Lecture 1: Main Models & Basics of Wasserstein Distance

Lecture 1: Main Models & Basics of Wasserstein Distance J. A. Carrillo ICREA - Universitat Autònoma de Barcelona Methods and Models of Kinetic Theory

Outline 1 Presentation of models Nonlinear diffusions Kinetic models for granular gases 2 Wasserstein Distance: Basics Definition Properties 3 Contractivity in 1D Diffusive Models Dissipative Models Towards any dimension

Leading examples Nonlinear Diffusions.- u t = um, (x R d, t > 0) Inelastic Dissipative Models: Nonlinear friction equations.- 1 : f t + v f x =»Z (v w) v w γ( v w )f (x, w, t) dw f (x, v, t) v R Inelastic Dissipative Models: Nonlinear Boltzmann-type kinetic equations.- f + (v x)f =Qe(f, f ) t Conservation of mass and center of mass/mean velocity. Spreading (diffusions) versus Concentration (dissipative models). 1 Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997); Benedetto, D., Caglioti, E., Carrillo, J. A., Pulvirenti, M., J. Stat. Phys. (1998).

Nonlinear diffusions Outline 1 Presentation of models Nonlinear diffusions Kinetic models for granular gases 2 Wasserstein Distance: Basics Definition Properties 3 Contractivity in 1D Diffusive Models Dissipative Models Towards any dimension

Nonlinear diffusions Nonlinear diffusions Self-similar solution: Barenblatt profile 2.- An explicit self-similar solution that is integrable for m > (d 2)/d: B m(x, t) = t d/λ C 2m x 2 1 m t 2/λ «1/(1 m) for m 1 where λ = d(m 1) + 2 and C > 0 is determined to have unit mass. It verifies that B m(x, t) converges weakly-* as measures towards δ 0 as t 0 +. + 2 Zeldovich, Ya. B., Barenblatt, G. I. Doklady, USSR Academy of Sciences (1958).

Nonlinear diffusions Nonlinear diffusions Asymptotic behaviour: Comparison methods.- Friedman & Kamin (1980) (completed by Vázquez (1997)). Given an initial data in the class then, for any (d 2)/d < m X 0 = {u 0 L 1 (R d ) : u 0 0}, lim t u(, t) Bm(, t) L 1 = 0, and lim t td/λ u(, t) B m(, t) L = 0.

Nonlinear diffusions Nonlinear diffusions ρ t = div(xρ + ρm ), (x R d, t > 0) ρ(x, t = 0) = ρ 0(x) 0, (x R d ) ρ(x, t) = e dt u(e t x, 1 λ (eλt 1)) λ = d(m 1) + 2 u t = um, (x R d, t > 0) u(x, t = 0) = u 0(x) 0, (x R d ) 3 Exponential decay to equilibria translates into algebraic decay to Barenblatt profiles. 3 J. 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.

Nonlinear diffusions Nonlinear diffusions Generic finite speed of propagation/moving free boundary for degenerate diffusion equations and creation of thick tails for fast diffusion equations. No better rate of convergence can be established under the generality u 0 a probability density. At which rate does this self-similarity take over? General nonlinear diffusion equation: 8 >< >: No explicit source-type solutions, so... What is the typical asymptotic profile? u t = P(u), (x Rd, t > 0), u(x, t = 0) = u 0(x) 0 (x R d )

Kinetic models for granular gases Outline 1 Presentation of models Nonlinear diffusions Kinetic models for granular gases 2 Wasserstein Distance: Basics Definition Properties 3 Contractivity in 1D Diffusive Models Dissipative Models Towards any dimension

Kinetic models for granular gases Rapid Granular Flows Pattern formation in a vertically oscillated granular layer. 4 4 Bizon, C., Shattuck, M. D., Swift, J.B., Swinney, H.L., Phys. Rev. E (1999); Carrillo, J.A., Poschel, T., Salueña, C., in preparation (2006).

