Spatially Inhomogeneous Evolutionary Games€¦ · Spatially Inhomogeneous Evolutionary Games Massimo Fornasier Fakult at fur Mathematik Technische Universit at Munchen [email protected]

Spatially Inhomogeneous Evolutionary Games

Massimo Fornasier

Fakultät für MathematikTechnische Universität München

[email protected]://www-m15.ma.tum.de/

Applied Mathematics Seminar, FAUMay 12, 2020

Joint work with Luigi Ambrosio, Nathanael Bosch, Marco Morandotti, and

Giuseppe Savaré

Game equilibria: why are they so important?

I Physical systems naturally tend to minimize the potentialenergy ⇒ steady states;

I Game theorists have focused on the characterization of gameequilibria;

I Nash equilibria (1951): building on von Neumann’s notion ofmixed strategy (necessary to ensure existence of saddle pointsin zero sum games with two players);

I Morgenstern and von Neumann pointed out in their classicaltreatise on game theory (1947) the desirability of a“dynamical” approach to complement their “static” gamesolution concept;

I In physical systems evolutions towards minima of the potentialenergy are explained according to Newton’s law ⇒ it is not atall clear whether and how in dynamical games equilibria canemerge.

I In physical systems evolutions towards minima of the potentialenergy are explained according to Newton’s law

⇒ it is not atall clear whether and how in dynamical games equilibria canemerge.

What would be a good notion of evolutionary game?

I One of most advocated mechanisms of dynamical choice ofstrategies is based on a selection principle, inspired byDarwinian evolution concepts;

I The main idea is to re-interpret the probability of picking acertain strategy with the distribution of a population ofplayers adopting those strategies;

I The emergence of steady mixed strategies would be the resultof an evolutionary selection: at discrete times players meetrandomly, interact according to their strategies, and obtain apayoff;

I This payoff determines how the frequencies in the strategieswill evolve.

Replicator dynamicsI Players can adopt pure strategies u1, . . . , uN ∈ U (for us U is

a compact set in a suitable metric space), where U is the setof strategies;

I Denote with σi the frequency with which players pick thestrategy ui ;

I The payoff of playing strategy ui against uj will be denoted byJ(ui , uj), where J : U × U → R;

I The relative success of the strategy ui with respect to thestrategies played by the population is measured by

∆N(ui ) =N∑j=1

J(ui , uj)σj−N∑`=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N;

I The relative rate of change of usage of the strategy ui is thendescribed by

σ̇iσi

= ∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N

∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N;

σ̇iσi

= ∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N

∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N;

σ̇iσi

= ∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N

∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N;

σ̇iσi

= ∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N

∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N;

σ̇iσi

= ∆N(ui ) =N∑j=1

N∑j=1

σ`J(u`, uj)σj , i = 1, . . . ,N

Folk theorem of evolutionary game theoryω-limits (the set of accumulation points of the dynamics) andsteady states are closely related to the Nash equilibria of the gamedescribed by the payoff matrix A = (J(ui , uj))ij :

Theorem

(a) if σ̄ is a Nash equilibrium, then it is a rest point;

(b) if σ̄ is a strict Nash equilibrium, then it is asymptotically stable;

(c) if the rest point σ̄ is the limit of an interior orbit of the symplex (an orbitσ(t) ∈ int(SN)), then σ̄ is a Nash equilibrium; and

(d) if the rest point σ̄ is stable, then it is a Nash equilibrium.

Theorem

Mean-field approximation for N →∞Consider σNt :=

∑Nj=1 σj ,tδuj ∈P(U),

and its evolution accordingto

σ̇Nt =

(∫U

J(·, u′)dσNt (u′)−∫U×U

J(w , u′) dσNt (w)dσNt (u

′)

)σNt .

For any σ ∈P(U), we may denote

∆σ(u) :=

(∫U

J(u, u′)dσ(u′)−∫U×U

J(w , u′)dσ(w)dσ(u′)

),

so that ∆σN (ui ) = ∆N(ui ). By assuming that the initial conditionsσN0 ⇀ σ̄ for a given σ̄ ∈P(U), one can show that σNt ⇀ σt forN →∞ for any t, where σ is the solution to

d

dt

∫Uϕ(u)dσt(u) =

∫Uϕ(u)∆σ(u)dσt(u), σ(0) = σ̄,

for any ϕ ∈ C(U).

