Lecture 1: Main Models & Basics of Wasserstein Distancecalvino.polito.it/~mmkt/ed2006/Carrillo1.pdf · university-logo Presentation of models Wasserstein Distance: Basics Contractivity

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Presentation of models Wasserstein Distance: Basics Contractivity in 1D

Lecture 1: Main Models & Basics ofWasserstein Distance

J. A. Carrillo

ICREA - Universitat Autònoma de Barcelona

Methods and Models of Kinetic Theory

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Outline

1 Presentation of modelsNonlinear diffusionsKinetic models for granular gases

2 Wasserstein Distance: BasicsDefinitionProperties

3 Contractivity in 1DDiffusive ModelsDissipative ModelsTowards any dimension

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Leading examples

Nonlinear Diffusions.-

∂u∂t

= ∆um, (x ∈ Rd, t > 0)

Inelastic Dissipative Models: Nonlinear friction equations.- 1:

∂f∂t

+ v∂f∂x

=∂

∂v

»ZR(v− w)|v− w|γ(|v− w|)f (x, w, t) dw f (x, v, t)

–Inelastic Dissipative Models: Nonlinear Boltzmann-type kinetic equations.-

∂f∂t

+ (v · ∇x)f =Qe(f , f )

→ Conservation of mass and center of mass/mean velocity.

→ Spreading (diffusions) versus Concentration (dissipative models).

1Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997); Benedetto, D., Caglioti, E., Carrillo, J. A., Pulvirenti, M., J. Stat.

Phys. (1998).

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Leading examples


∂u∂t

= ∆um, (x ∈ Rd, t > 0)


∂f∂t

+ v∂f∂x

=∂

∂v



∂f∂t

+ (v · ∇x)f =Qe(f , f )




Phys. (1998).

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Leading examples


∂u∂t

= ∆um, (x ∈ Rd, t > 0)


∂f∂t

+ v∂f∂x

=∂

∂v



∂f∂t

+ (v · ∇x)f =Qe(f , f )




Phys. (1998).

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Leading examples


∂u∂t

= ∆um, (x ∈ Rd, t > 0)


∂f∂t

+ v∂f∂x

=∂

∂v



∂f∂t

+ (v · ∇x)f =Qe(f , f )




Phys. (1998).

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Leading examples


∂u∂t

= ∆um, (x ∈ Rd, t > 0)


∂f∂t

+ v∂f∂x

=∂

∂v



∂f∂t

+ (v · ∇x)f =Qe(f , f )




Phys. (1998).

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Nonlinear diffusions

Outline




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→ Self-similar solution: Barenblatt profile2.- An explicit self-similar solution thatis integrable for m > (d − 2)/d:

Bm(x, t) = t−d/λ

„C − 2m

1− m|x|2

t2/λ

«1/(1−m)

+

for m 6= 1 where λ = d(m− 1) + 2 and C > 0 is determined to have unit mass.It verifies that Bm(x, t) converges weakly-* as measures towards δ0 as t → 0+.

2Zeldovich, Ya. B., Barenblatt, G. I. Doklady, USSR Academy of Sciences (1958).

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→ Asymptotic behaviour: Comparison methods.- Friedman & Kamin (1980)(completed by Vázquez (1997)). Given an initial data in the class

X0 = u0 ∈ L1(Rd) : u0 ≥ 0 ,

then, for any (d − 2)/d < m

limt→∞

‖u(·, t)−Bm(·, t)‖L1 = 0, and limt→∞

td/λ‖u(·, t)−Bm(·, t)‖L∞ = 0.

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∂ρ

∂t= div(xρ +∇ρm), (x ∈ Rd, t > 0)

ρ(x, t = 0) = ρ0(x) ≥ 0, (x ∈ Rd)

??

ρ(x, t) = edtu(etx,1λ

(eλt − 1))

λ = d(m− 1) + 2

∂u∂t

= ∆um, (x ∈ Rd, t > 0)

u(x, t = 0) = u0(x) ≥ 0, (x ∈ Rd)3

→ Exponential decay to equilibria translates into algebraic decay to Barenblattprofiles.

3J. A. Carrillo, G. Toscani, Indiana Math. Univ. J. (2000); F. Otto, Comm. PDE (2001); J. Dolbeault, M. del Pino, J. Math. Pures Appl. (2002); J.L.

Vázquez, J. Evol. Eq. 2003.

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∂ρ

∂t= div(xρ +∇ρm), (x ∈ Rd, t > 0)

ρ(x, t = 0) = ρ0(x) ≥ 0, (x ∈ Rd)

??

