## Nonlinear Elliptic Systems and Mean Field Games

^{∗}

Martino Bardi and Ermal Feleqi Dipartimento di Matematica

Universit`a di Padova via Trieste, 63 I-35121 Padova, Italy

e-mail: bardi@math.unipd.it and feleqi@math.unipd.it March 19, 2015

Abstract

We consider a class of quasilinear elliptic systems of PDEs consisting of N Hamilton- Jacobi-Bellman equations coupled with N divergence form equations, generalising to N > 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solu- tions to these systems: either the Hamiltonians are required to be at most quadratic in the gradients, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton-Jacobi-Bellman equations of the theory of N -person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some N -person games.

Keywords: Nonlinear elliptic systems, stochastic differential games, N-person games, Nash equilibria, Mean Field Games.

### 1 Introduction

This paper deals with systems of partial differential equations of the following type

L^{i}v_{i}+ H^{i}(x, Dv_{i}) + λ_{i} = V^{i}[m] in Q,
L^{i∗}m_{i}− div g^{i}(x, Dv_{i})m_{i} = 0 in Q := T^{d},
R

Qmi(x)dx = 1, mi > 0, R

Qvi(x)dx = 0, i = 1, . . . , N,

(1)

∗Partially supported by the Fondazione CaRiPaRo Project ”Nonlinear Partial Differential Equations: mod- els, analysis, and control-theoretic problems” and the European Project Marie Curie ITN ”SADCO - Sensitivity Analysis for Deterministic Controller Design”. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

where the unknowns are the constants λ = (λ_{1}, . . . , λ_{N}), the functions v = (v_{1}, . . . , v_{N})
and the densities of probability measures m = (m1, . . . , mN), at least continuous on the
d-dimensional torus Q := T^{d}. The operators

L^{i}:= −tr(a^{i}(x)D^{2}), i = 1, . . . , N,

are second-order uniformly elliptic with Z^{d}-periodic and Lipschitz coefficients a^{i}(·), i.e., for
some ν, C > 0,

a^{i}(x) ≥ νId, a^{i}(x) = a^{i}(x + k) ∀k ∈ Z^{d}, |a^{i}(x) − a^{i}(x + y)| ≤ C|y| ∀x, y ∈ R^{d}, (2)
where Id is the identity d × d matrix, and

L^{i∗}v := −X

h,k

D_{hk}^{2} (a^{i}_{hk}(x)v)

are their formal adjoints. The Hamiltonians H^{i} : R^{d}× R^{d} → R are Z^{d}-periodic in x and
satisfy either one of the following sets of conditions.

C1. For all i = 1, . . . , N, H^{i}= H^{i}(x, p) is locally Lipschitz, superlinear in p uniformly in x,
i.e.,

x∈Qinf |H^{i}(x, p)|/|p| → +∞ as |p| → ∞, (3)
and ∃θ^{i} ∈ (0, 1), C > 0, such that

tr(a^{i})DxH^{i}· p + θ^{i}(H^{i})^{2} ≥ −C|p|^{2} for |p| large, and for a.e. x ∈ Q. (4)
C2. For all i = 1, . . . , N , for some α ∈ (0, 1), H^{i} is locally α-H¨older continuous and grows

at most quadratically

|H^{i}(x, p)| ≤ C1|p|^{2}+ C2 ∀x ∈ Q, p ∈ R^{d}, (5)
for some C_{1}, C_{2} > 0, the so-called natural growth condition.

We denote by C(Q) the set of Z^{d}-periodic continuous functions on R^{d}, by P (Q) the set of
probability measures on Q endowed with the weak^{∗}-topology of the dual C(Q)^{∗} of C(Q), and
assume that the given operators V^{i} : P (Q)^{N} → C(Q) satisfy

V^{i}[mn] → V^{i}[m] uniformly in Q for all mn→ m, m_{n}, m ∈ C(Q)^{N}∩ P (Q)^{N}, (6)
and

kV^{i}[m]k_{C}0,β(Q)≤ C ∀m ∈ C(Q)^{N}∩ P (Q)^{N}, (7)
where β = 1 if H^{i} verifies condition C1 above, and β = α if condition C2 holds. Finally, we
assume that

g^{i} : Q × R^{d}→ R^{d} are measurable, locally bounded, and continuous in p. (8)
Under the above conditions we show the existence of a solution λi ∈ R, vi ∈ C^{2,α}(Q), mi ∈
W^{1,p}(Q), for all 1 ≤ p < ∞, i = 1, . . . , N , to the system (1), where α is the H¨older exponent
appearing in condition C2 on H^{i} if such condition holds, and it is any number in (0, 1) if

condition C1 is assumed instead. We do not address the uniqueness of the solution here. It is
known to hold if N = 1 under some additional conditions (in particular, some monotonicity
of V^{i}) introduced by Lasry and Lions [34, 36]. If N > 1 one may expect uniqueness only
in some very special cases, see Cirant [21, 22] where Neumann boundary conditions are also
treated.

There are two main motivations for studying systems of the form (1). The first is the theory of stationary Mean Field Games (briefly, MFG) as formulated by J-M. Lasry, P-L.

Lions [34, 36]. In fact, for N = 1, L^{i} = −∆, H^{i} = H smooth and g^{i} = DpH, (1) reduces
to the stationary MFG PDEs introduced and studied in [34, 36]. This is a model of the
equilibrium distribution of a large population of identical agents whose dynamics is subject to
white noise and seeking to minimise a given individual cost functional. For N > 1 the system
(1) with g^{i} = DpH^{i} can be associated to Mean Field Games with N different populations
of players, see Section 3.6 and [21] for further motivations. Our assumptions allow a rather
general controlled dynamics for the generic agent of each population; in particular, condition
C2 on the Hamiltonians does not require any coercivity and allows the control set to be
bounded, which in several applications can be more realistic than the model problem treated
in [34, 36], see Sections 3.3 and 3.4 for some examples. Mean-Field Games were also introduced
independently by Huang, Caines, and Malham´e and studied by different methods [31, 32], see
Section 3.6.

The second motivation is the synthesis of Nash equilibria in feedback form for N -person stochastic differential games with cost functionals of ergodic type. This is usually based on the solution of a strongly coupled system of N Hamilton-Jacobi-Bellman (briefly, HJB) equations, the system (42) in Section 3.1, following a classical observation of Friedman [25]. A systematic study of such nonlinear elliptic systems was pursued by Bensoussan and Frehse starting with [10], see [12] for problems with ergodic cost and their book [13] for more references. In [34, 36] Lasry and Lions propose instead a system of N HJB and N Kolmogorov-Fokker- Planck equations of the form (1) for games where the players are coupled only via the cost functionals. An advantage of this system is the weaker coupling among the HJB equations.