∑Nj=1 σj ,tδuj ∈P(U), and its evolution according

to

σ̇Nt =

(∫U

′)

)σNt .

∆σ(u) :=

(∫U

J(u, u′)dσ(u′)−∫U×U

),

d

dt

∫Uϕ(u)dσt(u) =

∫Uϕ(u)∆σ(u)dσt(u), σ(0) = σ̄,

to

σ̇Nt =

(∫U

′)

)σNt .

∆σ(u) :=

(∫U

J(u, u′) dσ(u′)−∫U×U

),

so that ∆σN (ui ) = ∆N(ui ).

By assuming that the initial conditionsσN0 ⇀ σ̄ for a given σ̄ ∈P(U), one can show that σNt ⇀ σt forN →∞ for any t, where σ is the solution to

d

dt

∫Uϕ(u)dσt(u) =

∫Uϕ(u)∆σ(u)dσt(u), σ(0) = σ̄,

to

σ̇Nt =

(∫U

′)

)σNt .

∆σ(u) :=

(∫U

J(u, u′) dσ(u′)−∫U×U

),

d

dt

∫Uϕ(u) dσt(u) =

∫Uϕ(u)∆σ(u) dσt(u), σ(0) = σ̄,

Hellinger-Kakutani distancesA reaction distance via dynamic interpolation:

H2(µ0, µ1) = inf

{1

4

∫ 10

∫|wt |2dµtdt : µ ∈ C([0, 1],M+U),

∂tµt = wtµt , µt=i = µi

}

=

∫(√

p0 −√

p1)2 dµ∗, pi :=

dµidµ∗

.

In order to establish a similar distance for probability measures,one introduces the so-called spherical Hellinger-Kakutani distance

HS2(σ0, σ1) = inf

{1

4

∫ 10

∫|wt |2dσtdt : σ ∈ C([0, 1],P(U)),

∂tσt =

(wt−

(∫wtdσt

))σt , σt=i = σi

}

= arccos

(1− H

2(σ0, σ1)2

2

)= arcsin

(√1−

∫ √p0√

p1dσ∗

)2

H2(µ0, µ1) = inf

{1

4

∫ 10

∫|wt |2dµtdt : µ ∈ C([0, 1],M+U),

}

=

∫(√

p0 −√

p1)2 dµ∗, pi :=

dµidµ∗

.

HS2(σ0, σ1) = inf

{1

4

∫ 10

∫|wt |2dσtdt : σ ∈ C([0, 1],P(U)),

∂tσt =

(wt−

(∫wtdσt

))σt , σt=i = σi

}

= arccos

(1− H

2(σ0, σ1)2

2

)= arcsin

(√1−

∫ √p0√

p1dσ∗

)2

H2(µ0, µ1) = inf

{1

4

∫ 10

∫|wt |2dµtdt : µ ∈ C([0, 1],M+U),

}

=

∫(√

p0 −√

p1)2 dµ∗, pi :=

dµidµ∗

.

HS2(σ0, σ1) = inf

{1

4

∫ 10

∫|wt |2dσtdt : σ ∈ C([0, 1],P(U)),

∂tσt =

(wt−

(∫wtdσt

))σt , σt=i = σi

}

= arccos

(1− H

2(σ0, σ1)2

2

)= arcsin

(√1−

∫ √p0√

p1dσ∗

)2

H2(µ0, µ1) = inf

{1

4

∫ 10

∫|wt |2dµtdt : µ ∈ C([0, 1],M+U),

}

=

∫(√

p0 −√

p1)2 dµ∗, pi :=

dµidµ∗

.