ρ(x, t) = edtu(etx,1λ

(eλt − 1))

λ = d(m− 1) + 2

∂u∂t

= ∆um, (x ∈ Rd, t > 0)

u(x, t = 0) = u0(x) ≥ 0, (x ∈ Rd)3

→ Exponential decay to equilibria translates into algebraic decay to Barenblattprofiles.

3J. A. Carrillo, G. Toscani, Indiana Math. Univ. J. (2000); F. Otto, Comm. PDE (2001); J. Dolbeault, M. del Pino, J. Math. Pures Appl. (2002); J.L.

Vázquez, J. Evol. Eq. 2003.

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→ Generic finite speed of propagation/moving free boundary for degeneratediffusion equations and creation of thick tails for fast diffusion equations.

→ No better rate of convergence can be established under the generality u0 aprobability density.At which rate does this self-similarity take over?

→ General nonlinear diffusion equation:8><>:∂u∂t

= ∆P(u), (x ∈ Rd, t > 0),

u(x, t = 0) = u0(x) ≥ 0 (x ∈ Rd)

No explicit source-type solutions, so...What is the typical asymptotic profile?

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= ∆P(u), (x ∈ Rd, t > 0),

u(x, t = 0) = u0(x) ≥ 0 (x ∈ Rd)


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= ∆P(u), (x ∈ Rd, t > 0),

u(x, t = 0) = u0(x) ≥ 0 (x ∈ Rd)


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Kinetic models for granular gases

Outline




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Rapid Granular Flows

Pattern formation in a vertically oscillated granular layer. 4

4Bizon, C., Shattuck, M. D., Swift, J.B., Swinney, H.L., Phys. Rev. E (1999); Carrillo, J.A., Poschel, T., Salueña, C., in preparation (2006).

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Rapid Granular Flows

Schock waves in supersonic sand. 5

5Rericha, E., Bizon, C., Shattuck, M. D., Swinney, H.L., Phys. Rev. Letters (2002).

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Inelastic Collisions

Binary Collisions:

Spheres of diameter r > 0. Given (x, v) and (x− rn, w), where n ∈ S2 is the unitvector along the impact direction, the post-collisional velocities are found assumingconservation of momentum and a loose of normal relative velocity after the collision:

(v′ − w′) · n = −e((v− w) · n)

where 0 < e ≤ 1 is called the restitution coefficient.

Postcollisional velocities:

v′ =12(v + w) +

u′

2

w′ =12(v + w)− u′

2

where u′ = u− (1 + e)(u · n)n, u = v− w and u′ = v′ − w′.

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Binary Collisions:

Spheres of diameter r > 0. Given (x, v) and (x− rn, w), where n ∈ S2 is the unitvector along the impact direction, the post-collisional velocities are found assumingconservation of momentum and a loose of normal relative velocity after the collision:

(v′ − w′) · n = −e((v− w) · n)

where 0 < e ≤ 1 is called the restitution coefficient.

Postcollisional velocities:

v′ =12(v + w) +

u′

2

w′ =12(v + w)− u′

2

where u′ = u− (1 + e)(u · n)n, u = v− w and u′ = v′ − w′.

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Postcollisional velocities: NewParameterization

v′ =12(v + w) +

u′

2

w′ =12(v + w)− u′

2

where

u′ =1− e

4u +

1 + e4|u|σ,

u = v− w and u′ = v′ − w′.

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Boltzmann equation for granular gasesHypothesis:

Binary, localized in t and x and inelastic collisions.

Molecular chaos.

Boltzmann equation for inelastic particles:

∂f∂t

+(v ·∇x)f =Qe(f , f )=1

4π

ZR3

ZS2+

((v−w) · n)

»1e2 f (v∗)f (w∗)− f (v)f (w)

–dndw.a

Weak form of the Boltzmann equation:

< ϕ, Qe(f , f ) >=1

4π

ZR3

ZR3

ZS2|v− w|f (v)f (w)

hϕ(v′)− ϕ(v)

idσ dv dw

aJenkins, J. T., Richman, M. W., Arch. Rat. Mech. Anal. (1985).

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Molecular chaos.


∂f∂t

+(v ·∇x)f =Qe(f , f )=1

4π

ZR3

ZS2+

((v−w) · n)

»1e2 f (v∗)f (w∗)− f (v)f (w)

–dndw.a


< ϕ, Qe(f , f ) >=1

4π

ZR3

ZR3


hϕ(v′)− ϕ(v)

idσ dv dw


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Molecular chaos.