Kinetic models for granular gases Rapid Granular Flows Schock waves in supersonic sand. 5 5 Rericha, E., Bizon, C., Shattuck, M. D., Swinney, H.L., Phys. Rev. Letters (2002).

Kinetic models for granular gases Inelastic Collisions Binary Collisions: Spheres of diameter r > 0. Given (x, v) and (x rn, w), where n S 2 is the unit vector along the impact direction, the post-collisional velocities are found assuming conservation 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 = 1 u (v + w) + 2 2 w = 1 u (v + w) 2 2 where u = u (1 + e)(u n)n, u = v w and u = v w.

Kinetic models for granular gases Inelastic Collisions Postcollisional velocities: New Parameterization where v = 1 u (v + w) + 2 2 w = 1 u (v + w) 2 2 u = 1 e 4 u + 1 + e 4 u σ, u = v w and u = v w.

Kinetic models for granular gases Boltzmann equation for granular gases Hypothesis: Binary, localized in t and x and inelastic collisions. Molecular chaos. Boltzmann equation for inelastic particles: f t Z Z» 1 1 + (v x)f =Qe(f, f )= ((v w) n) 4π R 3 S+ 2 e f 2 (v )f (w ) f (v)f (w) dndw. a Weak form of the Boltzmann equation: < ϕ, Q e(f, f ) >= 1 Z Z Z h i v w f (v)f (w) ϕ(v ) ϕ(v) dσ dv dw 4π S 2 R 3 R 3 a Jenkins, J. T., Richman, M. W., Arch. Rat. Mech. Anal. (1985).

Kinetic models for granular gases Basic Properties 6 f t = Qe(f, f ) Mass and momentum are preserved while energy decreases: v 2 + w 2 v 2 w 2 = 1 e2 u 2 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 4 What are the typical asymptotic profiles? Do self-similar solutions exist? If so At which rate does this self-similarity take over? θ 3 2, 6 J.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).

Kinetic models for granular gases Simplified Granular media model Another dissipative models (toy models) keeping the same properties: conservation of mass, mean velocity and dissipation of energy are of the form: where f + (v x)f = I(f, f ) t I(f, f ) = div v (v w) v w f (x, w, t) dw f (x, v, t).»zr 3 This dissipative collision in the one dimensional case simplifies to 7 : I(f, f ) =»Z (v w) v w f (x, w, t) dw f (x, v, t) v R 7 Benedetto, D., Caglioti, E., Pulvirenti, M., Math. Mod. and Num. An. (1997).

Kinetic models for granular gases Summary: Spreading versus Concentration Diffusive models - Spreading: u t = P(u), (x Rd, t > 0) Typical asymptotic profiles for P(u) = u m (Barenblatt): with θ m(t). B m(x, t) = θ m(t) d/2 B m(θ m(t) 1/2 x, t o,m) Homogeneous Kinetic Dissipative models - Concentration: 8 >< Q e(f, f ) f t = Z >: div v»f (v, t) W(v w)f (w, t) dw R d Typical asymptotic profiles (Homogeneous Cooling States), if they exist: with θ hc(t) 0. f hc(v, t) = ρ θ 2 3 hc (t) g ((v u) θ 2 1 hc (t))

Definition Outline 1 Presentation of models Nonlinear diffusions Kinetic models for granular gases 2 Wasserstein Distance: Basics Definition Properties 3 Contractivity in 1D Diffusive Models Dissipative Models Towards any dimension

Definition Definition of the distance 8 Transporting measures: Given T : R d R d mesurable, we say that T transports µ P onto ν P or that ν is the push-forward of µ through T, ν = T#µ, if ν[k] := µ[t 1 (K)] for all mesurable sets K R d, equivalently Z Z ϕ dν = (ϕ T) dµ R d R d for all ϕ C o(r d ). Random variables: Say that X is a random variable with law given by µ, is to say X : (Ω, A, P) (R d, B d) is a mesurable map such that X#P = µ, i.e., Z Z ϕ(x) dµ = (ϕ X) dp = E [ϕ(x)]. R d Ω Coming back to the transport of measures: if X and Y are random variables with law µ and ν respectively, then ν = T#µ is equivalent to Y = T(X). 8 C. Villani, AMS Graduate Texts (2003).

