On the speed in Wasserstein distances of heat distributions

On the speed in Wasserstein distances of heat distributions

Kazumasa Kuwada

(Tokyo Institute of Technology)

7th International Conference on Stochastic Analysis and its Applications (Seoul National University) Aug. 6–11, 2014

1. Introduction

Speed in transportation cost

(X_t)_t≥0: stochastic process on M

⇓ ⇑

(µ_t)_t≥0: curve in P(M), µ_t := PPP◦ X_t⁻¹

Q. T_c(µ_t, µ_s) ≈ ? (s → t) T_c(µ, ν) := inf

(Z

M×M

c dπ

π: coupling of µ and ν

)

(Optimal transportation cost for a cost function c)

Speed in transportation cost

(X_t)_t≥0: stochastic process on M

⇓ ⇑

(µ_t)_t≥0: curve in P(M), µ_t := PPP◦ X_t⁻¹ Q. T_c(µ_t, µ_s) ≈ ? (s → t)

T_c(µ, ν) := inf (Z

M×M

c dπ

π: coupling of µ and ν

)

(Optimal transportation cost for a cost function c)

Background

µt = e^t∆µ0: heat dist. on a met. meas. sp. (M, d, v)

⇒ “∂tµt = −∇Entv(µt)” w.r.t. W2 = (T_d²)^1/2 [Jordan, Kinderlehrer & Otto ’98]

[Ambrosio, Gigli & Savar´e ’05, . . .]

“∂_tEnt_v(µ_t) = −h∇Ent_v, ∂_tµ_ti = −|∂_tµ_t|²”

⇓ lims↓t

W2(µs, µt) s − t

2

= Z

M

|∇ρ_t|²

ρ_t dv =: I(µ_t) (Fisher information)

Background

µt = e^t∆µ0: heat dist. on a met. meas. sp. (M, d, v)

⇒ “∂tµt = −∇Entv(µt)” w.r.t. W2 = (T_d²)^1/2 [Jordan, Kinderlehrer & Otto ’98]

[Ambrosio, Gigli & Savar´e ’05, . . .]

“∂_t Ent_v(µ_t) = −h∇Ent_v, ∂_tµ_ti = −|∂_tµ_t|²”

⇓ lims↓t

W₂(µ_s, µ_t) s − t

²

= Z

M

|∇ρ_t|²

ρ_t dv =: I(µ_t) (Fisher information)

Background

µt = e^t∆µ0: heat dist. on a met. meas. sp. (M, d, v)

⇒ “∂tµt = −∇Entv(µt)” w.r.t. W2 = (T_d²)^1/2 [Jordan, Kinderlehrer & Otto ’98]

[Ambrosio, Gigli & Savar´e ’05, . . .]

“∂_t Ent_v(µ_t) = −h∇Ent_v, ∂_tµ_ti = −|∂_tµ_t|²”

⇓ lims↓t

W₂(µ_s, µ_t) s − t

²

= Z

M

|∇ρ_t|²

ρ_t dv =: I(µ_t) (Fisher information)

Background

“Hess Ent_v ≥ K” (⇔ “Ric ≥ K”) for K ∈ RRR

⇓

W₂(µ⁽⁰⁾_t , µ⁽¹⁾_t ) ≤ e^−KtW₂(µ⁽⁰⁾₀ , µ⁽¹⁾₀ )

⇓

W₂(µ_t, µ_t+s) ≤ e^−KtW₂(µ₀, µ_s)

⇓

I(µ_t) ≤ e^−2KtI(µ₀)

(⇒ log Sobolev ineq. (when K > 0))

Background

“Hess Ent_v ≥ K” (⇔ “Ric ≥ K”) for K ∈ RRR

⇓

W₂(µ⁽⁰⁾_t , µ⁽¹⁾_t ) ≤ e^−KtW₂(µ⁽⁰⁾₀ , µ⁽¹⁾₀ )

⇓

W₂(µ_t, µ_t+s) ≤ e^−KtW₂(µ₀, µ_s)

⇓

I(µt) ≤ e^−2KtI(µ0)

(⇒ log Sobolev ineq. (when K > 0))

Questions

What happens for other transportation costs?

