# A priori estimates and large population limits for some nonsymmetric Nash
systems with semimonotonicity

Marco Cirant  and  Davide Francesco Redaelli

###### Abstract.

We address the problem of regularity of solutions
$u^{i}(t,x^{1},\ldots,x^{N})$ to a family of semilinear parabolic systems of
$N$ equations, which describe closed-loop equilibria of some $N$-player
differential games with quadratic Lagrangian in the velocity, running costs
$f^{i}(x)$ and final costs $g^{i}(x)$. By global (semi)monotonicity
assumptions on the data $f,g$, and assuming that derivatives of $f^{i},g^{i}$
in directions $x_{j}$ are of order $1/N$ for $j\neq i$, we prove that
derivatives of $u^{i}$ enjoy the same property. The estimates are uniform in
the number of players $N$. Such behaviour of the derivatives of $f^{i},g^{i}$
arise in the theory of Mean Field Games, though here we do not make any
symmetry assumption on the data.

Then, by the estimates obtained we address the convergence problem
$N\to\infty$ in a “heterogeneous” Mean Field framework, where players all
observe the empirical measure of the whole population, but may react
differently from one another.

###### Contents

  1. 1 Introduction
  2. 2 Recurring general notation
  3. 3 Summary of main results
    1. 3.1 Two notions of (semi)monotonicity
    2. 3.2 Estimates on the Nash system
    3. 3.3 Nonsymmetric Mean-Field Games
  4. 4 Estimates under the one-sided Lipschitz condition on the drift
    1. 4.1 A Lipschitz estimate for linear equations
    2. 4.2 First-order derivatives of the value functions
    3. 4.3 Second-order derivatives of the value functions
    4. 4.4 Third-order derivatives of the value functions
  5. 5 The $\mathscr{D}$*semimonotone case
  6. 6 The $\mathscr{L}$*semimonotone case
  7. 7 Analysis of the $N\to\infty$ limit
    1. 7.1 Characterisation of the limit
    2. 7.2 A continuum of MFG systems
  8. A An existence and uniqueness theorem for the Nash system
  9. B An estimate on the space-time regularity for the heat equation

##  1\. Introduction

Consider the following semilinear (backward) parabolic system for the unknowns
$u^{i}=u_{N}^{i}:[0,T]\times({\mathbb{R}}^{d})^{N}\to{\mathbb{R}}$,
$i=1,\ldots,N$,

(1.1)
$\begin{cases}-\partial_{t}u^{i}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i})+\dfrac{1}{2}|D_{i}u^{i}|^{2}+\displaystyle\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D_{j}u^{j}\cdot
D_{j}u^{i}=f^{i}\vspace{-0.3cm}\\\ u^{i}|_{t=T}=g^{i}\end{cases}$

stated in
$[0,T]\times({\mathbb{R}}^{d})^{N}\ni(t,\boldsymbol{x})=(t,x^{1},\dots,x^{N})$.
The main goal of this paper is to derive estimates on $u^{i}$ and their
space/time derivatives that are stable with respect to the number of
equations; that is, uniformly in $N$.

The data are the maps
$f^{i},g^{i}\colon({\mathbb{R}}^{d})^{N}\to{\mathbb{R}}$, the parameters
$\sigma>0$ and $\beta\geqslant 0$ and the horizon $T\geqslant 0$. The symbol
$\boldsymbol{\mathsf{I}}=\mathsf{I}_{Nd}$ denotes the $Nd$-dimensional
identity matrix and
$\boldsymbol{\mathsf{J}}=\mathsf{J}_{N}\otimes\mathsf{I}_{d}$, where
$\mathsf{J}_{N}=1_{N}\otimes 1_{N}$ is the $N$-dimensional matrix of
ones.111We use $\otimes$ to denote both the Kronecker product between matrices
and the outer product between vectors. $D^{2}=D^{2}_{\boldsymbol{x}}$ denotes
the Hessian with respect to $\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}$ and
$D_{j}=D_{x^{j}}$ denotes the gradient with respect to
$x^{j}\in{\mathbb{R}}^{d}$. Finally, the symbol $\cdot$ denotes the Euclidean
inner product, and $|\cdot|$ denotes the Euclidean norm. Hence, (1.1) is a
compact formulation of