They give an existence result in the case L^{i} = −ν^{i}∆, ν^{i} > 0, H^{i} smooth, satisfying (3) and
(4) with C = 0, and g^{i} = DpH^{i}, and show how to synthesize a feedback Nash equilibrium
for the N -person game from the solution of such system. In Section 3.1 we show a more
precise connection among the classical systems of [10, 12] and systems of the form (1), which
is related to the adjoint methods for Hamilton-Jacobi equation explored by Evans [23]. In
Section 3 we also generalise the result in [34, 36] on the synthesis of the equilibrium and give
several examples of classes of differential games to which our abstract results apply.

We recall that the pioneering papers [34, 36] prove also the convergence, in a suitable sense, of N -person games to a Mean-Field Game with a single population as the number of players tends to infinity. Some estimates of the present paper were used by Feleqi [24] to prove the same result in the case of several interacting populations and under more general conditions.

The proof of the existence of solutions to (1) under condition C1 follows the suggestion in [34, 36] about getting a priori estimates for Dvi by the Bernstein method (see also [1], [37] and [38]). On the other hand, rather than relying on estimates for Bellman equations, we mostly use more classical local a priori estimates for linear equations. We also included some of their proofs for the reader’s convenience when we were not able to find an explicit reference in the literature, in the attempt to make the paper reasonably self-contained and readable by a wide audience. The proof in this case allows some variants to conditions C1

and C2 and can be adapted, for instance, to Hamiltonian behaving like |p_{1}| + |p_{2}|^{γ}, γ > 1, as
p = (p1, p2) ∈ R^{d}^{1}× R^{d}^{2} tends to infinity, see Remark 2.4 and Example 3.9.

Under condition C2 the proof is different: we approximate the additive eigenvalue problem (1) with systems with zero-th order terms, as in Bensoussan and Frehse [11, 13], and for such systems we employ some deep estimates by Ladyzhenskaya and Uraltseva [33] and other tools of the elliptic theory [27]. In this generality our results are new even in the scalar case N = 1.

We conclude with some bibliographical remarks. In [14] Bensoussan and Frehse also weaken the standard quadratic growth condition on the Hamiltonians but their results do not apply to our system. The difficulties arising from constraints on the controls were treated by Mannucci [39] for parabolic equations. The existence of Nash equilibria for some stochastic N - person differential games was also proved by probabilistic methods, see, e.g., [18, 26] and the references therein. For a general presentations of Mean-Field Games and their applications we refer to the lecture notes by Gu´eant, Lasry, and Lions [30] and Cardaliaguet [19] and the survey paper by Gomes and S´aude [29]. Evolutive MFG were first studied in [35, 36] and [31, 32]. The justification of stationary MFG via long-time asymptotics is in [20], extended stationary MFG were studied in [28] and numerical methods in [2], see also the references therein. For other recent contributions on MFG see also the courses of P.-L. Lions at Coll`ege de France, the monograph [15] and the two special issues of Dynamic Games and Applications [6, 7].

The paper is organized as follows. Section 2 contains the statements and proofs of the existence results for (1) and for related systems with zero-th order terms arising from dis- counted infinite horizon problems. In Section 3 we describe the connections of the system (1) with N-person and Mean-Field games. We also apply the results of Section 2 to show the existence of Nash equilibria for such games in the cases of long-time-average costs and discounted infinite horizon costs. Finally, the Appendix contains the proofs of some technical lemmas.

### 2 Existence of solutions to elliptic systems

We will use the notations tr b for the trace of a square matrix b, and a·b = tr ab^{t}, |b| := (b·b)^{1/2}.
The adjoint operator L^{i∗} will be interpreted in the sense of distributions:

< L^{i∗}v, φ >=

Z

Q

vLφdx ∀φ ∈ C^{∞}(Q).

2.1 The additive eigenvalue problem

Theorem 2.1. Assume (2) and (8), H^{i} satisfy condition C1, and V^{i} verify (6) and (7) with
β = 1. Then there exist λ_{1}, . . . , λ_{N} ∈ R, v1, . . . , v_{N} ∈ C^{2,α}(Q), m_{1}, . . . , m_{N} ∈ W^{1,p}(Q), for
all 0 < α < 1, 1 ≤ p < ∞, which solve the system (1).

We need the following two lemmas for linear equations. We believe they are well-known, but for lack of a precise reference we give their proofs in the Appendix.

Lemma 2.2. Let

L = −a_{hk}(x)D_{hk}+ b_{h}(x)D_{h}

be a second-order uniformly elliptic linear differential operator in Q = T^{d} with coefficients
ahk, bh, ∈ C^{α}(Q), h, k = 1, . . . , d, 0 < α < 1. Then, for any f ∈ C^{α}(Q), the problem

Lv + λ = f

R

Qv(x)dx = 0 (9)

has a unique solution (v, λ) ∈ C^{2,α}(Q) × R. Moreover,

|λ| ≤ kf k_{∞}, (10)

kvk_{C}1,α(Q) ≤ Ckf k_{∞}, (11)

kvk_{C}2,α(Q)≤ Ckf k_{C}α(Q) (12)

for some constant C > 0 which depends only on (the coefficients of ) L.

Lemma 2.3. Let

L = −a_{hk}(x)D_{hk}, g : Q → R^{d}

be a (symmetric) second-order uniformly elliptic linear differential operator in Q = T^{d} with
coefficients a_{hk}∈ C^{0,1}(Q), h, k = 1, . . . , d, and a bounded measurable vector field, respectively.

Then the problem

L^{∗}m − div(g(x)m) = 0
R

Qm(x)dx = 1 (13)

has a unique solution m ∈ W^{1,p}(Q) for all 1 ≤ p < ∞. Moreover, m is positive and

kmk_{W}1,p(Q)≤ C(kgk_{∞}), (14)

for some constant C(kgk∞) which depends (continuously) on g only through kgk∞ (and also
on p and the coefficients a_{hk}).

Proof of Theorem 2.1. The proof is based on Schauder’s fixed point theorem (see for
instance [41, Theorem 4.1.1, p. 25] or [27, Corollary 11.2, p. 280]) and on a priori estimates
for the gradients of v_{i} that are obtained by Bernstein’s method, as suggested in [34, 36].

We first assume, instead of (3) and (4), that the Hamiltonians are bounded, that is,

∃M > 0 such that |H^{i}(x, p)| ≤ M ∀i, x, p.