Definition Definition of the distance Euclidean Wasserstein jzz Distance: ff 1/2 n W 2(µ, ν)= inf x y 2 dπ(x, y) = inf E h X Y 2io 1/2 π R d R d (X,Y) where the transference plan π runs over the set of joint probability measures on R d R d with marginals f and g P 2(R d ) and (X, Y) are all possible couples of random variables with µ and ν as respective laws. Monge s optimal mass transport problem: Find jz ff 1/2 I := inf x T(x) 2 dµ(x); ν = T#µ. T R d Take γ T = (1 R d T)#µ as transference plan π.

Definition Definition of the distance Wasserstein Distances: Given 1 p <, we define jzz ff 1/p W p(µ, ν)= inf x y p dπ(x, y) = inf {E [ X π R d R d (X,Y) Y p ]} 1/p where the transference plan π runs over the set of joint probability measures on R d R d with marginals f and g P p(r d ) and (X, Y) are all possible couples of random variables with µ and ν as respective laws. We define the -Wasserstein distance as W (µ, ν) = lim Wp(µ, ν) = inf {esssup { x T(x), x supp(µ)} with ν = T#µ} p T for measures with all moments bounded.

Properties Outline 1 Presentation of models Nonlinear diffusions Kinetic models for granular gases 2 Wasserstein Distance: Basics Definition Properties 3 Contractivity in 1D Diffusive Models Dissipative Models Towards any dimension

Properties Euclidean Wasserstein Distance Basic Properties 1 Convexity: f 1, f 2, g 1, g 2 P 2(R d ) and α [0, 1], then W 2 2 (αf 1 + (1 α)f 2, αg 1 + (1 α)g 2) αw 2 2 (f 1, g 1) + (1 α)w 2 2 (f 2, g 2). 2 Relation to Temperature: f P 2(R d ) and a R 3, then Z W2 2 (f, δ a) = v a 2 df (v). R d 3 Scaling: Given f in P 2(R d ) and θ > 0, let us define S θ [f ] = θ d/2 f (θ 1/2 v) for absolutely continuous measures with respect to Lebesgue or its corresponding definition by duality for general measures; then for any f and g in P 2(R d ), we have W 2(S θ [f ], S θ [g]) = θ 1/2 W 2(f, g).

Properties Euclidean Wasserstein Distance Convergence Properties 1 Convergence of measures: W 2(f n, f ) 0 is equivalent to f n f weakly-* as measures and convergence of second moments. 2 Weak lower semicontinuity: Given f n f and g n g weakly-* as measures, then W 2(f, g) lim inf W2(fn, gn). n 3 Completeness: The space P 2(R d ) endowed with the distance W 2 is a complete metric space. Moreover, the set j Z ff M θ = µ P 2(R d ) such that v 2 df (v) = θ R 3, i.e. the sphere of radius θ in P 2(R d ), is a complete metric space.

Properties Wasserstein Distances Properties 1 Convexity of W p p 2 Scaling: for any f and g in P p(r d ), we have W p(s θ [f ], S θ [g]) = θ 1/2 W p(f, g). 3 Completeness and weak lower semicontinuity. 4 Convergence weak-* convergence as measures and convergence of p-moments.

Properties One dimensional Case Distribution functions: In one dimension, denoting by F(x) the distribution function of µ, F(x) = we can define its pseudo inverse: F 1 (η) = inf{x : F(x) > η} for η (0, 1), we have F 1 : ((0, 1), B 1), dη) (R, B 1) is a random variable with law µ, i.e., F 1 #dη = µ Z Z 1 ϕ(x) dµ = ϕ(f 1 (η)) dη = E [ϕ(x)]. R 0 Z x dµ,