What happens when there is no gradient flow structure?

Outline of the talk 1. Introduction

2. Heat distributions on backward Ricci flow

3. Idea of the proof

4. Further problems

Outline of the talk

1. Introduction

2. Heat distributions on backward Ricci flow

3. Idea of the proof

4. Further problems

Framework

(M, g(t)): cpl. Riem. mfds., t ∈ [0, T]

∂_tg(t) = 2 Ric_t (backward Ricci flow)

((X(t))_t≥0,(PPP^x)_x∈M): g(t)-Brownian motion

!!! ∆_g(t): generator µt = PPP^µ0 ◦X(t)⁻¹: heat dist.

v_t: g(t)-volume meas., µ_t = ρ_tv_t

F ∂tvt = Rtvt (Rt: g(t)-scalar curv.) Ass. sup

t

|Rm_t|_g(t) < ∞ (Rm_t: g(t)-curv. tensor)

∂

_t

µ

_t

6= −∇ Ent

_vt

(µ

_t

)

Ent_vt(µ_t) :=

Z

M

ρ_tlogρ_tdv_t = Z

M

logρ_tdµ_t

F ∂tµt = ∆tµt (weakly)

⇒ ∂tEntv_t(µt) = − Z

M

|∇ρ_t|²

ρ²_t + Rt

dµt

=: −F(µ_t) (F-functional)

⇒ No monotonicity of Ent_vt(µ_t)!

W

₂

-contraction

Observation

When g(t) ≡ g₀, ∂_tg(t) = 2 Ric_t ⇒ Ric ≡ 0

F T_d²

t(µ⁽⁰⁾_t , µ⁽¹⁾_t ) & [McCann & Topping ’10], [Arnaudon, Coulibaly & Thalmaier ’09], [K. ’12]

W

₂

-contraction

Observation

When g(t) ≡ g₀, ∂_tg(t) = 2 Ric_t ⇒ Ric ≡ 0

F T_d²

t(µ⁽⁰⁾_t , µ⁽¹⁾_t ) & [McCann & Topping ’10], [Arnaudon, Coulibaly & Thalmaier ’09], [K. ’12]

W

₂

-contraction

Observation

When g(t) ≡ g₀, ∂_tg(t) = 2 Ric_t ⇒ Ric ≡ 0

F T_d²

t(µ⁽⁰⁾_t , µ⁽¹⁾_t ) & [McCann & Topping ’10], [Arnaudon, Coulibaly & Thalmaier ’09], [K. ’12]

⇒/ T_d²

t(µ_t, µ_t+s) & (time-inhomogeneity)

W

₂

-contraction

Observation

When g(t) ≡ g₀, ∂_tg(t) = 2 Ric_t ⇒ Ric ≡ 0

F T_d²

t(µ⁽⁰⁾_t , µ⁽¹⁾_t ) & [McCann & Topping ’10], [Arnaudon, Coulibaly & Thalmaier ’09], [K. ’12]

F T_L^t,t+s

0 (µ_t, µ_t+s) & in t [Lott ’09], [Amaba & K.]





L^t,t₀ ⁰(x, y) := inf

γ(t)=x, γ(t⁰)=y

"

Z t⁰ t

|γ(r)|˙ ²_r + R_r dr

#







(L₀-distance; introduced by Lott)

Monotonicity of F

Theorem 1

Suppose Entv₀(µ0) < ∞ and F(µ₀) < ∞

⇒ lim

s↓0

T_L^t,t+s

0 (µ_t, µ_t+s)

s = F(µ_t) a.e. t ∈ [0, T] Corollary 2

F(µ_t) &

Rem: g(t) ≡ g, Ric ≥ 0 ⇒ I(µ_t) &

[Lott ’09] when M: cpt.

by Eulerian calculus (requires smoothness)