Let

B =

u = (u1, . . . , uN) ∈ (C^{1,α}(Q))^{N} :
Z

Q

udx = 0

,

which is a Banach space as a closed subspace of C^{1,α}(Q)^{N}. We define an operator
T : B → B,

according to the scheme

u 7→ m 7→ (v, λ) 7→ v,

as follows. Given u = (u1, . . . , uN) ∈ B, we plug it in place of v in the second N scalar linear
equations of the system (1) and solve those equations for the unknowns m = (m1, . . . , m_{N}),
requiring that these unknowns satisfy conditions in the third line of (1). That these {m_{i}}^{N}_{i=1}

exist, are uniquely defined and have the required properties, is a consequence of Lemma 2.3.

Then, with these {mi}, we solve the N scalar linear equations obtained from the first equations
in (1) after plugging u_{i} into H^{i}in place of v_{i}, i.e., by solving the N uncoupled linear equations

L^{i}v_{i}+ H^{i}(x, Du_{i}) + λ_{i} = V^{i}[m]

for the unknowns v = (v_{1}, . . . , v_{N}), λ = (λ_{1}, . . . , λ_{N}) ∈ R^{N}. Since we require in addition
that the vi have zero mean, then v = (v1, . . . , vN) ∈ C^{2,α}(Q) and λ = (λ1, . . . , λN) ∈ R^{N} are
uniquely defined, as a consequence of Lemma 2.2. We set T u = v. By (10), (12), (14), and
a standard embedding theorem, T is continuous and compact. Moreover, the C^{1,α}-estimate
(11) and the boundedness of H^{i} and V^{i} gives

kvk_{C}1,α(Q) ≤ C

for some C > 0 independent of v; thus T B is bounded. Therefore, by Schauder’s fixed point theorem, T has a fixed point (in the convex hull of the closure of T B).

Now we turn to consider Hamiltonians H^{i} that satisfy the assumptions of the theorem.

We introduce the truncated Hamiltonians H_{R}^{i} defined as follows

H_{R}^{i}(x, p) =(H^{i}(x, p), if |p| ≤ R,
H^{i}

x, R_{|p|}^{p}

, if |p| > R, x ∈ Q, p ∈ R^{d}, (15)
where the parameter R > 0 is to be fixed in the sequel sufficiently large. Let R_{1}> 0 be such
that (4) is verified for all x ∈ Q, |p| ≥ R_{1}. Then

x∈Qinf (tr a^{i})DxH_{R}^{i} · p + θ^{i}(H_{R}^{i})^{2} ≥ −C|p|^{2} for all R ≥ |p| ≥ R1, (16)
with the same θ^{i} (i = 1, . . . , N ) and C as in (4). Clearly the H_{R}^{i} are bounded and Lipschitz
continuous. So let λ^{R}_{1}, . . . , λ^{R}_{N} ∈ R, v^{R}_{1}, . . . , v_{N}^{R} ∈ C^{2,α}(Q), m^{R}_{1}, . . . , m^{R}_{N} ∈ W^{1,p}(Q) (0 < α <

1, 1 ≤ p < ∞) be a solution of (1) with H_{R}^{i} in the place of H^{i}.

The crucial step of the proof is an a priori estimate for kDv_{i}^{R}k_{∞} uniform in R, obtained
by Bernstein’s method. We drop the indices i and R in the following estimates. Let w = Dv
and ψ = (1/2)|w|^{2}. We have these identities

Dψ = wD^{2}v (17)

D^{2}ψ =X

h

w_{h}D^{2}w_{h}+ (D^{2}v)^{2} (18)

Lψ = −a · D^{2}ψ = −X

h

wh(a · D^{2}wh) − a · (D^{2}v)^{2}. (19)

Let us assume temporarily that u is of class C^{3} and a_{hk}, H, F are of class C^{1} (clearly the
truncation of H above could have been done smoothly). We apply to the first equations in (1)
the operator w · D and use (19), (17) to obtain

Lψ + a · (D^{2}v)^{2}− δa · D^{2}v + D_{x}H · w + D_{p}H · Dψ = G · w, (20)
where

δa := (δa_{hk})h,k=1,...,d with δa_{hk}:=X

l

D_{l}a_{hk}w_{l} (21)

and G is a function whose L^{∞}-norm does not exceed a universal constant which does not
depend on v (recall the assumptions on operators V^{i}). We use the following inequalities which
are simple consequences of Cauchy-Schwarz inequality: for any a, b symmetric matrices with
a ≥ 0 and c, e ∈ M^{d×m}^{i}

(a · b)^{2}≤ (a · b^{2})tr a and (tr ce^{t}b)^{2} ≤ |e|^{2}cc^{t}· b^{2}. (22)
Assume that

a = 1

2σσ^{t}, (23)

where the matrix σ(x) is Lipschitz; such a decomposition is always possible for a(x) is Lipschitz and its smallest eigenvalue (which is positive) is bounded away from zero uniformly for x ∈ Q.

By (23) and (21) we have

δa = (δσ)σ^{t}.

Using this identity, the second of inequalities (22) and (23), we obtain
δa · D^{2}v ≤ |δσ| σσ^{t}· (D^{2}v)^{2}1/2

=√

2|δσ| a · (D^{2}v)^{2}1/2

≤ εa · (D^{2}v)^{2}+ 1

2ε|δσ|^{2}, (24)

for any 0 < ε < 1. On the other hand, using the first of inequalities (22) and the first equations in (1), we have

(a · (D^{2}v)^{2})tr a ≥ (a · D^{2}v)^{2}= (Lv)^{2}

≥ (λ + H − V [m])^{2}

≥ ωH^{2}− c_{ω} (25)

for any 0 < ω < 1 and a constant cω independent of R, The last inequality is obtained by the boundedness of operators V and constants λ. In fact, looking at the minima and maxima of v in the first equations of (1) we obtain

|λ| ≤ sup

x∈Q

(|H(x, 0)| + |V [m](x)|).