HS2(σ0, σ1) = inf

{1

4

∫ 10

∫|wt |2dσtdt : σ ∈ C([0, 1],P(U)),

∂tσt =

(wt−

(∫wtdσt

))σt , σt=i = σi

}

= arccos

(1− H

2(σ0, σ1)2

2

)= arcsin

(√1−

∫ √p0√

p1dσ∗

)2

Replicator dynamics as Hellinger flow

For a functional J : P(U)→ R we consider the minimizingmovement scheme:

given τ > 0 and σ0τ = σ0, recursively define σnτ

as the minimizing solution of

minσ∈P(U)

1

2τHS2(σ, σn−1τ ) + J (σ).

Hence, σt = limN→∞ σNt/N solves the equation

∂tσt = −(δJδσ

(u, σt)−∫δJδσ

(w , σt)dσt(w)

)σt

If J (σ) = −12∫U

∫U J(u, u

′)dσ(u)dσ(u′) (and symmetric J) thenone obtains again the replication dynamics.

For a functional J : P(U)→ R we consider the minimizingmovement scheme: given τ > 0 and σ0τ = σ0, recursively define σ

nτ

minσ∈P(U)

1

2τHS2(σ, σn−1τ ) + J (σ).

(w , σt)dσt(w)

)σt

If J (σ) = −12∫U

∫U J(u, u

For a functional J : P(U)→ R we consider the minimizingmovement scheme: given τ > 0 and σ0τ = σ0, recursively define σ

nτ

minσ∈P(U)

1

2τHS2(σ, σn−1τ ) + J (σ).

(w , σt)dσt(w)

)σt

If J (σ) = −12∫U

∫U J(u, u

Spatially inhomogeneous evolutionary games

Idea of a spatially inhomogeneous replicator dynamics:

I A population of players is distributed over a position space Rdand they are each endowed with probability distributions ofstrategies (mixed strategies) σ ∈P(U), from which theydraw at random pure strategies to evolve their positions.

I Each player’s probability distribution of strategies σ ∈P(U)evolves in time according to a replicator dynamics, which takesinto account all other players and respective mixed strategies.

Spatially inhomogeneous evolutionary games

Idea of a spatially inhomogeneous replicator dynamics:

I A population of players is distributed over a position space Rdand they are each endowed with probability distributions ofstrategies (mixed strategies) σ ∈P(U), from which theydraw at random pure strategies to evolve their positions.

I Each player’s probability distribution of strategies σ ∈P(U)evolves in time according to a replicator dynamics, which takesinto account all other players and respective mixed strategies.

A combined dynamics: informal derivation

Fix a Lipschitz payoff function J : (Rd × U)2 → R, a time interval[0,T ], a time step h = T/n.

At time t = ih we consider anempirical distribution of N players µNt =

1N

∑Nj=1 δxj (t) ∈P(R

d),

so that each player x ∈ supp(µNt ) is endowed with a correspondingdistribution of strategies σx ,t ∈P(U) and ηt = σx ,tµNt is thedisintegration of ηt ∈P(Rd × U). We define formally thedeterministic/stochastic evolution of the individual player and itsmixed strategies by

σx ,t+h = (1 + h∆ηt (x , u))σx ,t

x(t + h) = x(t) + he(x(t), u), for u ∼ σx ,t+h,

where

∆ηt (x , u) =

∫Rd×U

J(x , u, x ′, u′)dηt(x′, u′)−

∫U

∫Rd×U

J(x , u, x ′, u′)dηt(x′, u′)dσx,t(u).

Then one formally sets σx+h,t+h := σx,t+h.

Fix a Lipschitz payoff function J : (Rd × U)2 → R, a time interval[0,T ], a time step h = T/n. At time t = ih we consider anempirical distribution of N players µNt =

1N

d),

so that each player x ∈ supp(µNt ) is endowed with a correspondingdistribution of strategies σx ,t ∈P(U) and ηt = σx ,tµNt is thedisintegration of ηt ∈P(Rd × U).

