PDF Wasserstein controls for heat distributions

Wasserstein controls for heat distributions

Kazumasa Kuwada

(Tokyo Institute of Technology)

Stochastic Processes, Analysis and Mathematical Physics In honor of Professor Masatoshi Fukushima’s Sanju

(Kansai University) Aug. 25-29, 2014

1. Introduction & Overview

What’s Wasserstein control?

µ_t = e^tLµ₀: law at time t of diffusion process

on met. meas. sp. (M, d, v) generated by L W_p(µ, ν) = inf{kdk_L^p_(π) | π: coupling of µ and ν}

(L^p-Wasserstein distance (p ∈ [1,∞])) Wasserstein control

A (Lipschitz type) upper bound of W_p(µ_t, µ⁰_t) by W_p(µ₀, µ⁰₀)

and “difference” between t & s

(µ_t = e^tLµ₀, µ⁰_t = e^tLµ⁰₀)

What’s Wasserstein control?

µ_t = e^tLµ₀: law at time t of diffusion process

on met. meas. sp. (M, d, v) generated by L W_p(µ, ν) = inf{kdk_L^p_(π) | π: coupling of µ and ν}

(L^p-Wasserstein distance (p ∈ [1,∞])) Space-time Wasserstein control

A (Lipschitz type) upper bound of W_p(µ_s, µ⁰_t) by W_p(µ₀, µ⁰₀) and “difference” between t & s

(µ_t = e^tLµ₀, µ⁰_t = e^tLµ⁰₀)

3 / 19

Coupling by parallel transport

On a Riem. mfd M:

d dtd

(^∃∃∃X_t,^∃∃∃X_t⁰ ): coupling of BMs by parallel transp.

⇓

Ric ≥ K ⇒ Wp(µt, µ⁰_t) ≤ e^−KtWp(µ0, µ⁰₀)

(^∀∀∀p ∈ [1,∞]) Similarly, on RRR^m, dXt = dWt +∇V(Xt)dt,

HessV ≥ K ⇒ the same bound

Coupling by parallel transport

On a Riem. mfd M: d

dtd(

∃∃∃

X_t,

∃

∃∃

X_t⁰ ) ≤ − Z X_t⁰

X_t

Ric

⇓

Ric ≥ K ⇒ W_p(µ_t, µ⁰_t) ≤ e^−KtW_p(µ₀, µ⁰₀)

(^∀∀∀p ∈ [1,∞]) Similarly, on RRR^m, dX_t = dW_t +∇V(X_t)dt,

HessV ≥ K ⇒ the same bound

4 / 19

Duality with gradient estimate

W_p(P_t^∗µ, P_t^∗µ⁰) ≤ CW_p(µ, µ⁰)

⇔ (^∀∀∀p ∈ [1,∞])

|∇P_tf| ≤ CP_t(|∇f|^p∗)^1/p∗

[K. ’10,’13 / Bakry, Gentil & Ledoux]

Ric ≥ K on a Riem. mfd

⇔ 1

2∆|∇f|² − h∇f,∇∆fi ≥ K|∇f|²

⇔ |∇P_tf| ≤ e^−KtP_t(|∇f|^p∗)^1/p∗

⇔ W_p(P_t^∗µ, P_t^∗µ⁰) ≤ e^−KtW_p(µ, µ⁰)

Duality with gradient estimate

W_p(P_t^∗µ, P_t^∗µ⁰) ≤ CW_p(µ, µ⁰)

⇔ (^∀∀∀p ∈ [1,∞])

|∇P_tf| ≤ CP_t(|∇f|^p∗)^1/p∗

[K. ’10,’13 / Bakry, Gentil & Ledoux]

Ric ≥ K on a Riem. mfd

⇔ 1

2∆|∇f|² − h∇f,∇∆fi ≥ K|∇f|²

⇔ |∇P_tf| ≤ e^−KtP_t(|∇f|^p∗)^1/p∗

⇔ W_p(P_t^∗µ, P_t^∗µ⁰) ≤ e^−KtW_p(µ, µ⁰)

[Bakry & ´Emery ’84 / von Renesse & Sturm ’05]

