of Ptf

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1. Introduction

2. Known results for lower Ricci bounds 3. Curvature-dimension conditions

4. Proofs & extensions 4.1 Duality

4.2 Questions

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1. Introduction

4.2 Questions

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lower Ricci curv. bound

For K ∈RRR, TFAE ([von Renesse & Sturm ’05] etc.):

(i) Ric ≥ K

(ii) W2(P_t^∗µ0, P_t^∗µ1) ≤ e⁻^KtW2(µ0, µ1), (iii) |∇P_tf|² ≤ e⁻^2KtP_t(|∇f|²)

(iv) 1

2(∆|∇f|² − 2h∇f,∇∆fi) ≥ K|∇f|² (v) Ent is K-convex w.r.t. W₂

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How important?

(iii)(iv) has rich applications

in functional ineq. & diﬀerential geometry, e.g. quantitative Lipschitz regularization of P_t

[Bakry & ´Emery etc.]

F More applications if “dimM” is involved

(e.g. Harnack inequalities) (v) makes sense even on met. meas. sp.’s

& stable (e.g., under Gromov-Hausdorﬀ conv.) [Sturm ’06, Lott & Villani ’09]

⇒ extension of (ii)(iii)(iv) to singular spaces

[Ambrosio, Gigli & Savar´e] etc.

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How important?

(e.g. Harnack inequalities)

(v) makes sense even on met. meas. sp.’s

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How important?

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How important?

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Implications

On non-smooth sp.:

(i) Ric ≥ K

⇔

Bochner-Weitzenb¨ock formula

[Bakry & ´Emery ’84] (iv) 1

2(∆|∇f|² − 2h∇f,∇∆fi) ≥ K|∇f|²

⇔ [Bakry & ´Emery ’84]

(iii) |∇Ptf| ≤ e⁻^KtPt(|∇f|²)^1/2

⇔ [K. ’10 / K.]

(ii) W₂(P_t^∗µ₀, P_t^∗µ₁) ≤ e⁻^KtW₂(µ₀, µ₁)

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Implications

On non-smooth sp.:

(i) Ric ≥ K

⇔

[Bakry & ´Emery ’84]

(iv) 1

2(∆|∇f|² − 2h∇f,∇∆fi) ≥ K|∇f|²

(iii) |∇Ptf| ≤ e⁻^KtPt(|∇f|²)^1/2

⇔ [K. ’10 / K.]

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Implications

On non-smooth sp.:

(i) Ric ≥ K

⇔

(iv) 1

2(∆|∇f|² − 2h∇f,∇∆fi) ≥ K|∇f|²

(iii) |∇Ptf| ≤ e⁻^KtPt(|∇f|²)^1/2

⇔ [K. ’10 / K.]

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Implications

On non-smooth sp.:

(i) Ric ≥ K

⇔

(iv) 1

2(∆|∇f|² − 2h∇f,∇∆fi) ≥ K|∇f|²

(iii) |∇Ptf| ≤ e⁻^KtPt(|∇f|²)^1/2

⇔ [K. ’10 / K.]

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Implications

On non-smooth sp.:

(i) Ric ≥ K

⇔

(iv) 1

2(∆|∇f|² − 2h∇f,∇∆fi) ≥ K|∇f|²

(iii) |∇Ptf| ≤ e⁻^KtPt(|∇f|²)^1/2

⇔ [K. ’10 / K.]

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Implications

On non-smooth sp.:

(v) Ent is K-convex w.r.t. W₂

⇓

Identiﬁcation of P_t^∗µ with the gradient ﬂow of Ent in (P(M), W2)

Linearity of heat ﬂow w.r.t. initial data [Gigli, K. & Ohta]: cpt. Alexandrov sp. [Ambrosio, Gigli & Savar´e]: met. meas. sp. [Koskela & Zhou]

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Implications

On non-smooth sp.:

⇓

Linearity of heat ﬂow w.r.t. initial data

[Gigli, K. & Ohta]: cpt. Alexandrov sp. [Ambrosio, Gigli & Savar´e]: met. meas. sp. [Koskela & Zhou]

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Implications

On non-smooth sp.:

⇓

Linearity of heat ﬂow w.r.t. initial data [Gigli, K. & Ohta]: cpt. Alexandrov sp.

[Ambrosio, Gigli & Savar´e]: met. meas. sp.

[Koskela & Zhou]

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Implications

On non-smooth sp.:

⇓

Identiﬁcation of P_t^∗µ with the gradient ﬂow of Ent in (P(M),W2)

Linearity of heat ﬂow w.r.t. initial data [Gigli, K. & Ohta]: cpt. Alexandrov sp.

[Ambrosio, Gigli & Savar´e]: met. meas. sp.