We define formally thedeterministic/stochastic evolution of the individual player and itsmixed strategies by

σx ,t+h = (1 + h∆ηt (x , u))σx ,t

where

∆ηt (x , u) =

∫Rd×U

J(x , u, x ′, u′)dηt(x′, u′)−

∫U

∫Rd×U

Fix a Lipschitz payoff function J : (Rd × U)2 → R, a time interval[0,T ], a time step h = T/n. At time t = ih we consider anempirical distribution of N players µNt =

1N

d),

so that each player x ∈ supp(µNt ) is endowed with a correspondingdistribution of strategies σx ,t ∈P(U) and ηt = σx ,tµNt is thedisintegration of ηt ∈P(Rd × U). We define formally thedeterministic/stochastic evolution of the individual player and itsmixed strategies by

σx ,t+h = (1 + h∆ηt (x , u))σx ,t

where

∆ηt (x , u) =

∫Rd×U

J(x , u, x ′, u′)dηt(x′, u′)−

∫U

∫Rd×U

A combined dynamics: informal derivationGiven now a bounded test function Φ: Rd × U → R∫

Φ(x , u)d(ηt+h − ηt)(x , u)

=1

N

N∑j=1

(∫Φ(xj(t + h), u)dσxj (t+h),t+h(u)−

∫Φ(xj(t), u)dσxj (t),t(u)

)

=1

N

N∑j=1

(∫(Φ(xj(t + h), u)− Φ(xj(t), u))dσxj (t+h),t+h(u)

+

∫Φ(xj(t), u)d(σxj (t+h),t+h(u)− σxj (t),t(u))

)

≈ hN

N∑j=1

(∫∇xΦ(xj(t), u) ·

(∫e(xj(t), v)dσxj (t),t(v)

)dσxj (t+h),t+h(u)

+

∫Φ(xj(t), u)∆ηt (xj(t), u)dσxj (t),t(u)

)

= h

∫ (∇xΦ(x , u) ·

(∫e(x , v)dσx,t(v)

)+ Φ(x(t), u)∆ηt (x(t), u)

)dηt(x , u)

=1

N

N∑j=1

)

=1

N

N∑j=1

+

)

≈ hN

N∑j=1

(∫∇xΦ(xj(t), u) ·

)dσxj (t+h),t+h(u)

+

)

= h

∫ (∇xΦ(x , u) ·

)+ Φ(x(t), u)∆ηt (x(t), u)

)dηt(x , u)

=1

N

N∑j=1

)

=1

N

N∑j=1

+

)

≈ hN

N∑j=1

(∫∇xΦ(xj(t), u) ·

)dσxj (t+h),t+h(u)

+

)

= h

∫ (∇xΦ(x , u) ·

)+ Φ(x(t), u)∆ηt (x(t), u)

)dηt(x , u)

=1

N

N∑j=1

)

=1

N

N∑j=1

+

)

≈ hN

N∑j=1

(∫∇xΦ(xj(t), u) ·

)dσxj (t+h),t+h(u)

+

)

= h

∫ (∇xΦ(x , u) ·

)+ Φ(x(t), u)∆ηt (x(t), u)

)dηt(x , u)

=1

N

N∑j=1

)

=1

N

N∑j=1

+

)

≈ hN

N∑j=1

(∫∇xΦ(xj(t), u) ·

)dσxj (t+h),t+h(u)

+

)

= h

∫ (∇xΦ(x , u) ·

)+ Φ(x(t), u)∆ηt (x(t), u)

)dηt(x , u)

An evolution of measures

For h→ 0 and N →∞ we formally derive the following PDE for η:

∂tηt +∇x[(∫

e(x , v)dσx ,t(v)

)ηt

]−∆ηt (x , u)ηt(x , u) = 0.

By integrating against tests functions Φ(x , u) = Φ(x) one woulddeduce the equation for the marginal µt

∂tµt+∇x[(∫

e(x , v)dσx ,t(v)

)µt

]︸︷︷︸

transport term

−(∫

∆ηt (x , v)dσt(x , v)

)︸︷︷︸

≡0

µt = 0.

How can we get a modellistically consistent evolution, whichcombines replicator dynamics and player’s moves? We need tomake a further abstraction step ...

∂tηt +∇x[(∫

e(x , v)dσx ,t(v)

)ηt

]−∆ηt (x , u)ηt(x , u) = 0.

∂tµt+∇x[(∫

e(x , v)dσx ,t(v)

)µt

]︸︷︷︸

transport term

−(∫

)︸︷︷︸

≡0

µt = 0.