5 / 19

Duality with gradient estimate

W_p(P_t^∗µ, P_t^∗µ⁰) ≤ CW_p(µ, µ⁰)

⇔ (^∀∀∀p ∈ [1,∞])

|∇P_tf| ≤ CP_t(|∇f|^p∗)^1/p∗

[K. ’10,’13 / Bakry, Gentil & Ledoux]

Ric ≥ K on a Riem. mfd

⇔ 1

2∆|∇f|² − h∇f,∇∆fi ≥ K|∇f|²

⇔ |∇P_tf| ≤ e^−KtP_t(|∇f|^p∗)^1/p∗

⇔ W_p(P_t^∗µ, P_t^∗µ⁰) ≤ e^−KtW_p(µ, µ⁰)

Duality with gradient estimate

W_p(P_t^∗µ, P_t^∗µ⁰) ≤ CW_p(µ, µ⁰)

⇔ (^∀∀∀p ∈ [1,∞])

|∇P_tf| ≤ CP_t(|∇f|^p∗)^1/p∗

[K. ’10,’13 / Bakry, Gentil & Ledoux]

Ric ≥ K on a Riem. mfd

⇔ 1

2∆|∇f|² − h∇f,∇∆fi ≥ K|∇f|²

⇔ |∇P_tf| ≤ e^−KtP_t(|∇f|^p∗)^1/p∗

⇔ W_p(P_t^∗µ, P_t^∗µ⁰) ≤ e^−KtW_p(µ, µ⁰)

[Bakry & ´Emery ’84 / von Renesse & Sturm ’05]

5 / 19

Otto calculus

ν_t = P_t^∗µ₀

⇔ “∂_tν_t = −∇Ent(ν_t)” on (P(M), W₂)

[Jordan, Kinderlehrer & Otto ’98, Otto ’01]

[Ambrosio, Gigli & Savar´e ’05, ...]

F “Hess Ent ≥ K” (w.r.t. W2)

⇒ W2(νt, ν_t⁰) ≤ e^−KtW2(ν0, ν₀⁰)

F “Hess Ent ≥ K” ⇔ Ric ≥ K on a Riem. mfd [von Renesse & Sturm ’05]

Otto calculus

ν_t = P_t^∗µ₀

⇔ “∂_tν_t = −∇Ent(ν_t)” on (P(M), W₂)

[Jordan, Kinderlehrer & Otto ’98, Otto ’01]

[Ambrosio, Gigli & Savar´e ’05, ...]

F “Hess Ent ≥ K” (w.r.t. W2)

⇒ W2(νt, ν_t⁰) ≤ e^−KtW2(ν0, ν₀⁰)

F “Hess Ent ≥ K” ⇔ Ric ≥ K on a Riem. mfd [von Renesse & Sturm ’05]

6 / 19

Further topics

Coupling method

⇒ Heat dist.’s on (backward) Ricci flows

[Arnaudon, Koulibaly & Thalmaier ’09 / K. ’12]

Monotonicity of L/L₀-transportation cost

[K. & Philipowski ’11 / Amaba & K.]

Duality

⇒ W_p-controls for subelliptic diffusions (on Lie gr.) [Melcher ’08 / Eldredge ’10] & [K. ’10]

Further topics

Coupling method

⇒ Heat dist.’s on (backward) Ricci flows

[Arnaudon, Koulibaly & Thalmaier ’09 / K. ’12]

Monotonicity of L/L₀-transportation cost

[K. & Philipowski ’11 / Amaba & K.]

Duality

⇒ W_p-controls for subelliptic diffusions (on Lie gr.) [Melcher ’08 / Eldredge ’10] & [K. ’10]

7 / 19

Further topics

∂_tµ_t = −∇Ent(µ_t) on (P(M), W₂)

† “Hess Ent ≥ K” as a definition of “Ric ≥ K”

[Sturm ’06 / Lott & Villani ’09]

† Unifying “Ric ≥ K”

on Riemannian metric measure sp.’s [Ambrosio, Gigli & Savar´e ’14 / AGS / AGS / Ambrosio, Gigli, Mondino & Rajala]

Further topics

Extenstion to “Ric ≥ K & dim ≤ N” on mm sp.