Multiplying (20) by tr a, and using (24), (25), we get

(tr a)(Lψ + DpHDψ + DxH · w) + (1 − ε)ωH^{2}≤ (tr a)

G · w + 1
2ε|δσ|^{2}

+ cω

If we choose ε and ω such that (1 − ε)ω > θ, where θ is the constant appearing in (16), using (16) we get

(tr a)(Lψ + DpHDψ) + ((1 − ε)ω − θ)H^{2}≤ (tr a)

G · w + 1
2ε|δσ|^{2}

+ C|w|^{2}+ cω (26)
for R ≥ |w| > R_{1}. At a maximum point of ψ^{R}_{i} , (now we reintroduce i and R in order to avoid
any possible confusion), say x^{i}_{R}, taking into account that |δσ^{i}| is at most linear in w_{i}^{R}, we
have

((1 − ε^{i})ω^{i}− θ^{i})(H_{R}^{i})^{2}(x^{i}_{R}, w_{i}^{R}(x^{i}_{R})) ≤ C(|w_{i}^{R}(x^{i}_{R})|^{2}+ 1) (27)

for some C > 0 (independent of R) and R ≥ |w^{R}_{i} (x^{i}_{R})| ≥ R_{1}. But by (3), the left-hand
side above is superlinear in |w_{i}^{R}(x^{i}_{R})| and thus |w_{i}^{R}(x^{i}_{R})| must be bounded by some constant
independent of R.

Thus, we have shown that

kDu^{R}_{i} k_{∞}≤ R_{2} (28)

for some R_{2} > 0 independent of R > R_{1}. So if take any R > max{R_{1}, R_{2}} in (15), we discover
that λ^{R}_{1}, . . . , λ^{R}_{N} ∈ R, v^{R}1, . . . , v_{N}^{R} ∈ C^{2,α}(Q), m^{R}_{1}, . . . , m^{R}_{N} ∈ W^{1,p}(Q) (0 < α < 1, 1 ≤ p < ∞)
is also a solution of the original system of PDEs (1).

To complete the proof under our general assumptions we observe that equation (20) still
holds a.e. in R^{d}. In fact u ∈ W^{3,p}, p > n (by classical elliptic regularity theory for linear
equations), hence u is three times differentiable in the usual sense a.e. and also the coefficients,
being Lipschitz continuous, are differentiable a.e. A sort of chain rule usable for our purposes
holds by a result in [4], and the maximum principle to be used is that of [17]. Alternatively,
and more simply, one can proceed by regularizing the data {(a^{i}_{hk})_{1≤h,k≤d} : i = 1, . . . , N },
V^{i}[m], for m ∈ P (Q)^{N}, and H^{i}, for i = 1, . . . , N , via smooth approximations to the identity
{ρ_{ε}}_{ε>0}, { ˆρ_{ε}}_{ε>0}given by ρ_{ε}(x) = ε^{−d}ρ_{1}(x/ε) and ˆρ_{ε}(x, p) = ρ_{ε}(x)ρ_{ε}(p) for all x ∈ R^{d}, p ∈ R^{d},
ε > 0, where ρ1is some mollification kernel in R^{d}(that is, a nonnegative function of class C^{∞}
with support in the unit ball B of R^{d} and R

Bρ_{1}(z) dz = 1). Noting that kρ_{ε}? a^{i}_{hk}k_{C}1(Q) →
ka^{i}_{hk}k_{C}0,1(Q), kρ_{ε} ? V^{i}[m]k_{C}^{1}_{(Q)} → kV^{i}[m]k_{C}^{0,1}_{(Q)} and inf D_{x}( ˆρ_{ε}? H^{i}) → ess inf D_{x}H^{i} as
ε → 0 for all h, k, i, m ∈ P (Q)^{N}, one deduces estimates of type (28) with R_{2} independent of
ε for ε small enough.

Remark 2.4. (i) In dimension d = 1 condition (4) is not needed. Note that ^{d}_{dx}^{2}^{v}2^{i} is bounded
from below if H^{i} is bounded from below; since it has zero mean in (0, 1), it is bounded in L^{1}
norm. Therefore ^{dv}_{dx}^{i} is bounded.

(ii) Theorem 2.1 still holds if (3) is weakened to

∃ν^{i}> 0 such that lim inf

|p|→∞

infx∈Q|H^{i}(x, p)|

|p| = ν^{i}> 0, (29)

provided that we substitute (4) with the stronger condition: ∃θ^{i}, η^{i}∈ (0, 1) such that
lim inf

|p|→∞

1

|p|^{2} inf

x∈Q

θ^{i}(tr a)D_{x}H^{i}· p + (1 − θ^{i})θ^{i}η^{i}(H^{i})^{2}− |p|^{2}(tr a^{i})|Dσ^{i}|^{2}
2

≥ 0, (30)
where a^{i} = (1/2)σσ^{t}, Dσ^{i} = (Dσ^{i}_{hk}) is a matrix (whose entries are vectors) and |Dσ^{i}|^{2} =
P

h,k|Dσ_{hk}^{i} |^{2}. To see this take ε^{i} = θ^{i} in (24), multiply (26) by θ^{i} and choose ω^{i} in (25)
so that ω^{i} > η^{i}. Let s > 0, R_{s} > 0 be such that the quantity under the lim inf sign in the
left-hand side of (30) is > −s for all |p| ≥ R_{s}. Noting that |δσ^{i}| ≤ |Dσ^{i}||w_{i}^{R}|, we deduce

(ω^{i}− η^{i})(1 − θ^{i})θ^{i}(H_{R}^{i})^{2}(x^{i}_{R}, w_{i}^{R}(x^{i}_{R})) − s|w^{R}_{i} (x^{i}_{R})|^{2} ≤ C(|w_{i}^{R}(x^{i}_{R})| + 1)

for some C > 0 (independent of R and s) and for R ≥ |w^{R}_{i} (x^{i}_{R})| ≥ R_{s}. Now if we choose s > 0
so small that s < ν^{i}(ω^{i}− η^{i})(1 − θ^{i})θ^{i}, by (29) we find that |w^{R}_{i} (x^{i}_{R})| is bounded uniformly
in R.

(iii) The arguments of the proof of Theorem 2.1, with some obvious modifications, prove also the solvability of the problem

Lv + λ + H(x, Dv) = 0

for the unknowns v ∈ C^{2}(Q), λ ∈ R, where L satisfies (2) and the Hamiltonian H is locally
Lipschitz continuous and satisfies (3) and (4) or, alternatively, any condition mentioned in
the remarks above. This result is known for Dirichlet and Neumann boundary value problems
with H satisfying similar conditions, see [37]. If L is degenerate elliptic the existence of a
viscosity solution v ∈ C^{0,1}(Q) can be found in [38].

These existence results for (1), under condition C1, may be interpreted as follows. The
Hamiltonians H^{i} can grow arbitrarily provided that they “do not oscillate too much in x”

which rigorously means that they should satisfy the technical condition (4). On the other hand, if the Hamiltonians have at most quadratic growth as in condition C2 of the Introduc- tion, the so called natural growth condition, we do not need any additional assumption of the aforementioned type in order to ensure existence. Indeed, we have the following result.

