A generalization of the Hopf-Cole transformation for stationary Mean Field Games systems
aa r X i v : . [ m a t h . A P ] M a y A generalization of the Hopf-Coletransformation for stationary Mean FieldGames systems
Marco Cirant,
Dipartimento di Matematica “F. Enriques”Universit`a di MilanoVia C. Saldini, 5020133–Milano, Italy
Abstract
In this note we propose a transformation which decouples stationary Mean FieldGames systems with superlinear Hamiltonians of the form | p | r ′ , r ′ >
1, and turnsthe Hamilton-Jacobi-Bellman equation into a quasi-linear equation involving the r -Laplace operator. Such a transformation requires an assumption on solutions ofthe system, which is satisfied for example in space dimension one or if solutions areradial. Key words:
Stationary Mean Field Games, p -Laplacian, Hopf-Cole transformation. Mean Field Games (briefly MFG) is a branch of Dynamic Games which hasbeen proposed independently by Lasry, Lions [7], [8], [9], [11] and Caines,Huang, Malham´e [6], and aims at modeling and analyzing decision processesinvolving a very large number of indistinguishable rational agents. In MFG,every agent belonging to a population of infinite individuals has the goal ofminimizing some cost which depends on his own state and on the average
Email address: [email protected] (Marco Cirant).1 . The author is supported by a Post-Doc Fellowship from the Universit`a degliStudi di Milano.
Preprint submitted to 7 septembre 2018 istribution of the other players. Suppose that the state of a typical player isdriven by the stochastic differential equation dX s = − α s ds + √ ν dB s ∈ Ω , where α s is the control, B s is a Brownian motion, ν > ⊆ R d , d ≥
1, is the so-called state space. Suppose also that the cost functionalhas the long-time-average form J ( X , α ) = lim inf T →∞ T Z T E [ L ( α s ) + f ( X s , ˆ m s )] ds, where the (convex) Lagrangian function L ( α ) is associated to the cost paid bythe player to change his own state, and the term involving f , that we assumeto be a C (Ω × [0 , ∞ )) function, is the cost paid for being at state x ∈ Ω,and it depends on the empirical density ˆ m s of the other players. Then, underthe assumption that players are indistinguishable, it has been shown thatequilibria of the game (in the sense of Nash) are captured by the followingsystem of non-linear elliptic equations (HJB) − ν ∆ u ( x ) + H ( Du ( x )) + λ = f ( x, m ( x )) in Ω(K) − ν ∆ m ( x ) − div( DH ( Du ( x )) m ( x )) = 0 in Ω R Ω m ( x ) dx = 1 , m ≥ . (1)Here, H denotes the Legendre transform of L . The two unknowns u, λ in theHamilton-Jacobi-Bellman equation (1)-(HJB) provide respectively the optimalcontrol of a typical player, given in feedback form by α ∗ : x
7→ − DH ( Du ( x )),and the average cost J ( X , α ∗ ). On the other hand, the solution m of theKolmogorov equation (1)-(K) is the stationary distribution of players imple-menting the optimal strategy, that is the long time behavior of the wholepopulation playing in an optimal way. The two equations are coupled via thecost function f . Note that a set of boundary conditions is usually associated to(1), for example u, m can be assumed to be periodic if Ω = (0 , d or Neumannconditions are imposed if Ω is bounded and X s is subject to reflection at ∂ Ω.In some models Ω is the whole R d .A relevant class of MFG models assumes that the Lagrangian function L hasthe form L ( α ) = l r | α | r , l > , r >
1. Consequently, the Hamiltonian function H becomes H ( p ) = h r ′ | p | r ′ , h = l − r ′ > , r ′ = rr − > . (2)In the particular situation where L and H are quadratic (namely r = r ′ = 2),it has been pointed out (see [7], [11]) that the so-called Hopf-Cole transfor-mation decouples (1), and reduces it to a single elliptic semilinear equation2f generalized Hartree type. Precisely, let ϕ := ce − u ν , c >