[Koskela & Zhou]

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Implications

On non-smooth sp.:

† TFAE [Ambrosio, Gigli & Savar´e]

(v)* Ent is K-convex w.r.t.W₂ & linearity of heat ﬂow (vi) ^∀µ₀, ^∃sol (µ_t)_t_≥₀ to EVI_K of Ent: for ^∀ν,

d dt

(W₂(µ_t, ν)² 2

)

+ K

2 W₂(µ_t, ν)²

+ Ent(µ_t) ≤ Ent(ν)

F (iii) ⇒ (v) under a suitable ass.(incl. linearity)

[Ambrosio, Gigli & Savar´e]

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Implications

On non-smooth sp.:

† TFAE [Ambrosio, Gigli & Savar´e]

(v)* Ent is K-convex w.r.t.W₂ & linearity of heat ﬂow (vi) ^∀µ₀, ^∃sol (µ_t)_t_≥₀ to EVI_K of Ent: for ^∀ν,

d dt

(W₂(µ_t, ν)² 2

)

+ K

2 W₂(µ_t, ν)²

+ Ent(µ_t) ≤ Ent(ν) F (iii) ⇒ (v) under a suitable ass.(incl. linearity)

[Ambrosio, Gigli & Savar´e]

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Summary of implications (when the heat ﬂow is linear) (vi) EVI

(ii) Wass.

(v) Ent

(iii) ∇-est. (iv) Γ₂

%9

ey

ey %9 EY ey %9

◦ ◦

What we did for Ric ≥ K & dim ≤ N:

Formulate a missing condition corresponding to (ii) Extension of the implication (ii) ⇔ (iii)

(even in an abstract setting) Another approach based on a coupling method

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Summary of implications (when the heat ﬂow is linear) (vi) EVI

(ii) Wass.

(v) Ent

(iii) ∇-est. (iv) Γ₂

%9

ey

ey %9 EY ey %9

◦ ◦

What we did for Ric ≥ K & dim ≤ N:

Formulate a missing condition corresponding to (ii) Extension of the implication (ii) ⇔ (iii)

(even in an abstract setting) Another approach based on a coupling method

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1. Introduction

4.2 Questions

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Known conditions

(i)

’

Ric ≥ K

& dimM ≤ N

⇔

(iv)

’

1

2∆(|∇f|²) − h∇f,∇∆fi ≥ K|∇f|²

+ 1

N(∆f)²

⇔

[F.-Y. Wang ’11]

(iii)

’

|∇Ptf|² ≤ e⁻^2KtPt(|∇f|²)

−1 − e⁻^2Kt

NK (∆P_tf)² (i)’⇔ (v)’: CDCDCD(K, N) [Sturm ’06 / Lott & Villani ’09]

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Known conditions

(i)’ Ric ≥ K & dimM ≤ N

⇔

(iv)

’

1

2∆(|∇f|²) − h∇f,∇∆fi ≥ K|∇f|²

+ 1

N(∆f)²

⇔

[F.-Y. Wang ’11]

(iii)

’

|∇Ptf|² ≤ e⁻^2KtPt(|∇f|²)

−1 − e⁻^2Kt

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Known conditions

(i)’ Ric ≥ K & dimM ≤ N

(iv)’ 1

2∆(|∇f|²) − h∇f,∇∆fi ≥ K|∇f|² + 1

N(∆f)²

⇔

[F.-Y. Wang ’11]

(iii)’ |∇Ptf|² ≤ e⁻^2KtPt(|∇f|²)

−1 − e⁻^2Kt

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Known conditions

(i)’ Ric ≥ K & dimM ≤ N

(iv)’ 1

2∆(|∇f|²) − h∇f,∇∆fi ≥ K|∇f|² + 1

N(∆f)²

⇔ [F.-Y. Wang ’11]

(iii)’ |∇Ptf|² ≤ e⁻^2KtPt(|∇f|²)

−1 − e⁻^2Kt

NK (∆P_tf)²

(i)’⇔ (v)’: CDCDCD(K, N) [Sturm ’06 / Lott & Villani ’09]

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Known conditions

(i)’ Ric ≥ K & dimM ≤ N

(iv)’ 1

2∆(|∇f|²) − h∇f,∇∆fi ≥ K|∇f|² + 1

N(∆f)²

⇔ [F.-Y. Wang ’11]

(iii)’ |∇Ptf|² ≤ e⁻^2KtPt(|∇f|²)

−1 − e⁻^2Kt

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.

Theorem 1 ([K.])

.

.. .

.

For K ∈RRR and N ∈ [2,∞),

(iii)’ is equivalent to the following (ii)’:

(ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ (

−

∫ t s

e^Krξ(dr) )−2

W₂(µ₀, µ₁)²+ N

2 ξ([s,t])² where ξ(dr) =

( 2K 1 −e⁻^2Kr

)₋1/2

dr

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The case K = 0

.