[Erbar, K. & Sturm]

† “Hess Ent − 1

N∇Ent^⊗2 ≥ K”

† ¹

2∆|∇f|²− h∇f,∇∆fi ≥ K|∇f|² + (∆f)² N

† |∇P_tf|² ≤ e^−2KtP_t(|∇f|²)− 1−e^−2Kt

N K (∆P_tf)²

[Bakry & Ledoux ’06]

† W₂(P_s^∗µ, P_t^∗µ⁰)²

≤ e(−2Kt)∨(−2Ks)

W₂(µ, µ⁰)²+ Z t

s

s

N K e^2Kr −1dr

!²

9 / 19

Further topics

Extenstion to “Ric ≥ K & dim ≤ N” on mm sp.

[Erbar, K. & Sturm]

† “Hess Ent − 1

N∇Ent^⊗2 ≥ K”

† ¹

2∆|∇f|²− h∇f,∇∆fi ≥ K|∇f|² + (∆f)² N

† |∇P_tf|² ≤ e^−2KtP_t(|∇f|²)− 1−e^−2Kt

N K (∆P_tf)²

[Bakry & Ledoux ’06]

† W₂(P_s^∗µ, P_t^∗µ⁰)²

≤ e(−2Kt)∨(−2Ks)

W (µ, µ⁰)²+

Z ts

N K dr

!²

Further topics

Extenstion to “Ric ≥ K & dim ≤ N” on mm sp.

[Erbar, K. & Sturm]

† “Hess Ent − 1

N∇Ent^⊗2 ≥ K”

† ¹

2∆|∇f|²− h∇f,∇∆fi ≥ K|∇f|² + (∆f)² N

† |∇P_tf|² ≤ e^−2KtP_t(|∇f|²)− 1−e^−2Kt

N K (∆P_tf)²

[Bakry & Ledoux ’06]

† W₂(P_s^∗µ, P_t^∗µ⁰)²

≤ e(−2Kt)∨(−2Ks)

W₂(µ, µ⁰)²+ Z t

s

s

N K e^2Kr −1dr

!²

9 / 19

Further topics

Extenstion to “Ric ≥ K & dim ≤ N” on mm sp.

[Erbar, K. & Sturm]

† “Hess Ent − 1

N∇Ent^⊗2 ≥ K”

† ¹

2∆|∇f|²− h∇f,∇∆fi ≥ K|∇f|² + (∆f)² N

† |∇P_tf|² ≤ e^−2KtP_t(|∇f|²)− 1−e^−2Kt

N K (∆P_tf)²

[Bakry & Ledoux ’06]

† W₂(P_s^∗µ, P_t^∗µ⁰)²

≤ e(−2Kt)∨(−2Ks)

W (µ, µ⁰)²+

Z ts

N K dr

!²

Further topics

Extenstion to “Ric ≥ K & dim ≤ N” on mm sp.

[Erbar, K. & Sturm]

† “Hess Ent − 1

N∇Ent^⊗2 ≥ K”

† ¹

2∆|∇f|²− h∇f,∇∆fi ≥ K|∇f|² + (∆f)² N

† |∇P_tf|² ≤ e^−2KtP_t(|∇f|²)− 1−e^−2Kt

N K (∆P_tf)²

[Bakry & Ledoux ’06]

† W₂(P_s^∗µ, P_t^∗µ⁰)²

≤ e(−2Kt)∨(−2Ks)

W₂(µ, µ⁰)²+ Z t

s

s

N K e^2Kr −1dr

!²

9 / 19

1. Introduction & Overview

2. Duality in space-time estimates

3. Wp-space-time control on Riem. mfds

1. Introduction & Overview

2. Duality in space-time estimates

3. Wp-space-time control on Riem. mfds

Framework

(M, d): Polish met. sp., d: geodesic metric P_t(x,·) ∈ P(M): semigroup of Markov kernels

Ass. ^∀∀∀f ∈ C_b^Lip(M), ^∀∀∀x ∈ M, ^∀∀∀t > 0

∃∃∃LP_tf(x) = lim

s→0

P_t+sf(x) −P_tf(x) s

Hopf-Lax semigr. Q_rf(x) := inf

y∈M

f(y) +r

d(x, y) r

^p

(p ∈ (1,∞))

Framework

(M, d): Polish met. sp., d: geodesic metric P_t(x,·) ∈ P(M): semigroup of Markov kernels

Ass. ^∀∀∀f ∈ C_b^Lip(M), ^∀∀∀x ∈ M, ^∀∀∀t > 0

∃∃∃LP_tf(x) = lim

s→0

P_t+sf(x) −P_tf(x) s

Hopf-Lax semigr.