Theorem 2.5. Assume (2) and (8), H^{i} satisfy condition C2, and V^{i} verify (6) and (7) with
β = α. Then there exist λ_{1}, . . . , λ_{N} ∈ R, v1, . . . , v_{N} ∈ C^{2,α}(Q), m_{1}, . . . , m_{N} ∈ W^{1,p}(Q), for
all 1 ≤ p < ∞, which solve the system (1).

The proof is obtained as a limit in a system of equations with zero-th order terms, based on Theorem 2.6, and it is therefore postponed to the next section.

2.2 Equations with zero-th order terms

Theorem 2.6. Assume (2) and (8), H^{i} satisfy condition C2, and V^{i} verify (6) and (7)
with β = α. Let ρ^{1}, . . . , ρ^{N} be positive constants. Then there exist v1, . . . , vN ∈ C^{2,α}(Q),
m1, . . . , m_{N} ∈ W^{1,p}(Q), for all 1 ≤ p < ∞, i = 1, . . . , N , which solve

L^{i}v_{i}+ H^{i}(x, Dv_{i}) + ρ^{i}v_{i} = V^{i}[m] in Q,
L^{i∗}m_{i}− div g^{i}(x, Dv_{i})m_{i} = 0 in Q,
R

Q^{d}mi(x)dx = 1, mi> 0, i = 1, . . . , N,

(31)

For the proof we need the following lemma which is proved in the Appendix.

Lemma 2.7. Let

L = −a_{hk}(x)D_{hk}+ b_{h}(x)D_{h}+ c(x)

be a second-order uniformly elliptic linear differential operator in Q = T^{d} with coefficients
ahk, bh, c ∈ C^{α}(Q), h, k = 1, . . . , d, 0 < α < 1. Assume also that c > 0. Then, for any
f ∈ C^{α}(Q), the equation

Lv = f (32)

has one and only one solution v ∈ C^{2,α}(Q). Moreover, for some constant C > 0 which
depends only on (the coefficients of ) L,

kvk_{C}2,α(Q)≤ Ckf k_{C}α(Q). (33)

Proof of Theorem 2.6. We define an operator

T : C^{1,α}(Q)^{N} → C^{1,α}(Q)^{N},

u 7→ m 7→ v,

in the following way. Given u = (u1, . . . , uN), we solve the second N equations in (31) with ui

plugged into g^{i} in place of vi and with the corresponding normalisation conditions, and find
m = (m_{1}, . . . , m_{N}). With these m_{i}and the u_{i}plugged into the Hamiltonians H^{i}, i = 1, . . . , N,
in place of vi we solve the first N linear equations of (31), that is,

L^{i}vi+ ρ^{i}vi+ H^{i}(x, Dui) = V^{i}[m], i = 1, . . . , N (34)
and find v = (v1, . . . , vN) ∈ C^{2,α}(Q)^{N}. This is possible in virtue of Lemma 2.7. We set
T u = v.

It is standard to verify that T : C^{1,α}(Q) → C^{1,α}(Q), u → T u is continuous and com-
pact. By Schaefer’s version of Leray-Schauder theorem (see [41, Theorem 4.3.2, p 29] or [27,
Theorem 11.3, p. 280]), we need only look at the set of the fixed points of the operators sT ,
0 ≤ s ≤ 1, that is,

{u ∈ C^{1,α}(Q)^{N} : sT u = u for some 0 ≤ s ≤ 1}, (35)
and prove that it is bounded in C^{1,α}(Q)^{N}.

To prove such an estimate first note that if u = sT u for some 0 ≤ s ≤ 1, then
ku_{i}k_{C(Q)}= s

ρ^{i} max

x∈Q(|H^{i}(x, 0)| + |F^{i}(x)|) ≤ C

for some C > 0 independent of u and s, as can be seen by looking at the extrema of ui

which satisfies equation (34) with u_{i} = v_{i} and H^{i}, F^{i} multiplied by s. We combine this
with a classical a priori interior estimate for the gradients of solutions of elliptic quasilinear
equations [33, Theorem 3.1, p. 266] to get

ku_{i}k_{C}1(Q)≤ C

for some C > 0 independent of u and s. Here we are using the growth assumption (5)
and assuming temporarily that the leading coefficients of the quasilinear equation are of
class C^{1}. This additional condition can be removed by approximation because the estimates
depend only on the L^{∞}-norm of the derivatives of the coefficients and not on their modulus
of continuity. The last estimate combined with [27, Theorem 8.32, p. 210] yields

ku_{i}k_{C}1,α(Q) ≤ C

for some C > 0 again independent of u and s. Thus T has at least one fixed point.

Proof of Theorem 2.5. Let v^{ρ}_{1}, . . . , v_{N}^{ρ} ∈ C^{2,α}(Q), m^{ρ}_{1}, . . . , m^{ρ}_{N} ∈ W^{1,p}(Q) (1 ≤ p < ∞,
i = 1, . . . , N ) be a solution of (31) with ρ^{i} = ρ > 0. Let < v^{ρ}_{i} >= R

Qv_{i}^{ρ}dx be the mean of
v_{i}^{ρ}. The crucial observation is that

kv_{i}^{ρ}− < v_{i}^{ρ}> k∞≤ C

for some C > 0 independent of ρ, which can be shown by using the techniques of [11]. Then
it is not difficult to complete the proof by showing that there exists a sequence ρ_{n} → 0 such
that

(v^{ρ}_{i}^{n}− < v^{ρ}_{i}^{n} >, ρnv_{i}^{ρ}^{n}, m^{ρ}_{i}^{n}) → (vi, λi, mi) in C^{2}(Q) × C(Q) × C(Q),
where (v_{i}, λ_{i}, m_{i}), i = 1, . . . , N, is a solution of (1).

Remark 2.8. An alternative existence result for the system (31) can be stated under condi- tion C1 instead of C2 if (7) holds with β = 1. This requires only a slight modification of the proof of Theorem 2.1.

### 3 Stochastic differential games

As an application of the previous results, in this section we show the existence of Nash equilibria for a class of N -person stochastic differential games with infinite horizon, such that the state of each player evolves independently from the states of the other players and the only coupling comes through the costs. Games of this type arise in many engineering and economic problems [31, 32, 36].