Corollary 2 ([K.])

.

.. .

.

For N ∈ [2,∞), TFAE:

(i)’ Ric ≥ 0 & dimM ≤ N (ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ W₂(µ₀, µ₁)² + 2N(√

t −√ s)²

(iii)’ |∇P_tf|² ≤ P_t(|∇f|²) − 2

N(∆P_tf)² Remark

[Bakry, Gentil & Ledoux]: (iv)’ ⇒ (iii)” ⇒ (ii)’

((iii)” an intertwining ineq. for P_t and Hopf-Lax semigr.)

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The case K = 0

(ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ W₂(µ₀, µ₁)² + 2N(√

t −√ s)²

⇓ µ₀ = δ_x₀, µ₁ = δ_x₁, s = 0

P_t(d(x₀,·)²)(x₁) ≤ d(x₀, x₁)² + 2N t

⇒ ∆(d(x₀,·)²)(x₁) ≤ 2N

(sharp Laplacian comparison)

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The case K = 0

(ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ W₂(µ₀, µ₁)² + 2N(√

t −√ s)²

⇓ µ₀ = δ_x₀, µ₁ = δ_x₁, s = 0

P_t(d(x₀,·)²)(x₁) ≤ d(x₀, x₁)² + 2N t

⇒ ∆(d(x₀,·)²)(x₁) ≤ 2N

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The case K = 0

(ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ W₂(µ₀, µ₁)² + 2N(√

t −√ s)²

⇓ µ₀ = δ_x₀, µ₁ = δ_x₁, s = 0

P_t(d(x₀,·)²)(x₁) ≤ d(x₀, x₁)² + 2N t

⇒ ∆(d(x₀,·)²)(x₁) ≤ 2N

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The case K = 0

(ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ W₂(µ₀, µ₁)² + 2N(√

t −√ s)²

⇓ µ₀ = δ_x₀, µ₁ = δ_x₁, s = 0

P_t(d(x₀,·)²)(x₁) ≤ d(x₀, x₁)² + 2N t

⇒ ∆(d(x₀,·)²)(x₁) ≤ 2N

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1. Introduction

4.2 Questions

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Idea of the proof

(ii)’ ⇒ (iii)’: Diﬀerentiation (iii)’ ⇒ (ii)’: Kantorovich duality

& analysis of the Hopf-Lax semigroup (cf. [K. ’10 / K.] when N = ∞)

⇒ Extension to more general setting

(i)’ ⇒ (ii)’: Coupling of Brownian motions with diﬀerent speeds

⇒ (possibly) sharper estimate

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Idea of the proof

⇒ Extension to more general setting

(i)’ ⇒ (ii)’: Coupling of Brownian motions with diﬀerent speeds

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Idea of the proof

⇒ Extension to more general setting (i)’ ⇒ (ii)’: Coupling of Brownian motions with

diﬀerent speeds

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Idea of the proof

diﬀerent speeds

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Idea of the proof

diﬀerent speeds

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1. Introduction

4.2 Questions

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Sketch of proof: (ii)’ ⇒ (iii)’

(ii)’ W2(P_s^∗µ0, P_t^∗µ1)²

≤ (

−

∫ t s

e^Krξ(dr) )−2

W₂(µ₀, µ₁)²+ N

2 ξ([s,t])² (iii)’ 2Ψ(t)

N (∆P_tf)² +|∇P_tf|² ≤ e⁻^2KtP_t(|∇f|²) where ξ(dr) =

( 2K 1 −e⁻^2Kr

)₋1/2

dr =: dr

√Ψ(r)

◦ ◦

For π: coupling of P_t^∗δx and P_s^∗δy, P_tf(x) − P_sf(y) =

∫

(f(z) −f(w))π(dzdw) Take t− s = a d(x, y) for “suitable” a ∈ RRR:

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Sketch of proof: (ii)’ ⇒ (iii)’

(iii)’ 2Ψ(t)

N (∆Ptf)² +|∇Ptf|² ≤ e⁻^2KtPt(|∇f|²)

◦ ◦

For π: coupling of P_t^∗δ_x and P_s^∗δ_y, P_tf(x) − P_sf(y) =

∫

(f(z) −f(w))π(dzdw)

Take t− s = a d(x, y) for “suitable” a ∈ RRR:

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Sketch of proof: (ii)’ ⇒ (iii)’

(iii)’ 2Ψ(t)

N (∆Ptf)² +|∇Ptf|² ≤ e⁻^2KtPt(|∇f|²)

◦ ◦

∫

(f(z) −f(w))π(dzdw) Take t− s = a d(x, y) for “suitable” a ∈RRR:

⇒ (LHS)

d(x, y) → a∆P_tf(x) +|∇P_tf|(x) as s → t

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Sketch of proof: (ii)’ ⇒ (iii)’

(iii)’ 2Ψ(t)

N (∆Ptf)² +|∇Ptf|² ≤ e⁻^2KtPt(|∇f|²)

◦ ◦

∫

(f(z) −f(w))π(dzdw) Take t− s = a d(x, y) for “suitable” a ∈RRR:

⇒ (RHS) =

∫ f(z) − f(w)

d(z, w) d(z, w)π(dzdw)

”≤” P_t(|∇f|²)(x)^1/2W₂(P_t^∗δ_x, P_s^∗δ_y) · · ·

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Sketch of proof: (iii)’ ⇒ (ii)’

Ingredients

Kantorovich duality:

W₂(ν, µ)²

2 = sup

f

[∫

Q₁f dµ −

∫

f dν ]

Hopf-Lax semigroup:

Q_rf(x) := inf

y∈M

[

f(y) + d(x, y)² 2r

]

? ∂rQrf = −1

2|∇Qrf|² (Hamilton-Jacobi eq.)

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Sketch of proof: (iii)’ ⇒ (ii)’

(ii)’ W2(P_s^∗µ0, P_t^∗µ1)²

≤ (

−

∫ t s

e^Krξ(dr) )−2

W₂(µ₀, µ₁)²+ N

2 ξ([s,t])² (iii)’ 2Ψ(t)

N (∆P_tf)² +|∇P_tf|² ≤ e⁻^2KtP_t(|∇f|²) where ξ(dr) =

( 2K 1 −e⁻^2Kr

)₋1/2

dr =: dr

√Ψ(r)

◦ ◦

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Sketch of proof: (iii)’ ⇒ (ii)’

(ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ (

−

∫ t s

e^Krξ(dr) )−2

W₂(µ₀, µ₁)²+ N

2 ξ([s,t])²

◦ ◦

For simplicity, µ₀ = δ_x₀, µ₁ = δ_x₁ W2(P_s^∗δx₀, P_t^∗δx₁)²

2 = sup

f

[PtQ1f(x1) − Psf(x0)]

Idea: give an upper bound of [· · ·] being uniform in f

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Sketch of proof: (iii)’ ⇒ (ii)’

(ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ (

−

∫ t s

e^Krξ(dr) )−2

W₂(µ₀, µ₁)²+ N

2 ξ([s,t])²

◦ ◦

γ : [0,1] → M: geod. joining x₀ & x₁

α : [0,1] → [s, t], η : [0,1] → [0,1]: %, surj.

(suitably chosen)

⇒ PtQ1f(x1) − Psf(x0)

= P_α(1)Q₁f(γ(η(1))) − P_α(0)Q₀f(γ(η(0)))

=

∫ 1 0

∂_rP_α(r)Q_rf(γ(η(r)))dr

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Sketch of proof: (iii)’ ⇒ (ii)’

(ii)’ W₂(P_s^∗µ₀, P_t^∗µ₁)²

≤ (

−

∫ t s

e^Krξ(dr) )−2

W₂(µ₀, µ₁)²+ N

2 ξ([s,t])²

◦ ◦

∂_rP_α(r)Q_rf(γ(η(r)))

≤ α⁰(r)∆P_α(r)Q_rf(γ(η(r)))

− 1

2P_α(r)(|∇Q_rf|²)(γ(η(r))) + η⁰(r)|∇P_α(r)Qrf|(γ(η(r)))

≤ · · ·

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Remarks

(ii) / (ii)’

Wass. contr.

diﬀerentiation

⇒

⇐

integration

(iii) / (iii)’

gradient est.

If an est. like (ii)’ is “inﬁnitesimally sharp”, then it implies (iii)’

⇒ An weaker est. than (ii)’ can be equiv. to (iii)’

⇒ Self-improvements in Wass. contr.’s

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Extended duality

.

Theorem 3 ([K.])

.

.. .

.

M: Polish geod. sp., P_t = e^t^L: Feller or str. Feller Then for a, b : [0,∞) → (0,∞), TFAE:

(A) W_p(P_s^∗µ₀, P_t^∗µ₁)²

≤ (

−

∫ t s

ξ(dr)

√a(r) )₋2

W_p(µ₀, µ₁)² +ξ([s, t])²

(B) |∇P_tf|² ≤ a(t) [

P_t(|∇f|^q)^2/q + b(t)(LP_tf)² ]

where p⁻¹ + q⁻¹ = 1, ξ(dr) := b(r)⁻^1/2dr,

|∇f|(x): loc. Lip. const. of f at x

24 / 26

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1. Introduction

4.2 Questions