∂f∂t

+(v ·∇x)f =Qe(f , f )=1

4π

ZR3

ZS2+

((v−w) · n)

»1e2 f (v∗)f (w∗)− f (v)f (w)

–dndw.a


< ϕ, Qe(f , f ) >=1

4π

ZR3

ZR3


hϕ(v′)− ϕ(v)

idσ dv dw


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Molecular chaos.


∂f∂t

+(v ·∇x)f =Qe(f , f )=1

4π

ZR3

ZS2+

((v−w) · n)

»1e2 f (v∗)f (w∗)− f (v)f (w)

–dndw.a


< ϕ, Qe(f , f ) >=1

4π

ZR3

ZR3


hϕ(v′)− ϕ(v)

idσ dv dw


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Basic Properties6

∂f∂t

= Qe(f , f )

Mass and momentum are preserved while energy decreases:

|v′|2 + |w′|2 − |v|2 − |w|2 =1− e2

2|u|2

Let us fix unit mass and zero mean velocity for the rest.

Temperature cools off down to 0 as t →∞ (Haff’s law):

dθ

dt≤ − 1− e2

4θ

32 ,

What are the typical asymptotic profiles? Do self-similar solutions exist?If so At which rate does this self-similarity take over?

6J.A. Carrillo, C. Cercignani, I.M. Gamba, Phys. Rev. E (2000); C. Cercignani, R. Illner, C. Stoica, J. Stat. Phys. (2001); A. Bobylev, C.

Cercignani, J. Stat. Phys. (2002); I.M. Gamba, V. Panferov, C. Villani, Comm. Math. Phys. (2004); C. Mouhot, S. Mischer, Martínez-Ricard, J. Stat.

Phys. (2006).

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Basic Properties6

∂f∂t

= Qe(f , f )

Mass and momentum are preserved while energy decreases:

|v′|2 + |w′|2 − |v|2 − |w|2 =1− e2

2|u|2

Let us fix unit mass and zero mean velocity for the rest.

Temperature cools off down to 0 as t →∞ (Haff’s law):

dθ

dt≤ − 1− e2

4θ

32 ,

What are the typical asymptotic profiles? Do self-similar solutions exist?If so At which rate does this self-similarity take over?

6J.A. Carrillo, C. Cercignani, I.M. Gamba, Phys. Rev. E (2000); C. Cercignani, R. Illner, C. Stoica, J. Stat. Phys. (2001); A. Bobylev, C.

Cercignani, J. Stat. Phys. (2002); I.M. Gamba, V. Panferov, C. Villani, Comm. Math. Phys. (2004); C. Mouhot, S. Mischer, Martínez-Ricard, J. Stat.

Phys. (2006).

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Simplified Granular media model

Another dissipative models (toy models) keeping the same properties: conservationof mass, mean velocity and dissipation of energy are of the form:

∂f∂t

+ (v · ∇x)f = I(f , f )

where

I(f , f ) = divv

»ZR3

(v− w)|v− w|f (x, w, t) dw f (x, v, t)–

.

This dissipative collision in the one dimensional case simplifies to7:

I(f , f ) =∂

∂v

»ZR(v− w)|v− w|f (x, w, t) dw f (x, v, t)

–

7Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997).

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∂f∂t

+ (v · ∇x)f = I(f , f )

where

I(f , f ) = divv

»ZR3

(v− w)|v− w|f (x, w, t) dw f (x, v, t)–

.


I(f , f ) =∂

∂v


–


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∂f∂t

+ (v · ∇x)f = I(f , f )

where

I(f , f ) = divv

»ZR3

(v− w)|v− w|f (x, w, t) dw f (x, v, t)–

.


I(f , f ) =∂

∂v


–


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Summary: Spreading versus ConcentrationDiffusive models - Spreading:

∂u∂t

= ∆P(u), (x ∈ Rd, t > 0)

Typical asymptotic profiles for P(u) = um (Barenblatt):

Bm(x, t) = θm(t)−d/2Bm(θm(t)−1/2x, to,m)

with θm(t) ∞.

Homogeneous Kinetic Dissipative models - Concentration:

∂f∂t

=

8><>:Qe(f , f )

divv

»f (v, t)

ZRd∇W(v− w)f (w, t) dw

–Typical asymptotic profiles (Homogeneous Cooling States), if they exist:

fhc(v, t) = ρ θ− 3

2hc (t) g∞((v− u) θ

− 12

hc (t))

with θhc(t) 0.

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Summary: Spreading versus ConcentrationDiffusive models - Spreading:

∂u∂t

= ∆P(u), (x ∈ Rd, t > 0)

Typical asymptotic profiles for P(u) = um (Barenblatt):

Bm(x, t) = θm(t)−d/2Bm(θm(t)−1/2x, to,m)

with θm(t) ∞.