Consider a control system driven by the stochastic differential equations

dX_{t}^{i} = f^{i}(X_{t}^{i}, α^{i}_{t})dt + σ^{i}(X_{t}^{i})dW_{t}^{i}, X_{0}^{i} = x^{i}∈ R^{d}, i = 1, . . . , N (36)
where {W_{t}^{i}}^{N}_{i=1} are N independent Brownian motions in R^{d}, d ≥ 1, A^{i} ⊆ R^{m} are closed,

f^{i}: R^{d}× A^{i}→ R^{d} σ^{i} : R^{d}→ R^{d×d}

are continuous, Z^{d}-periodic and Lipschitz continuous in x uniformly in α, the matrix σ^{i}(x)
is nonsingular for any value of x, α^{i}_{t} is an admissible control of the i-th player, that is, a
stochastic process taking values in A^{i} and adapted to W_{t}^{i}. In view of the assumed periodicity
in x^{i} of all data we will often consider functions as defined on Q = T^{d}.

3.1 N -person games with long-time-average cost

Consider a game where the i-th player seeks to minimize the long-time-average or ergodic cost

J^{i}(X, α^{1}, . . . , α^{N}) := lim inf

T →+∞

1 TE

Z T 0

L^{i}(X_{t}^{i}, α^{i}_{t}) + F^{i}(X_{t}^{1}, . . . , X_{t}^{N})dt

, (37)

where X = (x^{1}, . . . , x^{N}) is the initial position of the system (36). On the cost of the i-th
player (37) we assume

L^{i} : Q × A^{i} → R continuous, (38)

F^{i} : Q^{N} → R β-H¨older continuous (39)

for some β ∈ (0, 1). Define

H^{i}(x^{i}, p, α) := −p · f^{i}(x^{i}, α) − L^{i}(x^{i}, α), H^{i}(x^{i}, p) := sup

α∈A^{i}

H^{i}(x^{i}, p, α), p ∈ R^{d}, (40)
and suppose the sup is attained in the definition of H^{i} for all x^{i}, p. Moreover, set

a^{i} := σ^{i}(σ^{i})^{t}/2, L^{i} := −a^{i}· D^{2}. (41)
Following the classical theory initiated by Friedman [25] and continued by Bensoussan and
Frehse [10, 12, 13], we look at the Nash equilibria for the maximization of the pre-Hamiltonians
with parameters x^{j}, p^{j} ∈ R^{d}, j = 1, . . . , N , namely,

P^{i}(α^{1}, . . . , α^{N}) := −

N

X

j=1

f^{j}(x^{j}, α^{j}) · p^{j}− L^{i}(x^{i}, α^{i}).

Clearly (α^{1}, . . . , α^{N}) is a Nash equilibrium for the static game with payoffs P^{1}, . . . , P^{N} if and
only if α^{i} ∈ argmax_{α}iH^{i}(x^{i}, p^{i}, α^{i}) and the value of P^{i} at the equilibrium is

P^{i}(α^{1}, . . . , α^{N}) = H^{i}(x^{i}, p^{i}) −X

j6=i

f^{j}(x^{j}, α^{j}) · p^{j}.

Therefore the system of Bellman equations of ergodic type [12, 13] associated to the game considered here is

PN

j=1L^{j}v_{i}+ H^{i}(x^{i}, D_{x}iv_{i}) + λ_{i} = F^{i}(X) +P

j6=if^{j}(x^{j}, α^{j}) · D_{x}jv_{i} in R^{dN},
α^{i}∈ argmax_{α}iH^{i}(x^{i}, D_{x}iv_{i}, α^{i}), i = 1, . . . , N,

(42)

where the unknowns are the constants λ_{i} and the functions v_{i}(X), i = 1, . . . , N . Note that
this system is strongly coupled via the terms f^{j}(x^{j}, α^{j}) on the right hand side of the equation
for v_{i}, because α^{j} depends on D_{x}^{j}v_{j}.

Assume there exist functions α^{i} : R^{d}× R^{d}→ A^{i} such that

α^{i}(x, p) is a maximum point for α → H^{i}(x, p, α) ∀ x, p, (43)
α^{i} is locally Lipschitz and Z^{d}− periodic in x, (44)
and define

g^{i}(x, p) = −f^{i}(x, α^{i}(x, p)), i = 1, . . . N. (45)
Making this choice in (42) we get

N

X

j=1

L^{j}vi+ H^{i}(x^{i}, D_{x}^{i}vi) +X

j6=i

g^{j}(x^{j}, D_{x}^{j}vj) · D_{x}^{j}vi+ λi= F^{i}(X) in R^{dN}, (46)

i = 1, . . . , N . By a classical verification argument [12], if (vi, λi) ∈ C^{2}(T^{dN}) × R, i = 1, . . . , N ,
is a solution, then α^{i}(·, D_{x}iv_{i}(·)), i = 1, . . . , N , is a Nash equilibrium feedback. More precisely,
if one solves the stochastic differential equation

dX_{t}^{i}= f^{i}(X_{t}^{i}, α^{i}(X_{t}^{i}, D_{x}iv_{i}(X_{t})))dt + σ^{i}(X_{t}^{i})dW_{t}^{i}, X_{0}^{i} = x^{i}∈ R^{d}, i = 1, . . . , N. (47)
then α^{i}_{t} := α^{i}(X_{t}^{i}, D_{x}^{i}v_{i}(X_{t})), i = 1, . . . , N , is a Nash equilibrium for the cost functionals
(37), and λ_{i}, i = 1, . . . , N , are the values of the game corresponding to such equilibrium.

We want to derive from (46) a different system of elliptic equations of the form (1) from which we can still synthesize a Nash equilibrium feedback. We follow an idea introduced by Lasry and Lions in the seminal paper [34]. Given a solution of (46) consider the equilibrium process defined by (47). Let us assume that

D_{x}iv^{i} depends only on x^{i} for all i. (48)
Then the equations in (47) are decoupled and each of them defines a non-degenerate diffusion
X_{t}^{i} on the torus T^{d}, which has a unique ergodic invariant measure with density mi. It is
known that each mi ∈ C^{2}(T^{d}) solves the Kolmogorov-Fokker-Planck equation

L^{i∗}m_{i}− div_{x}i g^{i}(x^{i}, D_{x}iv_{i})m_{i} = 0,
Z

Q

m_{i}(x)dx = 1, i = 1, . . . , N, (49)
where L^{i∗} is the formal adjoint of L^{i}.

The next result exploits the adjoint structure of (46) and (49) to prove that multiplying the i-th equation in (46) byQ

j6=imj(x^{j}) and integrating over Q^{N −1}with respect to dx^{j}, j 6= i,
we arrive at

L^{i}v_{i}+ H^{i}(x, Dv_{i}) + λ_{i} = V^{i}[m] in R^{d}, (50)

where

V^{i}[m](x) =
Z

Q^{N −1}

F^{i}(x^{1}, . . . , x^{i−1}, x, x^{i+1}, . . . , x^{N})Y

j6=i

m_{j}(x^{j})dx^{j}. (51)
Note that (39) implies (7) for such V^{i} and then it is easy to see that also (6) holds.