∂tηt +∇x[(∫

e(x , v)dσx ,t(v)

)ηt

]−∆ηt (x , u)ηt(x , u) = 0.

∂tµt+∇x[(∫

e(x , v)dσx ,t(v)

)µt

]︸︷︷︸

transport term

−(∫

)︸︷︷︸

≡0

µt = 0.

How can we get a modellistically consistent evolution, whichcombines replicator dynamics and player’s moves?

We need tomake a further abstraction step ...

∂tηt +∇x[(∫

e(x , v)dσx ,t(v)

)ηt

]−∆ηt (x , u)ηt(x , u) = 0.

∂tµt+∇x[(∫

e(x , v)dσx ,t(v)

)µt

]︸︷︷︸

transport term

−(∫

)︸︷︷︸

≡0

µt = 0.

Inconsistency with finite particle approximation

In order to understand the inconsistency, let us consider thesimplest situation where η is concentrated in two distinct positionsx1, x2, e.g.

η =1

2δx1 ⊗ σ1 +

1

2δx2 ⊗ σ2, σi ∈P(U).

We thus find

∆η(xi , u) = Z (xi , u)−∫U

Z (xi , v) dσi (v) where

Z (xi , u) =

∫Rd×U

J(x , u, x ′, u′) dη(x ′, u′).

Clearly Z depends continuously on η w.r.t. the weak topology.

In order to understand the inconsistency, let us consider thesimplest situation where η is concentrated in two distinct positionsx1, x2, e.g.

η =1

2δx1 ⊗ σ1 +

1

2δx2 ⊗ σ2, σi ∈P(U).

We thus find

∆η(xi , u) = Z (xi , u)−∫U

Z (xi , v) dσi (v) where

Z (xi , u) =

∫Rd×U

J(x , u, x ′, u′) dη(x ′, u′).

Clearly Z depends continuously on η w.r.t. the weak topology.

However, when (x1, x2)→ (x , x) we get a discontinuous behaviourof ∆.

In fact, if x1 = x2 = x the previous equations read as

η = δx⊗σ, σ =1

2(σ1+σ2), ∆η(x , u) = Z (x , u)−

∫U

Z (x , v) dσ

but it is also

lim(x1,x2)→(x ,x)

∆η(xi , u) = Z (x , u)−∫U

Z (x , v)dσi (v) 6= ∆η(x , u).

However, when (x1, x2)→ (x , x) we get a discontinuous behaviourof ∆.In fact, if x1 = x2 = x the previous equations read as

η = δx⊗σ, σ =1

2(σ1+σ2), ∆η(x , u) = Z (x , u)−

∫U

Z (x , v) dσ

but it is also

lim(x1,x2)→(x ,x)

∆η(xi , u) = Z (x , u)−∫U

Z (x , v)dσi (v) 6= ∆η(x , u).

However, when (x1, x2)→ (x , x) we get a discontinuous behaviourof ∆.In fact, if x1 = x2 = x the previous equations read as

η = δx⊗σ, σ =1

2(σ1+σ2), ∆η(x , u) = Z (x , u)−

∫U

Z (x , v) dσ

but it is also

lim(x1,x2)→(x ,x)

∆η(xi , u) = Z (x , u)−∫U

Z (x , v) dσi (v) 6= ∆η(x , u).

The correct approach

I The space of pairs of positions and mixed strategies isC := Rd ×P(U), whose elements are denoted y =(x , σ)describing the state of a player.

I The system will be described by the evolution of a measureΣ ∈P(C ) = P(Rd ×P(U)) on our state space, whichrepresents a distribution of players with strategies.

I we define the new interaction potential

∆Σ,(x,σ)(u) :=

∫C

∫U

J(x , u, x ′, u′)dσ′(u′) dΣ(x ′, σ′)

−∫U

∫C

∫U

J(x ,w , x ′, u′) dσ′(u′)dΣ(x ′, σ′) dσ(w).