Homogeneous Kinetic Dissipative models - Concentration:

∂f∂t

=

8><>:Qe(f , f )

divv

»f (v, t)

ZRd∇W(v− w)f (w, t) dw

–Typical asymptotic profiles (Homogeneous Cooling States), if they exist:

fhc(v, t) = ρ θ− 3

2hc (t) g∞((v− u) θ

− 12

hc (t))

with θhc(t) 0.

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Definition

Outline




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Definition

Definition of the distance8

Transporting measures:

Given T : Rd −→ Rd mesurable, we say that T transports µ ∈ P onto ν ∈ P or thatν is the push-forward of µ through T , ν = T#µ, if ν[K] := µ[T−1(K)] for allmesurable sets K ⊂ Rd, equivalentlyZ

Rdϕ dν =

ZRd

(ϕ T) dµ

for all ϕ ∈ Co(Rd).

Random variables:

Say that X is a random variable with law given by µ, is to sayX : (Ω,A, P) −→ (Rd,Bd) is a mesurable map such that X#P = µ, i.e.,Z

Rdϕ(x) dµ =

ZΩ

(ϕ X) dP = E [ϕ(X)] .

Coming back to the transport of measures: if X and Y are random variables with lawµ and ν respectively, then ν = T#µ is equivalent to Y = T(X).

8C. Villani, AMS Graduate Texts (2003).

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Definition




Rdϕ dν =

ZRd

(ϕ T) dµ


Random variables:


Rdϕ(x) dµ =

ZΩ

(ϕ X) dP = E [ϕ(X)] .



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Definition




Rdϕ dν =

ZRd

(ϕ T) dµ


Random variables:


Rdϕ(x) dµ =

ZΩ

(ϕ X) dP = E [ϕ(X)] .



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Definition




Rdϕ dν =

ZRd

(ϕ T) dµ


Random variables:


Rdϕ(x) dµ =

ZΩ

(ϕ X) dP = E [ϕ(X)] .



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Definition

Definition of the distance

Euclidean Wasserstein Distance:

W2(µ, ν)= infπ

ZZRd×Rd

|x− y|2 dπ(x, y)ff1/2

= inf(X,Y)

nE

h|X − Y|2

io1/2

where the transference plan π runs over the set of joint probability measures onRd × Rd with marginals f and g ∈ P2(Rd) and (X, Y) are all possible couples ofrandom variables with µ and ν as respective laws.

Monge’s optimal mass transport problem:

Find

I := infT

ZRd|x− T(x)|2 dµ(x); ν = T#µ

ff1/2

.

Take γT = (1Rd × T)#µ as transference plan π.

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Definition



W2(µ, ν)= infπ

ZZRd×Rd

|x− y|2 dπ(x, y)ff1/2

= inf(X,Y)

nE

h|X − Y|2

io1/2



Find

I := infT


ff1/2

.


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Definition



W2(µ, ν)= infπ

ZZRd×Rd

|x− y|2 dπ(x, y)ff1/2

= inf(X,Y)

nE

h|X − Y|2

io1/2



Find

I := infT


ff1/2

.


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Definition


Wasserstein Distances:

Given 1 ≤ p < ∞, we define

Wp(µ, ν)= infπ

ZZRd×Rd

|x− y|p dπ(x, y)ff1/p

= inf(X,Y)

E [|X − Y|p]1/p

where the transference plan π runs over the set of joint probability measures onRd × Rd with marginals f and g ∈ Pp(Rd) and (X, Y) are all possible couples ofrandom variables with µ and ν as respective laws.

We define the∞-Wasserstein distance as

W∞(µ, ν) = limp∞

Wp(µ, ν) = infTesssup |x− T(x)|, x ∈ supp(µ) with ν = T#µ

for measures with all moments bounded.

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Definition




Wp(µ, ν)= infπ

ZZRd×Rd


= inf(X,Y)

E [|X − Y|p]1/p






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Definition




Wp(µ, ν)= infπ

ZZRd×Rd


= inf(X,Y)

E [|X − Y|p]1/p






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Properties

Outline




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Properties

Euclidean Wasserstein Distance

Basic Properties

1 Convexity: f1, f2, g1, g2 ∈ P2(Rd) and α ∈ [0, 1], then

W22 (αf1 + (1− α)f2, αg1 + (1− α)g2) ≤ αW2

2 (f1, g1) + (1− α)W22 (f2, g2).

2 Relation to Temperature: f ∈ P2(Rd) and a ∈ R3, then

W22 (f , δa) =

ZRd|v− a|2df (v).