Q_rf(x) := inf

y∈M

f(y) +r

d(x, y) r

^p

(p ∈ (1,∞))

11 / 19

Extended space-time duality

a_t, b_t > 0 conti., p ≥ β ≥ 1, J(du) = du

b_u: loc. finite meas. on [0,∞) Theorem 1

The following are equivalent:

(i) W_p(P_s^∗µ, P_t^∗µ⁰)^β ≤ Z

[s,t]

− a_uJ(du) β

W_p(µ, µ⁰)^β

+J([s, t])^β

+ J([s, t])^β

(ii) |∇P_tf|^β∗≤ a^β_t^∗

P_t(|∇f|^p∗)^β∗/p∗− b^β_t^∗|LP_tf|^β∗ for f = Q f˜ (δ > 0 & f˜∈ C^Lip(M))

Extended space-time duality

a_t, b_t > 0 conti., p ≥ β ≥ 1, J(du) = du

b_u: loc. finite meas. on [0,∞) Theorem 1

The following are equivalent:

(i) W_p(P_s^∗µ, P_t^∗µ⁰)^β ≤ Z

[s,t]

− a_uJ(du) β

W_p(µ, µ⁰)^β

+ J([s, t])^β

(ii) |∇P_tf|^β∗≤ a^β_t^∗

P_t(|∇f|^p∗)^β∗/p∗− b^β_t^∗|LP_tf|^β∗ for f = Qδf˜ (δ > 0 & f˜∈ C_b^Lip(M))

12 / 19

Rough idea of the proof

B (i) ⇒ (ii): Differentiation in space-time

P_sf(y) −P_tf(x) = Z

M×M

(f(z) − f(w))π(dzdw)

≈ Z

M×M

|∇f|(w)d(z, w)π(dzdw) for ^∀∀∀π: coupling of P_s^∗δ_y and P_t^∗δ_x

(LHS) → |∇P_tf|(x) + λLP_tf(x)

(y → x & s → t with λ(t−s) = d(x, y) (^∀∀∀λ ∈ RRR)) Key Fact: |∇Q_rf|: u.s.c.

Rough idea of the proof

B (i) ⇒ (ii): Differentiation in space-time

P_sf(y) −P_tf(x) = Z

M×M

(f(z) − f(w))π(dzdw)

≈ Z

M×M

|∇f|(w)d(z, w)π(dzdw)

for ^∀∀∀π: coupling of P_s^∗δ_y and P_t^∗δ_x (LHS) → |∇P_tf|(x) + λLP_tf(x)

(y → x & s → t with λ(t−s) = d(x, y) (^∀∀∀λ ∈ RRR))

Key Fact: |∇Q_rf|: u.s.c.

13 / 19

Rough idea of the proof

B (i) ⇒ (ii): Differentiation in space-time

P_sf(y) −P_tf(x) = Z

M×M

(f(z) − f(w))π(dzdw)

≈ Z

M×M

|∇f|(w)d(z, w)π(dzdw)

for ^∀∀∀π: coupling of P_s^∗δ_y and P_t^∗δ_x (LHS) → |∇P_tf|(x) + λLP_tf(x)

(y → x & s → t with λ(t−s) = d(x, y) (^∀∀∀λ ∈ RRR))

Rough idea of the proof

B (ii) ⇒ (i): Integration in space-time (µr)_r∈[0,1]: Wp-geod. from µ to µ⁰

& consider W_p-speed of P_ξ(r)^∗ µ_r (ξ: parametrization)

via the Kantorovich duality:

Wp(P_ξ(r)^∗ µs, P_ξ(r+s)^∗ µr+s)^p

ps^p ≈???