Properties One dimensional Case Wasserstein distance: Given µ and ν P 2(R d ) with distribution functions F(x) and G(y) then, ZZ W2 2 (µ, ν) = x y 2 dh(x, y) R d R d where H(x, y) = min(f(x), G(y)) is a joint distribution function and the above integral is defined in the Riemann-Stieljes sense. a In one dimension, it is easy to check that given two measures µ and ν with distribution functions F(x) and G(y) then, (F 1 G 1 )#dη has joint distribution function H(x, y) = min(f(x), G(y)). Therefore, in one dimension, the optimal plan is given by π opt(x, y) = (F 1 G 1 )#dη, and thus Z 1 W 2(µ, ν) = [F 1 (η) G 1 (η)] 2 dη 0 «1/2 a W. Hoeffding (1940); M. Fréchet (1951); A. Pulvirenti, G. Toscani, Annali Mat. Pura Appl. (1996).

Properties One dimensional Case Wasserstein distance: Given µ and ν P p(r), 1 p < then and Z 1 1/p W p(µ, ν) = [F 1 (η) G 1 (η)] dη«p = F 1 G 1 L p (R) 0 W (µ, ν) = F 1 G 1 L (R).

Properties Relation 1D-anyD Euclidean Wasserstein distance: If the probability measures µ i j on R are the successive one-dimensional marginals of the measure µ i on R d, for i = 1, 2 and j = 1,..., d, then dx W2 2 (µ 1 j, µ 2 j ) W2 2 (µ 1, µ 2 ). j=1

Properties Relation 1D-anyD Proof. Let π be a measure on R d x R d y with marginals µ 1 and µ 2, optimal in the sense that ZZ W2 2 (µ 1, µ 2 ) = x y 2 dπ(x, y). R d R d Then its marginal π j on R vj R wj has itself marginals µ 1 j and µ 2 j, so ZZ W2 2 (µ 1 j, µ 2 j ) v j w j 2 dπ j(v j, w j). R R The inequality follows by noting that dx v j w j 2 = v w 2. j=1

Diffusive Models Outline 1 Presentation of models Nonlinear diffusions Kinetic models for granular gases 2 Wasserstein Distance: Basics Definition Properties 3 Contractivity in 1D Diffusive Models Dissipative Models Towards any dimension

Diffusive Models 1D Nonlinear Diffusions 9 The flows of nonlinear diffusions are contractions in Wasserstein distance. u t = (P(u)) xx u(x, 0) = u 0(x) 0 L 1 (R) with P : R + 0 R, continuous, nondecreasing with P(0) = 0. Diccionary for F 1 (η): F 1 (η, t) = η F 1 t» 1 u (η, t) = (F 1 (η, t), t),» 1 (F 1 (η, t), t) F u t (F 1 (η, t), t) 2 F 1 h ux i (η, t) = (F 1 (η, t), t) η 2 u 3 Equation for F 1 : F 1 (η, t) =» P t η» F 1 η «1, 9 H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).

Diffusive Models 1D Nonlinear Diffusions 10 Contraction for nonlinear diffusions: Given any two solutions of the nonlinear diffusion equation W p(u 1(t), u 2(t)) W p(u 1(0), u 2(0)) for all 1 p. Proof. Assume 1 < p <, the other ones by approximation in p, then 1 d p(p 1) dt Z 1 0 Z 1 0 F 1 G 1 p 2 (F 1 η F 1 G 1 p dη = j G 1 η ) P» 1 Fη 1 P» ff 1 G 1 η dη 0 10 H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).

Diffusive Models 1D Nonlinear Diffusions 11 Strict Contraction for nonlinear diffusions with confining potential: Given any two probability density solutions of the nonlinear diffusion equation «dv ρ t = (t, x)ρ + (P(ρ)) dx xx where V(t, x) is a smooth strictly uniform convex potential ( d2 V dx 2 (t, x) α > 0), then for all 1 p. W p(ρ 1(t), ρ 2(t)) W p(ρ 1(0), ρ 2(0)) e αt x Proof. Exercise; F 1 (η, t) = dv t dx (t, F 1 )» P η» F 1 η «1, The case of V(x) = x 2 /2 and P(ρ) = ρ m comes by the scaling property of the Wasserstein distances and the previous non strict contraction. 11 H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).