3 Scaling: Given f in P2(Rd) and θ > 0, let us define

Sθ[f ] = θd/2f (θ1/2v)

for absolutely continuous measures with respect to Lebesgue or itscorresponding definition by duality for general measures; then for any f and gin P2(Rd), we have

W2(Sθ[f ],Sθ[g]) = θ−1/2 W2(f , g).

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Properties


Basic Properties


W22 (αf1 + (1− α)f2, αg1 + (1− α)g2) ≤ αW2

2 (f1, g1) + (1− α)W22 (f2, g2).


W22 (f , δa) =

ZRd|v− a|2df (v).


Sθ[f ] = θd/2f (θ1/2v)


W2(Sθ[f ],Sθ[g]) = θ−1/2 W2(f , g).

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Properties


Basic Properties


W22 (αf1 + (1− α)f2, αg1 + (1− α)g2) ≤ αW2

2 (f1, g1) + (1− α)W22 (f2, g2).


W22 (f , δa) =

ZRd|v− a|2df (v).


Sθ[f ] = θd/2f (θ1/2v)


W2(Sθ[f ],Sθ[g]) = θ−1/2 W2(f , g).

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Properties


Convergence Properties

1 Convergence of measures: W2(fn, f ) → 0 is equivalent to fn f weakly-* asmeasures and convergence of second moments.

2 Weak lower semicontinuity: Given fn f and gn g weakly-* as measures,then

W2(f , g) ≤ lim infn→∞

W2(fn, gn).

3 Completeness: The space P2(Rd) endowed with the distance W2 is a completemetric space. Moreover, the set

Mθ =

µ ∈ P2(Rd) such that

ZR3|v|2df (v) = θ

ff,

i.e. the sphere of radius√

θ in P2(Rd), is a complete metric space.

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Properties






W2(fn, gn).


Mθ =


ZR3|v|2df (v) = θ

ff,



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Properties






W2(fn, gn).


Mθ =


ZR3|v|2df (v) = θ

ff,



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Properties

Wasserstein Distances

Properties

1 Convexity of Wpp

2 Scaling: for any f and g in Pp(Rd), we have

Wp(Sθ[f ],Sθ[g]) = θ−1/2 Wp(f , g).

3 Completeness and weak lower semicontinuity.

4 Convergence⇐⇒ weak-* convergence as measures and convergence ofp-moments.

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Properties


Properties

1 Convexity of Wpp


Wp(Sθ[f ],Sθ[g]) = θ−1/2 Wp(f , g).



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Properties


Properties

1 Convexity of Wpp


Wp(Sθ[f ],Sθ[g]) = θ−1/2 Wp(f , g).



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Properties


Properties

1 Convexity of Wpp


Wp(Sθ[f ],Sθ[g]) = θ−1/2 Wp(f , g).



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Properties

One dimensional Case

Distribution functions:

In one dimension, denoting by F(x) the distribution function of µ,

F(x) =

Z x

−∞dµ,

we can define its pseudo inverse:

F−1(η) = infx : F(x) > η for η ∈ (0, 1),

we have F−1 : ((0, 1),B1), dη) −→ (R,B1) is a random variable with law µ, i.e.,F−1#dη = µ Z

Rϕ(x) dµ =

Z 1

0ϕ(F−1(η)) dη = E [ϕ(X)] .

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Properties


Distribution functions:

In one dimension, denoting by F(x) the distribution function of µ,

F(x) =

Z x

−∞dµ,

we can define its pseudo inverse:

F−1(η) = infx : F(x) > η for η ∈ (0, 1),

we have F−1 : ((0, 1),B1), dη) −→ (R,B1) is a random variable with law µ, i.e.,F−1#dη = µ Z

Rϕ(x) dµ =

Z 1

0ϕ(F−1(η)) dη = E [ϕ(X)] .

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Properties


Wasserstein distance:

Given µ and ν ∈ P2(Rd) with distribution functions F(x) and G(y) then,

W22 (µ, ν) =

ZZRd×Rd

|x− y|2 dH(x, y)

where H(x, y) = min(F(x), G(y)) is a joint distribution function and the aboveintegral is defined in the Riemann-Stieljes sense.a

In one dimension, it is easy to check that given two measures µ and ν withdistribution functions F(x) and G(y) then, (F−1 × G−1)#dη has joint distributionfunction H(x, y) = min(F(x), G(y)). Therefore, in one dimension, the optimal planis given by πopt(x, y) = (F−1 × G−1)#dη, and thus

W2(µ, ν) =

„Z 1

0[F−1(η)− G−1(η)]2 dη

«1/2

aW. Hoeffding (1940); M. Fréchet (1951); A. Pulvirenti, G. Toscani, Annali Mat. Pura Appl. (1996).