0, then (1)-(HJB)reads (setting for simplicity h = 1) − ν ∆ ϕ + ( f ( x, m ) − λ ) ϕ = 0 in Ω (3)for all c >
0. Moreover, if we set c = (cid:16)R Ω e − uν (cid:17) − , an easy computation showsthat ϕ is also a solution of (1)-(K), so if uniqueness of solutions for suchequation holds (that is true, for example, if suitable boundary conditions areimposed), then m = ϕ , and therefore ϕ becomes the only unknown in (3).This transformation can be exploited to study quadratic MFG systems, bothfrom the theoretical and numerical point of view. This strategy is adopted, forexample, in the works [1], [3], [4], [5].The aim of this note is to show that if r ′ = 2 there exists a similar change ofvariables, involving a suitable power of m , that in some cases decouples (1)and turns the Hamilton-Jacobi-Bellman equation (1)-(HJB) into a quasi-linearequation of the form − µ ∆ r ϕ + ( f ( x, ϕ r ) − λ ) ϕ r − = 0 in Ω , R Ω ϕ r dx = 1 , ϕ > , µ = ν (cid:16) νrh (cid:17) r − , (4)where ∆ r ϕ = div( | Dϕ | r − Dϕ ) is the standard r -Laplace operator ( r is theconjugate exponent of r ′ ). Such a transformation is in particular possible ifthe vector field νDm + DH ( Du ) m , which is divergence-free because of (1)-(K),is identically zero on Ω.Let us briefly recall in which sense ( u, m, λ ) solves (1). Definition
We say that a triple ( u, m, λ ) ∈ C (Ω) × W , (Ω) × R is a(local) solution of (1) if u, λ solve pointwise (1) - (HJB) and m solves (1) - (K) in the weak sense, namely ν Z Ω Dm · Dξ + Z Ω m DH ( Du ) · Dξ = 0 ∀ ξ ∈ C ∞ (Ω) . (5) We say that a couple ( ϕ, λ ) ∈ ( W ,r loc (Ω) ∩ L ∞ loc (Ω)) × R is a solution of (4) if µ Z Ω | Dϕ | r − Dϕ · Dξ + Z Ω ( f ( x, ϕ r ) − λ ) ϕ r − ξ = 0 ∀ ξ ∈ C ∞ (Ω) . Then, the transformation can be stated as follows.
Theorem 1.2
Suppose that H satisfies (2) . a) Let ( u, m, λ ) be a solution of (1) . If the following equality holds, νDm + h m | Du | r ′ − Du = 0 a.e. in Ω , (6)3 hen ϕ := m r in Ω (7) is a solution of (4) . b) Let ( ϕ, λ ) be a solution of (4) , and suppose that there exists u ∈ C (Ω) such that h ϕ | Du | r ′ − Du + νrDϕ = 0 in Ω . (8) Then, ( u, m, λ ) is a solution of (1) , where m := ϕ r in Ω . The proposed transformation reveals a connection between some (stationary)MFG systems with non-quadratic Hamiltonians of the form (2) and r -Laplaceequations, which have been widely studied in the literature and appear inmany other areas of interest. Apart from existence and uniqueness issues,this link might shed some light on MFG problems in general, which can betranslated into problems involving the r -Laplacian (e.g. qualitative propertiesof solutions, MFG in unbounded domains, vanishing viscosity limit ν → r ′ = 2, we have mentioned that if solutions of (1)-(K)are unique, then m = e − h uν / (cid:18)R Ω e − h uν (cid:19) − . In this case condition (6) is easilyverified, and the assertion of Theorem 1.2 is that ϕ = m / solves (4) with r = 2, which is precisely (3). In this sense our transformation can be seen asa generalization of the standard Hopf-Cole.Note that the change of variables m = ϕ r is local , in particular it is independentof boundary conditions that might be added to (1). However, in order to verify(6), (8) and therefore to apply Theorem 1.2, it is necessary to specify additionalinformation on the problem. Space dimension d = 1 and Neumann conditionsat the boundary, or u, m, ϕ enjoying radial symmetry are two possible scenarioswhere (6), (8) hold. Corollary 1.3
Suppose that H satisfies (2) and Ω = { x ∈ R d : | x | < R } for some R ∈ (0 , ∞ ] . Then, ( u, m, λ ) is a radial solution of (1) if and only if ( ϕ, λ ) is a radial solution of (4) , where ϕ = m /r . Corollary 1.4