∆Σ,(x,σ)(u) :=

∫C

∫U

J(x , u, x ′, u′)dσ′(u′) dΣ(x ′, σ′)

−∫U

∫C

∫U

∆Σ,(x,σ)(u) :=

∫C

∫U

J(x , u, x ′, u′)dσ′(u′) dΣ(x ′, σ′)

−∫U

∫C

∫U

I We fix a time interval [0,T ], a time step h = T/n and aninitial datum Σ̄ ∈P(Rd ×P(U)).

I For C = Rd ×P(U) we build a discrete solution

Mh ∈P(C([0,T];C))

concentrated on paths (x(t), σ(t)) : [0,T ]→ C , which arepiecewise affine (in the n intervals);

I If the player at time t = ih is in position x̄ , with mixedstrategy σ̄, first it upgrades t the probability replacing σ̄ by

σ̄′ := (1 + h∆Σt,h,(x̄ ,σ̄)) σ̄.

I The conditional probability relative to Mh|[0,t+h] of(x(t + h), σ(t + h)), given the information that at time t onehas (x , σ)(t) = (x̄ , σ̄), is ((x̄ + he(x̄ , ·))#σ̄′)× δσ̄′ .

I By iterating this process n times one can build Mh on [0,T ].

Mh ∈P(C([0,T];C))

σ̄′ := (1 + h∆Σt,h,(x̄ ,σ̄)) σ̄.

Mh ∈P(C([0,T];C))

σ̄′ := (1 + h∆Σt,h,(x̄ ,σ̄)) σ̄.

Mh ∈P(C([0,T];C))

σ̄′ := (1 + h∆Σt,h,(x̄ ,σ̄)) σ̄.

Mh ∈P(C([0,T];C))

σ̄′ := (1 + h∆Σt,h,(x̄ ,σ̄)) σ̄.

A combined dynamics: informal derivation II

For t = ih, 0 ≤ i ≤ n − 1, we denote Σt,h := (evt)#Mh, whereevt : C([0,T];C)→ C defined by evt(x , σ) := (x(t), σ(t)).

Givennow a bounded test function Φ: Rd ×P(U)→ R, one has∫

Φ(x , σ) d(Σt+h,h − Σt,h)(x , σ)

=

∫ [Φ(x(t + h), σ(t + h))− Φ(x(t), σ(t))

]dMh(x(·), σ(·))

=

∫ [Φ(x(t + h), (1 + h∆Σt,h,x(t),σ(t))σ(t))− Φ(x(t), σ(t))

]dMh(x(·), σ(·))

≈ h∫

DΦ(x(t), σ(t)) ·( ∫

e(x , u)dσ(t, u),∆Σt,h,x(t),σ(t)σ(t))dMh(x(·), σ(·))

= h

∫DΦ(x , σ) · bΣt,h (x , σ) dΣt,h(x , σ),

for the vector field

bΣ(x , σ) :=

(∫e(x , u)dσ(u),∆Σ,(x,σ) σ

).

For t = ih, 0 ≤ i ≤ n − 1, we denote Σt,h := (evt)#Mh, whereevt : C([0,T];C)→ C defined by evt(x , σ) := (x(t), σ(t)). Givennow a bounded test function Φ: Rd ×P(U)→ R, one has∫

=

∫ [Φ(x(t + h), σ(t + h))− Φ(x(t), σ(t))

]dMh(x(·), σ(·))

=

]dMh(x(·), σ(·))

≈ h∫

DΦ(x(t), σ(t)) ·( ∫

= h

bΣ(x , σ) :=

(∫e(x , u)dσ(u),∆Σ,(x,σ) σ

).

=

∫ [Φ(x(t + h), σ(t + h))− Φ(x(t), σ(t))

]dMh(x(·), σ(·))

=

]dMh(x(·), σ(·))

≈ h∫

DΦ(x(t), σ(t)) ·( ∫

= h

bΣ(x , σ) :=

(∫e(x , u)dσ(u),∆Σ,(x,σ) σ

).

=

∫ [Φ(x(t + h), σ(t + h))− Φ(x(t), σ(t))

]dMh(x(·), σ(·))

=

]dMh(x(·), σ(·))

≈ h∫

DΦ(x(t), σ(t)) ·( ∫

= h

bΣ(x , σ) :=

(∫e(x , u)dσ(u),∆Σ,(x,σ) σ

).