Proposition 3.1. Assume (vi, λi) ∈ C^{2}(T^{dN}) × R, i = 1, . . . , N , is a solution of (46) satis-
fying (48) and m_{1}, . . . , m_{N} solve (49). Then, for all i, x 7→ v_{i}(x^{1}, . . . , x^{i−1}, x, x^{i+1}, . . . , x^{N})
solves (50) for all x^{1}, . . . , x^{N}.

Proof. Multiply the i-th equation in (46) by Q

j6=im_{j}(x^{j}) and integrate over Q^{N −1} with
respect to dx^{j}, j 6= i. Observe that, for k 6= i, integrating by parts with respect to x^{k} we get

Z

Q^{N −1}

L^{k}vi+ g^{k}(x^{k}, D_{x}^{k}vk) · D_{x}^{k}vi

Y

j6=i

mj(x^{j})dx^{j} =
Z

Q^{N −2}

Z

Q

vi

L^{k ∗}m_{k}− div_{x}k

g^{k}(x^{k}, D_{x}kv_{k})m_{k}

dx^{k} Y

j6=i,k

mj(x^{j})dx^{j}.

Then (49) gives Z

Q^{N −1}

X

j6=i

L^{j}v_{i}+ g^{j}(x^{j}, D_{x}jv_{j}) · D_{x}jv_{i} Y

j6=i

m_{j}(x^{j})dx^{j} = 0.

On the other hand, using the assumption (48) and R

Qmj(x)dx = 1, we also have Z

Q^{N −1}

L^{i}vi+ H^{i}(x^{i}, Dvi) + λi

Y

j6=i

mj(x^{j})dx^{j} = L^{i}vi+ H^{i}(x^{i}, Dvi) + λi.

Now considering again the i-th equation in (46) multiplied by Q

j6=imj(x^{j}) and integrated
over Q^{N −1} with respect to dx^{j}, j 6= i, and plugging into it the last two identities, we get that
vi restricted to the variable x^{i} solves (50) with V^{i} given by (51).

Remark 3.2. If the maximum point α^{i}(x, p) of H^{i}(x, p, ·) is unique, then H^{i} is differentiable
with respect to p and D_{p}H^{i}(x, p) = −f^{i}(x, α^{i}(x, p)) = g^{i}(x, p). Then the resulting system of
HJB-KFP equations can be written in the same form as in the Lasry-Lions papers [34, 36],
namely,

L^{i}vi+ H^{i}(x, Dvi) + λi= V^{i}[m] in Q,
L^{i∗}mi− div D_{p}H^{i}(x, Dvi)mi = 0 in Q,
R

Q^{d}m_{i}(x)dx = 1, m_{i} > 0, R

Q^{d}v_{i}(x)dx = 0, i = 1, . . . , N.

(52)

These notations also show that the KFP equations are the linearizations of the HJB equations
around their solution v_{1}, . . . , v_{N}.

From the theory of the previous section we get the following existence result.

Corollary 3.3. In addition to (2) and the assumptions of this section suppose that H^{i} satisfy
either condition C1 or condition C2. Then there exist λ_{1}, . . . , λ_{N} ∈ R, v1, . . . , v_{N} ∈ C^{2}(Q),
m1, . . . , mN ∈ W^{1,p}(Q), 1 ≤ p < ∞, which solve (1) with V^{i} and g^{i} given by (51) and (45),
respectively.

3.2 Synthesis of Nash equilibria

Next we prove a verification result that produces Nash feedback equilibria for the N -person
differential game from any solution of the HJB-KFP system of PDEs. We recall that a
feedback for the i-th player is a Lipschitz map α^{i} : R^{d} → A^{i} that generates a process X_{t}^{i}
solving dX_{t}^{i} = f^{i}(X_{t}^{i}, α^{i}(X_{t}^{i}))dt+σ^{i}(X_{t}^{i})dW_{t}^{i}, X_{0}^{i} = x^{i}, and an admissible control α^{i}_{t}= α^{i}(X_{t}^{i}).

As usual, when we plug a feedback in the functional J^{i} we mean its associated admissible
control. A Nash equilibrium of the N -person game of Section 3.1 for the initial position
X = (x^{1}, . . . , x^{N}) is a vector of admissible controls (α^{1}, . . . , α^{N}) such that

J^{i}(X, α^{1}, . . . , α^{i−1}, α^{i}, α^{i+1}, . . . , α^{N}) ≥ J^{i}(X, α^{1}, . . . , α^{N}) for all α^{i} admissible control.

Theorem 3.4. Let λi, vi, mi, i = 1, . . . , N be a solution of the system (1) as in the preceding Corollary 3.3. Then

α^{i}(x) := α^{i}(x, Dvi(x)), x ∈ R^{d}, i = 1, . . . , N, (53)
define a feedback which is a Nash equilibrium for all initial positions X ∈ Q^{N} of the control
system (36). In addition, for each X = (x^{1}, . . . , x^{N}),

λi = J^{i}(X, α^{1}, . . . , α^{N}) = lim inf

T →+∞

1 TE

Z T 0

L^{i}(X^{i}_{t}, α^{i}(X^{i}_{t})) + F^{i}(X^{1}_{t}, . . . , X^{N}_{t} )dt

, (54)

where X^{i}_{t} is the process associated to the feedback α^{i} with X^{i}_{0}= x^{i}.

Proof. We follow the outline of proof in [34, 36]. Let us first check (54). The processes
X^{i}_{t} are independent diffusions on the torus, so each of them has a unique ergodic invariant
measure, which must be m_{i}, see Chapter 3, Section 3 of [16] or [3]. Then the joint process
(X^{1}_{t}, . . . , X^{N}_{t} ) is ergodic with invariant measure QN

i=1mi(x^{i}), and therefore the right hand
side of (54) is

J^{i}(X, α^{1}, . . . , α^{N}) =
Z

Q

L^{i}(x, α^{i}(x))dm_{i}(x) +
Z

Q^{N}

F^{i}(x^{1}, . . . , x^{N})

N

Y

j=1

dm_{j}(x^{j}).