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Properties



Given µ and ν ∈ P2(Rd) with distribution functions F(x) and G(y) then,

W22 (µ, ν) =

ZZRd×Rd

|x− y|2 dH(x, y)

where H(x, y) = min(F(x), G(y)) is a joint distribution function and the aboveintegral is defined in the Riemann-Stieljes sense.a

In one dimension, it is easy to check that given two measures µ and ν withdistribution functions F(x) and G(y) then, (F−1 × G−1)#dη has joint distributionfunction H(x, y) = min(F(x), G(y)). Therefore, in one dimension, the optimal planis given by πopt(x, y) = (F−1 × G−1)#dη, and thus

W2(µ, ν) =

„Z 1

0[F−1(η)− G−1(η)]2 dη

«1/2

aW. Hoeffding (1940); M. Fréchet (1951); A. Pulvirenti, G. Toscani, Annali Mat. Pura Appl. (1996).

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Properties



Given µ and ν ∈ Pp(R), 1 ≤ p < ∞ then

Wp(µ, ν) =

„Z 1

0[F−1(η)− G−1(η)]p dη

«1/p

= ‖F−1 − G−1‖Lp(R)

andW∞(µ, ν) = ‖F−1 − G−1‖L∞(R).

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Properties

Relation 1D-anyD

Euclidean Wasserstein distance:

If the probability measures µij on R are the successive one-dimensional marginals of

the measure µi on Rd, for i = 1, 2 and j = 1, . . . , d, then

dXj=1

W22 (µ

1j , µ

2j ) ≤ W2

2 (µ1, µ2).

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Properties

Relation 1D-anyD

Proof.

Let π be a measure on Rdx × Rd

y with marginals µ1 and µ2, optimal in the sense that

W22 (µ

1, µ2) =

ZZRd×Rd

|x− y|2 dπ(x, y).

Then its marginal πj on Rvj × Rwj has itself marginals µ1j and µ2

j , so

W22 (µ

1j , µ

2j ) ≤

ZZR×R

|vj − wj|2 dπj(vj, wj).

The inequality follows by noting thatdX

j=1

|vj − wj|2 = |v− w|2.

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Properties

Relation 1D-anyD

Proof.



W22 (µ

1, µ2) =

ZZRd×Rd

|x− y|2 dπ(x, y).


j , so

W22 (µ

1j , µ

2j ) ≤

ZZR×R



j=1

|vj − wj|2 = |v− w|2.

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Properties

Relation 1D-anyD

Proof.



W22 (µ

1, µ2) =

ZZRd×Rd

|x− y|2 dπ(x, y).


j , so

W22 (µ

1j , µ

2j ) ≤

ZZR×R



j=1

|vj − wj|2 = |v− w|2.

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Diffusive Models

Outline




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Diffusive Models

1D Nonlinear Diffusions9

The flows of nonlinear diffusions are contractions in Wasserstein distance.

ut = (P(u))xx

u(x, 0) = u0(x) ≥ 0 ∈ L1(R)

with P : R+0 → R, continuous, nondecreasing with P(0) = 0.

Diccionary for F−1(η):

∂F−1

∂t(η, t) = −

»1u

–(F−1(η, t), t)

∂F∂t

(F−1(η, t), t)

∂F−1

∂η(η, t) =

»1u

–(F−1(η, t), t) ,

∂2F−1

∂η2 (η, t) = −h ux

u3

i(F−1(η, t), t)

Equation for F−1:

∂F−1

∂t(η, t) = − ∂

∂η

»P

»„∂F−1

∂η

«−1––,

9H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).

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Diffusive Models



ut = (P(u))xx

u(x, 0) = u0(x) ≥ 0 ∈ L1(R)



∂F−1

∂t(η, t) = −

»1u

–(F−1(η, t), t)

∂F∂t

(F−1(η, t), t)

∂F−1

∂η(η, t) =

»1u

–(F−1(η, t), t) ,

∂2F−1

∂η2 (η, t) = −h ux

u3

i(F−1(η, t), t)

Equation for F−1:

∂F−1

∂t(η, t) = − ∂

∂η

»P

»„∂F−1

∂η

«−1––,


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Diffusive Models



ut = (P(u))xx

u(x, 0) = u0(x) ≥ 0 ∈ L1(R)



∂F−1

∂t(η, t) = −

»1u

–(F−1(η, t), t)

∂F∂t

(F−1(η, t), t)

∂F−1

∂η(η, t) =

»1u

–(F−1(η, t), t) ,

∂2F−1

∂η2 (η, t) = −h ux

u3

i(F−1(η, t), t)

Equation for F−1:

∂F−1

∂t(η, t) = − ∂

∂η

»P

»„∂F−1

∂η

«−1––,


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Diffusive Models


Contraction for nonlinear diffusions:

Given any two solutions of the nonlinear diffusion equation

Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))

for all 1 ≤ p ≤ ∞.