Key fact: ∂_r⁺Q_rϕ + |∇Q_rϕ|^p∗ p_∗ = 0

14 / 19

Rough idea of the proof

B (ii) ⇒ (i): Integration in space-time (µ_r)_r∈[0,1]: W_p-geod. from µ to µ⁰

& consider Wp-speed of P_ξ(r)^∗ µr (ξ: parametrization) via the Kantorovich duality:

W_p(µ, µ⁰)^p

ps^p−1 = sup

ϕ∈C_b^Lip(M)

Z

M

Qsϕdµ⁰ − Z

M

ϕdµ

Key fact: ∂_r⁺Q_rϕ + |∇Q_rϕ|^p∗ p_∗ = 0

Rough idea of the proof

B (ii) ⇒ (i): Integration in space-time (µr)_r∈[0,1]: Wp-geod. from µ to µ⁰

& consider W_p-speed of P_ξ(r)^∗ µ_r (ξ: parametrization) via the Kantorovich duality:

Wp(P_ξ(r)^∗ µs, P_ξ(r+s)^∗ µr+s)^p

ps^p ≈∂_s

Z

P_ξ(r+s)Qsϕ dµr+s

Key fact: ∂_r⁺Q_rϕ + |∇Q_rϕ|^p∗ p_∗ = 0

14 / 19

Rough idea of the proof

B (ii) ⇒ (i): Integration in space-time (µr)_r∈[0,1]: Wp-geod. from µ to µ⁰

& consider W_p-speed of P_ξ(r)^∗ µ_r (ξ: parametrization) via the Kantorovich duality:

Wp(P_ξ(r)^∗ µs, P_ξ(r+s)^∗ µr+s)^p

ps^p ≈∂_s

Z

P_ξ(r+s)Qsϕ dµr+s

Key fact: ∂_r⁺Q_rϕ + |∇Q_rϕ|^p∗ p_∗ = 0

Remarks

(i) ⇔ (ii) ⇔ (i)’:

(i)’ W_p(P_s^∗µ, P_t^∗µ⁰)^β ≤ A^β_s,tW_p(µ, µ⁰)^β +B_s,t^β

with At,t = at & B_s,t

|t− s| → 1

b_t as s → t (⇒ (i)’ enjoys self-improvement)

If P_t is strong Feller,

(ii) with f ∈ C_b^Lip(M) ⇔ (i)

The same holds for Pt defined on L^p-spaces (with a.e. umbiguity) under suitable assumptions

15 / 19

Remarks

(i) ⇔ (ii) ⇔ (i)’:

(i)’ W_p(P_s^∗µ, P_t^∗µ⁰)^β ≤ A^β_s,tW_p(µ, µ⁰)^β +B_s,t^β

with At,t = at & B_s,t

|t− s| → 1

b_t as s → t (⇒ (i)’ enjoys self-improvement)

If P_t is strong Feller,

(ii) with f ∈ C_b^Lip(M) ⇔ (i)

The same holds for Pt defined on L^p-spaces

Remarks

(i) ⇔ (ii) ⇔ (i)’:

(i)’ W_p(P_s^∗µ, P_t^∗µ⁰)^β ≤ A^β_s,tW_p(µ, µ⁰)^β +B_s,t^β

with At,t = at & B_s,t

|t− s| → 1

b_t as s → t (⇒ (i)’ enjoys self-improvement)

If P_t is strong Feller,

(ii) with f ∈ C_b^Lip(M) ⇔ (i)

The same holds for Pt defined on L^p-spaces (with a.e. umbiguity) under suitable assumptions

15 / 19

1. Introduction & Overview

2. Duality in space-time estimates

3. Wp-space-time control on Riem. mfds

Example of W

_p

-space-time control

M: cpl. Riem. mfd, dimM ≤ N, Ric ≤ K

p ≥ 2, bu = s

e^2Ku − 1

(N +p −2)K, J(du) = du b_u Theorem 2

The following holds: (a) W_p(P_s^∗µ, P_t^∗µ⁰)²≤

Z

[s,t]

− e^−KtJ(du) 2

W_p(µ, µ⁰)²

+J([s, t])²

(b) |∇P_tf|² ≤ e^−Kt

P_t(|∇f|^p∗)^2/p∗ − b²_t|LP_tf|² for f ∈ C_b^Lip(M)