On the other hand, the Ito-Dynkin formula gives E

h

vi(X^{i}_{T}) − vi(x^{i})
i

= E

Z T 0

−g^{i}(X^{i}_{t}, Dvi(X^{i}_{t})) · Dvi(X^{i}_{t}) − L^{i}vi(X^{i}_{t})

dt

By (43) and (45)

g^{i}(x, p) · p = H^{i}(x, p) + L^{i}(x, α(x, p)),
so the HJB equation (50) implies

Eh

v_{i}(X^{i}_{T}) − v_{i}(x^{i})i

=

λ_{i}T − E

Z T

0

L^{i}(X^{i}_{t}, α^{i}(X^{i}_{t})) +
Z

Q^{N −1}

F^{i}(x^{1}, . . . , x^{i−1}, X^{i}_{t}, x^{i+1}, . . . , x^{N})Y

j6=i

dm_{j}(x^{j})dt

.

We divide by T and let T → ∞. The left-hand side vanishes, whereas the right hand side
tends to λi− J^{i}(X, α^{1}, . . . , α^{N}) , which proves (54).

Next we check that the feedback law (53) defines a Nash equilibrium. We change the
control of the i-th player into an arbitrary admissible control α^{i}_{t} and get

1

TEv_{i}(X_{T}^{i}) − vi(x^{i}) = 1
TE

Z T 0

f^{i}(X_{t}^{i}, α^{i}_{t}) · Dvi(X_{t}^{i}) − L^{i}vi(X_{t}^{i}) dt

≥ λ_{i}− 1
TE

Z T

0

L^{i}(X_{t}^{i}, α^{i}_{t}) +
Z

Q^{N −1}

F^{i}(x^{1}, . . . , x^{i−1}, X_{t}^{i}, x^{i+1}, . . . , x^{N})Y

j6=i

dmj(x^{j})dt

.

By the ergodicity of the joint process (X^{1}_{t}, . . . , X^{i−1}_{t} , X_{t}^{i}, X^{i+1}_{t} , . . . , X^{N}_{t} ) the last term on the
right-hand side tends to J^{i}(X, α^{1}, . . . , α^{i−1}, α^{i}, α^{i+1}, . . . , α^{N}) as T → ∞, and then

J^{i}(X, α^{1}, . . . , α^{i−1}, α^{i}, α^{i+1}, . . . , α^{N}) ≥ λ_{i}= J^{i}(X, α^{1}, . . . , α^{N})
for all X ∈ Q^{N}.

3.3 Examples with unconstrained controls

In this section we consider A^{i}= R^{d} for all i and the system affine in the control, i.e.,
f^{i}(x, α) = ϕ^{i}(x) +

d

X

k=1

αkf_{k}^{i}(x) = ϕ^{i}(x) + Φ^{i}(x)α, (55)
where Φ^{i} is a square matrix whose columns are the vector fields f_{k}^{i}, k = 1, . . . , d, and all
vector fields are Lipschitz in Q. Then

H^{i}(x, p) := −p · ϕ^{i}(x) + sup

α∈R^{d}

{−p · Φ^{i}(x)α − L^{i}(x, α)}.

Assume L^{i} is β-H¨older in x, uniformly as α varies in any bounded subset, and

α→∞lim inf

x∈QL^{i}(x, α)/|α| = +∞. (56)

Then the sup in the definition of H^{i} is attained and

H^{i}(x, p) = L^{i}^{∗} x, −Φ^{i}(x)^{t}p − p · ϕ^{i}(x), L^{i}^{∗}(x, q) := max

α∈R^{d}

{q · α − L^{i}(x, α)}, (57)
i.e., L^{i}^{∗}(x, ·) is the convex conjugate of L^{i}(x, ·). Moreover H^{i} is locally Lipschitz in p and
β-H¨older in x. Now the other conditions of the existence theorems for the elliptic system and
of the verification Theorem 3.4 can be checked on the last expression of H^{i}.

Example 3.5. If for a γ > 0

L^{i}(x, α) ≥ γ(|α|^{2}− 1),
then L^{i}^{∗}(x, q) ≤ |q|^{2}/γ + γ and

|H^{i}(x, p)| ≤ C(1 + |p|^{2}), ∀x ∈ Q, p ∈ R^{d}.

Therefore H^{i} satisfy condition C2 and Theorems 2.5 and 2.6 can be used. If (43) and (44)
hold, then all the assumptions of Corollary 3.3 and Theorem 3.4 are verified.

It remains to check the conditions (43) and (44) on the existence of a Lipschitz argmax
for the Hamiltonians. A sufficient condition for them is that L^{i} be differentiable with respect
to α and D_{α}L^{i}(x, ·) be invertible (a fact related to the strict convexity of L^{i} in α) and locally
Lipschitz. Then the sup in the definition of H^{i} is attained at a unique value

α^{i}(x, p) = (D_{α}L^{i})^{−1} x, −Φ^{i}(x)^{t}p ,

which is a locally Lipschitz function of x and p. Next we give two examples where we can easily check all the conditions for the existence of a feedback Nash equilibrium and give a formula for it.

Example 3.6. Consider

L^{i}(x, α) = α^{t}B^{i}(x)α,

with positive definite matrices B^{i}(x) Lipschitz in Q, and the affine system (55). Then
L^{i}^{∗}(x, q) = q^{t}B^{i}(x)^{−1}q/4,

so H^{i} grows at most quadratically in p and satisfies condition C2. Moreover
α^{i}(x, p) = −1

2B^{i}(x)^{−1}Φ^{i}(x)^{t}p

verifies (43) and (44), so Remark 3.2 and Theorem 3.4 apply and the Nash equilibrium feedback is linear in p = Dvi.

If H^{i} do not satisfy the natural growth condition (5) we must check condition C1. By
assumption (56) L^{i}^{∗} is superlinear in q, so if the matrix Φ^{i}(x) is nonsingular also H^{i} is
superlinear in p, i.e., it satisfies (3). The technical condition (4) is harder to check in general
but can be easily seen in the next model problem.

Example 3.7. Consider

L^{i}(x, α) = c^{i}(x)|α|^{γ}
γ

for some Lipschitz c^{i} > 0 and γ > 1, and the affine system (55) with Φ^{i}(x) nonsingular for
all x. Then

L^{i}^{∗}(x, q) = c^{i}(x)^{1/(1−γ)}|q|^{γ/(γ−1)}γ − 1
γ ,

so we can compute DxH^{i} by (57) and see that (4) is satisfied for any choice of θ^{i} > 0. If,
moreover, F^{i} are Lipschitz, V^{i} defined by (51) verify (7) with β = 1, Then Theorem 2.1
applies to this case with g^{i} given by (45). Moreover

α^{i}(x, p) = c^{i}(x)^{1/(1−γ)}|q|(2−γ)/(γ−1)q, q = −Φ^{i}(x)^{t}p,
satisfies (43) and (44), so also Remark 3.2 and Theorem 3.4 apply.