Diffusive Models 1D Nonlinear Diffusions Convergence towards equilibrium: All probability solutions of the equation «dv ρ t = (t, x)ρ + (P(ρ)) dx xx converges exponentially fast towards the unique equilibrium ρ characterized by x V(x) + h(ρ (x)) = C ρh (ρ) = P (ρ). Here, P is strictly increasing for being h invertible.

Diffusive Models 1D Nonlinear Diffusions 12 Uniform Finite Speed of Propagation: Given any two probability density solutions of the degenerate nonlinear diffusion equation ρ t = (ρ m ) xx, we have the following control of the "relative speed" of the support spreading for any t > 0. inf{supp{ρ 1(t)}} inf{supp{ρ 2(t)}} W (ρ 1(0), ρ 2(0)) sup{supp{ρ 1(t)}} sup{supp{ρ 2(t)}} W (ρ 1(0), ρ 2(0)) 12 J.A. Carrillo, M.P. Gualdani, G. Toscani, CRAS (2004).

Dissipative Models Outline 1 Presentation of models Nonlinear diffusions Kinetic models for granular gases 2 Wasserstein Distance: Basics Definition Properties 3 Contractivity in 1D Diffusive Models Dissipative Models Towards any dimension

Dissipative Models 1D Dissipative Kinetic Equations 13 Given the 1D granular model f t =»Z (v w) v w γ f (w, t) dw f (v, t) v R Equation for F 1 : F 1 (η, t) = t Z 1 After some manipulations (good exercise) 0 F 1 (η, t) F 1 (p, t) γ (F 1 (η, t) F 1 (p, t)) dp. implying d 1 W2(f (t), g(t)) W2(f (t), g(t))1+γ dt 2γ 1 W 2(f (t), g(t)) 2 γ 1 «1/γ. γt + 2 γ 1 W 2(f (0), g(0)) γ 13 H. Li, G. Toscani, ARMA (2004);J.A. Carrillo, G. Toscani, proceedings conference in honor of S. Rionero (2005).

Towards any dimension Outline 1 Presentation of models Nonlinear diffusions Kinetic models for granular gases 2 Wasserstein Distance: Basics Definition Properties 3 Contractivity in 1D Diffusive Models Dissipative Models Towards any dimension

Towards any dimension Heat Equation 14 Contraction for the Heat Equation: Given any two probability density solutions u 1(x, t), u 2(x, t) of the heat equation, then W p(u 1(t), u 2(t)) W p(u 1(0), u 2(0)) for t 0, 1 p. Proof. Let γ o the optimal transference plan between u 1(0) y u 2(0) for W p, 1 p <. The solutions are u 1(t) = K(t, x) u 1(0) and u 2(t) = K(t, x) u 2(0). Let uszdefine γ t by duality: Z Z ϕ(x, y) dγ t(x, y) = ϕ(x + z, y + z)k(t, z) dz dγ o(x, y) R d R d R d R d R d Check that γ t is a transference plan between u 1(t) and u 2(t), and thus Wp p (u 1(t), u 2(t)) Z x y p dγ t(x, y)= R d R d Z x y p dγ o(x, y)=wp p (u 1(0), u 2(0)). R d R d 14 J.A. Carrillo, Boletín SEMA (2004).

Towards any dimension Homogeneous Boltzmann Equation 15 Convergence Towards Monokinetic State: Given f (v, t) P 2(R 3 ) with zero mean velocity then para t 0. Proof. Evolution of temperature: < v 2, Q e(f, f ) >= 1 e2 4 W 2(f (t), δ 0) C(1 + t) 2 Z R 3 Z R 3 v w 3 f (v)f (w) dv dw and thus using Jensen s and Hölder s inequality, we get the Haff s law: dθ dt 1 e2 4 Using the relation to temperature of the Euclidean Wasserstein distance, we conclude. θ 3 2. 15 J.A. Carrillo, Boletín SEMA (2004).