17 / 19

Example of W

_p

-space-time control

M: cpl. Riem. mfd, dimM ≤ N, Ric ≤ K p ≥ 2, bu =

s

e^2Ku − 1

(N +p −2)K, J(du) = du b_u Theorem 2

The following holds:

(a) Wp(P_s^∗µ, P_t^∗µ⁰)²≤ Z

[s,t]

− e^−KtJ(du) 2

Wp(µ, µ⁰)²

+J([s, t])²

(b) |∇P_tf|² ≤ e^−Kt

P_t(|∇f|^p∗)^2/p∗ − b²_t|LP_tf|²

Example of W

_p

-space-time control

M: cpl. Riem. mfd, dimM ≤ N, Ric ≤ K p ≥ 2, bu =

s

e^2Ku − 1

(N +p −2)K, J(du) = du b_u Theorem 2

The following holds:

(a) Wp(P_s^∗µ, P_t^∗µ⁰)²≤ Z

[s,t]

− e^−KtJ(du) 2

Wp(µ, µ⁰)²

+J([s, t])²

(b) |∇P_tf|² ≤ e^−Kt

P_t(|∇f|^p∗)^2/p∗ − b²_t|LP_tf|² for f ∈ C_b^Lip(M)

17 / 19

Method of the proof

Coupling by parallel transport of BMs with different speed

(⇒ (Rough) est. of W_p-speed)

(i)’ ⇒ (ii): still rough

(ii) ⇒ (c) below ⇒ (b) ⇒ (a) (c) ¹

2∆|∇f|² − h∇f,∇∆fi

≥ K|∇f|²+ (Lf)²

N + p−2 + p−2 4(p−1)

|∇|∇f|²|²

|∇f|²

(Bakry-´Emery theory)

Method of the proof

Coupling by parallel transport of BMs

with different speed (⇒ (Rough) est. of Wp-speed) (i)’ ⇒ (ii): still rough

(ii) ⇒ (c) below ⇒ (b) ⇒ (a) (c) ¹

2∆|∇f|² − h∇f,∇∆fi

≥ K|∇f|²+ (Lf)²

N + p−2 + p−2 4(p−1)

|∇|∇f|²|²

|∇f|²

(Bakry-´Emery theory)

18 / 19

Method of the proof

Coupling by parallel transport of BMs

with different speed (⇒ (Rough) est. of Wp-speed) (i)’ ⇒ (ii): still rough

(ii) ⇒ (c) below ⇒ (b) ⇒ (a) (c) ¹

2∆|∇f|² − h∇f,∇∆fi

≥ K|∇f|²+ (Lf)²

N + p−2 + p−2 4(p−1)

|∇|∇f|²|²

|∇f|²

(Bakry-´Emery theory)

Method of the proof

Coupling by parallel transport of BMs

with different speed (⇒ (Rough) est. of Wp-speed) (i)’ ⇒ (ii): still rough

(ii) ⇒ (c) below ⇒ (b) ⇒ (a) (c) ¹

2∆|∇f|² − h∇f,∇∆fi

≥ K|∇f|²+ (Lf)²

N + p−2 + p−2 4(p−1)

|∇|∇f|²|²

|∇f|²

(Bakry-´Emery theory)

18 / 19

Method of the proof

Coupling by parallel transport of BMs

with different speed (⇒ (Rough) est. of Wp-speed) (i)’ ⇒ (ii): still rough

(ii) ⇒ (c) below ⇒ (b) ⇒ (a) (c) ¹

2∆|∇f|² − h∇f,∇∆fi

≥ K|∇f|²+ (Lf)²

N + p−2 + p−2 4(p−1)

|∇|∇f|²|²

|∇f|²

(Bakry-´Emery theory)

Some remarks

∆ L = ∆ + Z (nonsymm.) is possible

Ric Ric − ∇Z − 1

N − mZ ⊗ Z

For Ric ≥ K & dim ≤ N ⇒ (c), no straightforward proof is known

19 / 19

Some remarks

∆ L = ∆ + Z (nonsymm.) is possible

Ric Ric − ∇Z − 1

N − mZ ⊗ Z

For Ric ≥ K & dim ≤ N ⇒ (c), no straightforward proof is known