Proof.

Assume 1 < p < ∞, the other ones by approximation in p, then

1p(p− 1)

ddt

Z 1

0|F−1 − G−1|pdη =Z 1

0|F−1 − G−1|p−2(F−1

η − G−1η )

P

»“F−1

η

”−1–− P

»“G−1

η

”−1–ff

dη ≤ 0


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Diffusive Models


Contraction for nonlinear diffusions:

Given any two solutions of the nonlinear diffusion equation

Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))

for all 1 ≤ p ≤ ∞.

Proof.

Assume 1 < p < ∞, the other ones by approximation in p, then

1p(p− 1)

ddt

Z 1

0|F−1 − G−1|pdη =Z 1

0|F−1 − G−1|p−2(F−1

η − G−1η )

P

»“F−1

η

”−1–− P

»“G−1

η

”−1–ff

dη ≤ 0


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Diffusive Models


Strict Contraction for nonlinear diffusions with confining potential:

Given any two probability density solutions of the nonlinear diffusion equation

ρt =

„dVdx

(t, x)ρ«

x

+ (P(ρ))xx

where V(t, x) is a smooth strictly uniform convex potential ( d2Vdx2 (t, x) ≥ α > 0), then

Wp(ρ1(t), ρ2(t)) ≤ Wp(ρ1(0), ρ2(0)) e−αt

for all 1 ≤ p ≤ ∞.

Proof.

Exercise;∂F−1

∂t(η, t) = −dV

dx(t, F−1)− ∂

∂η

»P

»„∂F−1

∂η

«−1––,

The case of V(x) = x2/2 and P(ρ) = ρm comes by the scaling property of theWasserstein distances and the previous non strict contraction.


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Diffusive Models


Strict Contraction for nonlinear diffusions with confining potential:

Given any two probability density solutions of the nonlinear diffusion equation

ρt =

„dVdx

(t, x)ρ«

x

+ (P(ρ))xx

where V(t, x) is a smooth strictly uniform convex potential ( d2Vdx2 (t, x) ≥ α > 0), then

Wp(ρ1(t), ρ2(t)) ≤ Wp(ρ1(0), ρ2(0)) e−αt

for all 1 ≤ p ≤ ∞.

Proof.

Exercise;∂F−1

∂t(η, t) = −dV

dx(t, F−1)− ∂

∂η

»P

»„∂F−1

∂η

«−1––,

The case of V(x) = x2/2 and P(ρ) = ρm comes by the scaling property of theWasserstein distances and the previous non strict contraction.


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Diffusive Models

1D Nonlinear Diffusions

Convergence towards equilibrium:

All probability solutions of the equation

ρt =

„dVdx

(t, x)ρ«

x

+ (P(ρ))xx

converges exponentially fast towards the unique equilibrium ρ∞ characterized by

V(x) + h(ρ∞(x)) = C ρh′(ρ) = P′(ρ).

Here, P is strictly increasing for being h invertible.

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Diffusive Models


Uniform Finite Speed of Propagation:

Given any two probability density solutions of the degenerate nonlinear diffusionequation

ρt = (ρm)xx ,

we have the following control of the "relative speed" of the support spreading

| infsuppρ1(t) − infsuppρ2(t)| ≤ W∞(ρ1(0), ρ2(0))

| supsuppρ1(t) − supsuppρ2(t)| ≤ W∞(ρ1(0), ρ2(0))

for any t > 0.

12J.A. Carrillo, M.P. Gualdani, G. Toscani, CRAS (2004).

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Dissipative Models

Outline




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Dissipative Models

1D Dissipative Kinetic Equations13

Given the 1D granular model

∂f∂t

=∂

∂v

»ZR(v− w)|v− w|γ f (w, t) dw f (v, t)

–

Equation for F−1:

∂F−1

∂t(η, t) = −

Z 1

0|F−1(η, t)− F−1(p, t)|γ(F−1(η, t)− F−1(p, t)) dp .

After some manipulations (good exercise)

ddt

W2(f (t), g(t)) ≤ − 12γ−1 W2(f (t), g(t))1+γ

implying

W2(f (t), g(t)) ≤„

2γ−1

γt + 2γ−1W2(f (0), g(0))−γ

«1/γ

.