1. Introduction

2. Heat distributions on backward Ricci flow

3. Idea of the proof

4. Further problems

Kantorovich duality

Observation: W₂ in the case g(t) ≡ g

W₂(µ_t, µ_t+s)²

2s = sup

ϕ∈Cb

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

Q_sϕ(x) := inf

y∈M

ϕ(y) + d(y, x)² 2s

= inf

γ(t)=y γ(t+s)=x

ϕ(γ(t)) + 1 2

Z t+s t

|γ(r˙ )|²dr

(Hopf-Lax semigroup) F ∂_sQ_sϕ+ 1

2|∇Q_sϕ|² = 0 (Hamilton-Jacobi eq.)

Kantorovich duality

Observation: W₂ in the case g(t) ≡ g

W₂(µ_t, µ_t+s)²

2s = sup

ϕ∈Cb

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

Q_sϕ(x) := inf

y∈M

ϕ(y) + d(y, x)² 2s

= inf

γ(t)=y γ(t+s)=x

ϕ(γ(t)) + 1 2

Z t+s

t

|γ(r˙ )|²dr

(Hopf-Lax semigroup)

F ∂_sQ_sϕ+ 1

2|∇Q_sϕ|² = 0 (Hamilton-Jacobi eq.)

Kantorovich duality

Observation: W₂ in the case g(t) ≡ g

W₂(µ_t, µ_t+s)²

2s = sup

ϕ∈Cb

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

Q_sϕ(x) := inf

y∈M

ϕ(y) + d(y, x)² 2s

= inf

γ(t)=y γ(t+s)=x

ϕ(γ(t)) + 1 2

Z t+s t

|γ(r˙ )|²dr

(Hopf-Lax semigroup) F ∂_sQ_sϕ + 1

2|∇Q_sϕ|² = 0 (Hamilton-Jacobi eq.)

Upper bound

W2(µt, µt+s)²

2s = sup

ϕ∈C_b

Z

M

Qsϕ dµt+s − Z

M

ϕ dµt

[· · ·] = Z s

0

∂_r Z

M

Q_rϕ dµ_t+r

dr

∂_r(···) = Z

−h∇Q_rϕ, ∇ρ_t+r

ρ_t+r i − 1

2|∇Q_rϕ|²

dµ_t+r

≤ 1

2I(µ_t+r)

lims↓t

W₂(µ_t, µ_t+s)²

s² ≤ lim

s↓t

1 s

Z t+s t

I(µ_r)dr = I(µ_t)

Upper bound

W2(µt, µt+s)²

2s = sup

ϕ∈C_b

Z

M

Qsϕ dµt+s − Z

M

ϕ dµt

[· · ·] = Z s

0

∂_r Z

M

Q_rϕ dµ_t+r

dr

∂_r(···) = Z

−h∇Q_rϕ, ∇ρ_t+r

ρ_t+r i − 1

2|∇Q_rϕ|²

dµ_t+r

≤ 1

2I(µ_t+r)

lims↓t

W₂(µ_t, µ_t+s)²

s² ≤ lim

s↓t

1 s

Z t+s t

I(µ_r)dr = I(µ_t)

Lower bound

W₂(µ_t, µ_t+s)²

2s = sup

ϕ∈Cb

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

lim

s↓t

1 s

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

“=”

Z

−h∇ϕ, ∇ρ_t

ρ_t i − 1

2|∇ϕ|²

dµ_t

⇓ ϕ = −logρ_t

lim

s↓t

W2(µt, µt+s)²

s² ≥ I(µt)

Lower bound

W₂(µ_t, µ_t+s)²

2s = sup

ϕ∈Cb

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

lim

s↓t

1 s

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

“=”

Z

−h∇ϕ, ∇ρ_t

ρ_t i − 1

2|∇ϕ|²

dµ_t

⇓ ϕ = −logρ_t

lim

s↓t

W2(µt, µt+s)²

s² ≥ I(µt)

