On the speed in transportation costs of heat distributions

On the speed in transportation costs of heat distributions

Kazumasa Kuwada

(Tokyo Institute of Technology)

UK-Japan Stochastic Analysis School The University of Warwick, Sept. 1–5, 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(γ(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)

L-transportation cost

L^t,t0(x, y) := inf

γ(t)=x, γ(t⁰)=y

"

Z t⁰

t

√r |γ(r)|˙ ²_r + R_r dr

#

(L-distance; introduced by Perelman) F For τ₁ < τ₂ fixed,

Ξ_τ1,τ2(t) := √

τ₂t− √ τ₁t

T_L^τ1t,τ2t(µ_τ1t, µ_τ2t)

−m(√

τ₂t −√ τ₁t)²

⇒ Ξτ₁,τ₂(t) & [Topping ’09], [K. & Philipowski ’11]

Monotonicity of W -entropy

Theorem 3

Suppose Ent_v0(µ₀) < ∞ and F(µ₀) < ∞

⇒ lim

s↓0

T_L^t,t+s(µ_t, µ_t+s)

s = √

tF(µ_t) a.e. t ∈ (0, T]

Corollary 4

t²F(µ_t) − mt

2 &. In particular, W(µ_t) &

W(t) := tF(µ_t) − Ent(µ_t) −m

1 + log(4πt) 2

[Topping ’09] when M: cpt. by optimal transport

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

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

The solution to the p_∗-heat equation

∂_tu = div(|∇u|^p∗−2∇u)

seems to fit better with Wp. (work in progress)

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

The solution to the p_∗-heat equation

∂_tu = div(|∇u|^p∗−2∇u)

seems to fit better with Wp. (work in progress)

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

The solution to the p_∗-heat equation

∂_tu = div(|∇u|^p∗−2∇u)

seems to fit better with Wp. (work in progress)

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 The solution to the p_∗-heat equation

∂_tu = div(|∇u|^p∗−2∇u)

seems to fit better with Wp. (work in progress)