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Dissipative Models



∂f∂t

=∂

∂v


–

Equation for F−1:

∂F−1

∂t(η, t) = −

Z 1



ddt

W2(f (t), g(t)) ≤ − 12γ−1 W2(f (t), g(t))1+γ

implying

W2(f (t), g(t)) ≤„

2γ−1

γt + 2γ−1W2(f (0), g(0))−γ

«1/γ

.


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Dissipative Models



∂f∂t

=∂

∂v


–

Equation for F−1:

∂F−1

∂t(η, t) = −

Z 1



ddt

W2(f (t), g(t)) ≤ − 12γ−1 W2(f (t), g(t))1+γ

implying

W2(f (t), g(t)) ≤„

2γ−1

γt + 2γ−1W2(f (0), g(0))−γ

«1/γ

.


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Dissipative Models



∂f∂t

=∂

∂v


–

Equation for F−1:

∂F−1

∂t(η, t) = −

Z 1



ddt

W2(f (t), g(t)) ≤ − 12γ−1 W2(f (t), g(t))1+γ

implying

W2(f (t), g(t)) ≤„

2γ−1

γt + 2γ−1W2(f (0), g(0))−γ

«1/γ

.


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Towards any dimension

Outline




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Heat Equation14

Contraction for the Heat Equation:

Given any two probability density solutions u1(x, t), u2(x, t) of the heat equation,then

Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))

for t ≥ 0, 1 ≤ p ≤ ∞.

Proof.

Let γo the optimal transference plan between u1(0) y u2(0) for Wp, 1 ≤ p < ∞.The solutions are u1(t) = K(t, x) ∗ u1(0) and u2(t) = K(t, x) ∗ u2(0).Let us define γt by duality:Z

Rd×Rdϕ(x, y) dγt(x, y) =

ZRd×Rd

ZRd

ϕ(x + z, y + z)K(t, z) dz dγo(x, y)

Check that γt is a transference plan between u1(t) and u2(t), and thus

Wpp (u1(t), u2(t))≤

ZRd×Rd

|x−y|p dγt(x, y)=

ZRd×Rd

|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).

14J.A. Carrillo, Boletín SEMA (2004).

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Heat Equation14



Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))

for t ≥ 0, 1 ≤ p ≤ ∞.

Proof.



ZRd×Rd

ZRd



Wpp (u1(t), u2(t))≤

ZRd×Rd


ZRd×Rd

|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).


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Heat Equation14



Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))

for t ≥ 0, 1 ≤ p ≤ ∞.

Proof.



ZRd×Rd

ZRd



Wpp (u1(t), u2(t))≤

ZRd×Rd


ZRd×Rd

|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).


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Heat Equation14



Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))

for t ≥ 0, 1 ≤ p ≤ ∞.

Proof.



ZRd×Rd

ZRd



Wpp (u1(t), u2(t))≤

ZRd×Rd


ZRd×Rd

|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).


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Heat Equation14



Wp(u1(t), u2(t)) ≤ Wp(u1(0), u2(0))

for t ≥ 0, 1 ≤ p ≤ ∞.

Proof.



ZRd×Rd

ZRd



Wpp (u1(t), u2(t))≤

ZRd×Rd


ZRd×Rd

|x−y|p dγo(x, y)=Wpp (u1(0), u2(0)).


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Homogeneous Boltzmann Equation15

Convergence Towards Monokinetic State:

Given f (v, t) ∈ P2(R3) with zero mean velocity then

W2(f (t), δ0) ≤ C(1 + t)−2

para t ≥ 0.

Proof.

Evolution of temperature:

< |v|2, Qe(f , f ) >=1− e2

4

ZR3

ZR3|v− w|3f (v)f (w) dv dw

and thus using Jensen’s and Hölder’s inequality, we get the Haff’s law:

dθ

dt≤ − 1− e2

4θ

32 .

Using the relation to temperature of the Euclidean Wasserstein distance, weconclude.


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W2(f (t), δ0) ≤ C(1 + t)−2

para t ≥ 0.

Proof.


< |v|2, Qe(f , f ) >=1− e2

4

ZR3



dθ

dt≤ − 1− e2

4θ

32 .



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W2(f (t), δ0) ≤ C(1 + t)−2

para t ≥ 0.

Proof.


< |v|2, Qe(f , f ) >=1− e2

4

ZR3



dθ

dt≤ − 1− e2

4θ

32 .



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Lecture 1: Main Models & Basics of Wasserstein Distancecalvino.polito.it/~mmkt/ed2006/Carrillo1.pdf · university-logo Presentation of models Wasserstein Distance: Basics Contractivity