Lower bound

W₂(µ_t, µ_t+s)²

2s = sup

ϕ∈Cb

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

lim

s↓t

1 s

Z

M

Q_sϕ dµ_t+s − Z

M

ϕ dµ_t

“=”

Z

−h∇ϕ, ∇ρ_t

ρ_t i − 1

2|∇ϕ|²

dµ_t

⇓ ϕ = −logρ_t

lim

s↓t

W2(µt, µt+s)²

s² ≥ I(µt)

Outline of the proof of Thm1

Kantorovich duality / Hopf-Lax semigr. for L^t,t+s₀ Difficulty: Dependency on t of geometry

No a priori integrability of ∇ρ_t & ∆_tρ_t

⇒ Approximation

† Use ^∃∃∃ of heat kernel & Gaussian (upper) bound

† Show Ent_vs(µ_s) −Ent_vt(µ_t) ≥ Z t

s

F(µ_r)dr (⇒ F(µ_r) < ∞ a.e. r)

† Show lim

s↓t

Entv_s(µs) − Entv_t(µt)

s − t ≤ F(µt)

Outline of the proof of Thm1

Kantorovich duality / Hopf-Lax semigr. for L^t,t+s₀ Difficulty: Dependency on t of geometry

No a priori integrability of ∇ρ_t & ∆_tρ_t

⇒ Approximation

† Use ^∃∃∃ of heat kernel & Gaussian (upper) bound

† Show Ent_vs(µ_s) −Ent_vt(µ_t) ≥ Z t

s

F(µ_r)dr (⇒ F(µ_r) < ∞ a.e. r)

† Show lim

s↓t

Entv_s(µs) − Entv_t(µt)

s − t ≤ F(µt)

Outline of the proof of Thm1

Kantorovich duality / Hopf-Lax semigr. for L^t,t+s₀ Difficulty: Dependency on t of geometry

No a priori integrability of ∇ρ_t & ∆_tρ_t

⇒ Approximation

† Use ^∃∃∃ of heat kernel & Gaussian (upper) bound

† Show Ent_vs(µ_s) −Ent_vt(µ_t) ≥ Z t

s

F(µ_r)dr (⇒ F(µ_r) < ∞ a.e. r)

† Show lim

s↓t

Entv_s(µs) − Entv_t(µt)

s − t ≤ F(µt)

Outline of the proof of Thm1

Kantorovich duality / Hopf-Lax semigr. for L^t,t+s₀ Difficulty: Dependency on t of geometry

No a priori integrability of ∇ρ_t & ∆_tρ_t

⇒ Approximation

† Use ^∃∃∃ of heat kernel & Gaussian (upper) bound

† Show Ent_vs(µ_s) −Ent_vt(µ_t) ≥ Z t

s

F(µ_r)dr (⇒ F(µ_r) < ∞ a.e. r)

† Show lim

s↓t

Entv_s(µs) − Entv_t(µt)

s − t ≤ F(µt)

Outline of the proof of Thm1

Kantorovich duality / Hopf-Lax semigr. for L^t,t+s₀ Difficulty: Dependency on t of geometry

No a priori integrability of ∇ρ_t & ∆_tρ_t

⇒ Approximation

† Use ^∃∃∃ of heat kernel & Gaussian (upper) bound

† Show Ent_vs(µ_s) −Ent_vt(µ_t) ≥ Z t

s

F(µ_r)dr (⇒ F(µ_r) < ∞ a.e. r)

† Show lim

s↓t

Entv_s(µs) − Entv_t(µt)

s − t ≤ F(µt)

Outline of the proof of Thm1

Kantorovich duality / Hopf-Lax semigr. for L^t,t+s₀ Difficulty: Dependency on t of geometry

No a priori integrability of ∇ρ_t & ∆_tρ_t

⇒ Approximation

† Use ^∃∃∃ of heat kernel & Gaussian (upper) bound

† Show Ent_vs(µ_s) −Ent_vt(µ_t) ≥ Z t

s

F(µ_r)dr

(⇒ F(µ_r) < ∞ a.e. r)