=

∫ [Φ(x(t + h), σ(t + h))− Φ(x(t), σ(t))

]dMh(x(·), σ(·))

=

]dMh(x(·), σ(·))

≈ h∫

DΦ(x(t), σ(t)) ·( ∫

= h

bΣ(x , σ) :=

(∫e(x , u)dσ(u),∆Σ,(x,σ) σ

).

A new master equation

This construction builds for h→ 0 a continuous mapΣ: [0,T ]→ C by Σt := (evt)#M, satisfying the equation

∂tΣt + divx,σ(bΣt Σt) = 0,

In particular, by disintegration Σt = πx ,tµt for µt ∈P(Rd) andπx ,t ∈P(P(U)), the marginal µt fulfills

∂tµt +∇x · (v(t, x)µt)︸︷︷︸transport

= 0,

where v(t, x) =∫P(U) e(x , u)dσ(u)dπx ,t(σ) is the “mean of the

mean” vector field of transport of µt .

A new master equation

This construction builds for h→ 0 a continuous mapΣ: [0,T ]→ C by Σt := (evt)#M, satisfying the equation

∂tΣt + divx,σ(bΣt Σt) = 0,

In particular, by disintegration Σt = πx ,tµt for µt ∈P(Rd) andπx ,t ∈P(P(U)), the marginal µt fulfills

∂tµt +∇x · (v(t, x)µt)︸︷︷︸transport

= 0,

where v(t, x) =∫P(U) e(x , u)dσ(u)dπx ,t(σ) is the “mean of the

mean” vector field of transport of µt .

ODEs in Banach spaces

I The characteristics are evolutions in the convex setC = Rd ×P(U).

I In order to take advantage of the theory of ODE, Bochner,and Frechét calculus in Banach spaces we embed C in theBanach space Y := Rd × F (U), whereF (U) := span(P(U))

‖·‖BLand the closure is taken in the

dual space (Lip(U))′;

I Notice that F (U) ⊂ (Lip(U))′, and that (Y , ‖ · ‖Y ), with‖y‖Y = ‖(x , σ)‖Y := |x |+ ‖σ‖BL, is a separable Banachspace. The space F (U) defined above, known in the literatureas the Arens-Eells space, is isometric to the predual of Lip(U).

I In order to take advantage of the theory of ODE, Bochner,and Frechét calculus in Banach spaces we embed C in theBanach space

Y := Rd × F (U), whereF (U) := span(P(U))

Evolution of the characteristics

for every continuous curve t ∈ [0,T ] 7→ Σt ∈P1(C ) we can thenassociate to bΣ the transition maps YΣ(t, s, y) induced by ODE inY

ẏt = bΣt (yt), ys = y .

A solution y = (x , σ) satisfies,

ẋt =

∫U

e(xt , u)dσt(u),

σ̇t =

(∫C

(∫U

J(xt , ·, x ′, u′)dσ′(u′)−∫U×U

J(xt ,w , x′, u′)dσ′(u′)dσt(w)

)dΣt(x

′, σ′)

)σt ,

Classical well-posedness of ODE in Banach spacesTheorem (Brézis, 1973)Let (Y , ‖ · ‖Y ) be a Banach space, C a closed convex subset of Y and letA(t, ·) : C → Y , t ∈ [0,T ], be a family of operators satisfying the followingproperties:

1. there exists a constant L ≥ 0 such that

‖A(t, y1)−A(t, y2)‖Y ≤ L‖y1−y2‖Y for every y1, y2 ∈ C and t ∈ [0,T ];

2. for every y ∈ C the map t 7→ A(t, y) is continuous in [0,T ];3. for every R > 0 there exists θ > 0 such that

y ∈ C , ‖y‖Y ≤ R ⇒ y + θA(t, y) ∈ C .

Then for every ȳ ∈ Y there exists a unique curve y : [0,T ]→ C of class C1satisfying yt ∈ C for all t ∈ [0,T ] and

d