† Show lim

s↓t

Entv_s(µs) − Entv_t(µt)

s − t ≤ F(µt)

Outline of the proof of Thm1

Kantorovich duality / Hopf-Lax semigr. for L^t,t+s₀ Difficulty: Dependency on t of geometry

No a priori integrability of ∇ρ_t & ∆_tρ_t

⇒ Approximation

† Use ^∃∃∃ of heat kernel & Gaussian (upper) bound

† Show Ent_vs(µ_s) −Ent_vt(µ_t) ≥ Z t

s

F(µ_r)dr (⇒ F(µ_r) < ∞ a.e. r)

† Show lim

s↓t

Entv_s(µs) − Entv_t(µt)

s − t ≤ F(µt)

Outline of the proof of Thm1

Kantorovich duality / Hopf-Lax semigr. for L^t,t+s₀ Difficulty: Dependency on t of geometry

No a priori integrability of ∇ρ_t & ∆_tρ_t

⇒ Approximation

† Use ^∃∃∃ of heat kernel & Gaussian (upper) bound

† Show Ent_vs(µ_s) −Ent_vt(µ_t) ≥ Z t

s

F(µ_r)dr (⇒ F(µ_r) < ∞ a.e. r)

† Show lim

s↓t

Entv_s(µs) − Entv_t(µt)

s − t ≤ F(µ_t)

1. Introduction

2. Heat distributions on backward Ricci flow

3. Idea of the proof

4. Further problems

T

_L

/W

_p

For Perelman’s L-distance? (backward Ricci flow)

† A monotonicity of transportation cost is known [Topping ’09], [K. & Philipowski ’11]

⇒ Monotonicity of W-functional

([Topping ’09]; M: cpt.)

T

_L

/W

_p

For Perelman’s L-distance? (backward Ricci flow)

† A monotonicity of transportation cost is known [Topping ’09], [K. & Philipowski ’11]

⇒ Monotonicity of W-functional

([Topping ’09]; M: cpt.)

T

_L

/W

_p

For Perelman’s L-distance? (backward Ricci flow)

† A monotonicity of transportation cost is known [Topping ’09], [K. & Philipowski ’11]

⇒ Monotonicity of W-functional

([Topping ’09]; M: cpt.)

T

_L

/W

_p

For Wp (p ∈ [1,∞)) in time-homogeneous case?

† Ric ≥ K ⇒ e^KtW_p(µ⁽⁰⁾_t , µ⁽¹⁾_t ) &

F lim

s↓0

W_p(µ_t, µ_t+s) s

^p

≤ Z

M

|∇ρ_t|^p ρ^p−1_t dv

⇓

Ric ≥ K > 0, v ∈ P(M)

⇒ W_p(µ, v)^p ≤ 1 K^p

Z

M

|∇ρ|^p ρ^p−1 dv

T

_L

/W

_p

For Wp (p ∈ [1,∞)) in time-homogeneous case?

† Ric ≥ K ⇒ e^KtW_p(µ⁽⁰⁾_t , µ⁽¹⁾_t ) &

F lim

s↓0

W_p(µ_t, µ_t+s) s

^p

≤ Z

M

|∇ρ_t|^p ρ^p−1_t dv

⇓

Ric ≥ K > 0, v ∈ P(M)

⇒ W_p(µ, v)^p ≤ 1 K^p

Z

M

|∇ρ|^p ρ^p−1 dv

T

_L

/W

_p

For Wp (p ∈ [1,∞)) in time-homogeneous case?

† Ric ≥ K ⇒ e^KtW_p(µ⁽⁰⁾_t , µ⁽¹⁾_t ) &

F lim

s↓0

W_p(µ_t, µ_t+s) s

^p

≤ Z

M

|∇ρ_t|^p ρ^p−1_t dv

⇓

Ric ≥ K > 0, v ∈ P(M)

⇒ W_p(µ, v)^p ≤ 1 K^p

Z

M

|∇ρ|^p ρ^p−1 dv