$\begin{dcases}\begin{multlined}-\partial_{t}u^{i}(t,\boldsymbol{x})-\sigma\sum_{1\leqslant
j\leqslant N}\Delta_{x^{j}}u^{i}(t,\boldsymbol{x})-\beta\sum_{1\leqslant
j,k\leqslant N}{\rm tr}\,D^{2}_{x^{j}x^{k}}u^{i}(t,\boldsymbol{x})\\\
+\dfrac{1}{2}|D_{x_{i}}u^{i}(t,\boldsymbol{x})|^{2}+\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D_{x^{j}}u^{j}(t,\boldsymbol{x})\cdot
D_{x^{j}}u^{i}(t,\boldsymbol{x})=f^{i}(t,\boldsymbol{x})\\\
\end{multlined}-\partial_{t}u^{i}(t,\boldsymbol{x})-\sigma\sum_{1\leqslant
j\leqslant N}\Delta_{x^{j}}u^{i}(t,\boldsymbol{x})-\beta\sum_{1\leqslant
j,k\leqslant N}{\rm tr}\,D^{2}_{x^{j}x^{k}}u^{i}(t,\boldsymbol{x})\\\
+\dfrac{1}{2}|D_{x_{i}}u^{i}(t,\boldsymbol{x})|^{2}+\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D_{x^{j}}u^{j}(t,\boldsymbol{x})\cdot
D_{x^{j}}u^{i}(t,\boldsymbol{x})=f^{i}(t,\boldsymbol{x})\\\ \\\\[-20.0pt]
u^{i}(T,\boldsymbol{x})=g^{i}(\boldsymbol{x}).\end{dcases}$

System (1.1) describes Markovian Nash equilibria in $N$-player differential
games, in particular it characterises the value function $u^{i}$ of the $i$-th
agent for each $i=1,\ldots,N$. In our setting, agents control via feedbacks
$\alpha^{i}=\alpha^{i}(t,\boldsymbol{x})$ their own states, which are driven
by the following ${\mathbb{R}}^{d}$-valued SDEs on $[0,T]$:

(1.2)
$\mathrm{d}X_{t}^{i}=\alpha^{i}(t,\boldsymbol{X}_{t})\,\mathrm{d}t+\sqrt{2\sigma}\,\mathrm{d}B_{t}^{i}+\sqrt{2\beta}\,\mathrm{d}W_{t},\quad
i\in\\{1,\dots,N\\},$

where the $B^{i}_{t}$’s and $W_{t}$ are $d$-dimensional independent Brownian
motions. The Brownian motions $B^{i}_{t}$ correspond to the individual noises,
while $W_{t}$ is the so-called common noise, as it is the same for all the
equations. The $i$-th agents aims at mimimising the following cost functional

$\alpha^{i}\mapsto\mathbb{E}\biggl{[}\,\int_{0}^{T}\Bigl{(}\,\dfrac{1}{2}|\alpha^{i}(s,\boldsymbol{X}_{s})|^{2}+f^{i}(X_{s})\Bigr{)}\,\mathrm{d}s+g^{i}(\boldsymbol{X}_{T})\biggr{]}.$

It is known that the choice
$\alpha^{*,i}(t,\boldsymbol{x})=-D_{i}u^{i}(t,\boldsymbol{x})$ characterises
Nash equilibria, see for instance [16, 23]. Moreover, since one expects
uniqueness of solutions to (1.1) by its (uniformly) parabolic structure, such
equilibria are unique.

For fixed $N$, existence and uniqueness results for semilinear systems of the
form (or close to) (1.1) are now classical, see for example [6, 7, 17, 23, 28,
34]. Nevertheless, the methods that are involved seem to be strongly sensitive
to the dimension $N$. The main reason is that one typically wants to employ
parabolicity to get existence, uniqueness and regularity, but its smoothing
effect may deteriorate as $N$ increases, while the strong transport term $\sum
D_{j}u^{j}\cdot D_{j}u^{i}$ may take over. In fact, it is now widely accepted
that (1.1) should be regarded as a nonlinear transport system for large $N$.
On top of that, the quadratic behavior in the gradients is critical: for some
quadratic systems, one may even have that solutions fail to be Hölder
continuous [22] (at least in the elliptic case). For the sake of self-
containedness, we included an existence theorem for (1.1) in the appendix (for
any fixed $N$).

Symmetric players, MFG and monotonicity. If interactions are assumed to be
symmetric (that is,
$f^{i},g^{i}(\boldsymbol{x})=f,g(x^{i},\boldsymbol{x}^{-i})$ and $f,g$ are
symmetric in the variable $\boldsymbol{x}^{-i}$, which is just the vector
$\boldsymbol{x}$ with the $i$-th component omitted, one expects, borrowing
intuitions from Statistical Mechanics, that a simplified limit model could be
derived in the limit $N\to\infty$. In this direction, the theory of Mean Field
Games, originated by the works [27, 35], has seen during the last fifteen
years extensive development in many fields of mathematics and applied
sciences. Within this framework, the problem can be embedded into the space of
probability measures $\mathcal{P}({\mathbb{R}}^{d})$: one may write

$f^{i}(\boldsymbol{x})=F(x^{i},m_{\boldsymbol{x}^{-i}}),$

where $m_{\boldsymbol{x}^{-i}}\vcentcolon=\frac{1}{N-1}\sum_{i\neq
j}\delta_{x^{j}}$ is the empirical measure and
$F:{\mathbb{R}}^{d}\times\mathcal{P}({\mathbb{R}}^{d})\to{\mathbb{R}}$, and
the same for $g^{i}$ (everything that will be said on $f^{i}$ in this
introduction will hold for $g^{i}$ also). Note that not only one is assuming
symmetry, but also that interactions between players are “small”, since a
variation in $x^{j}$ produces a variation in $F$ (and then in $f^{i}$) of
order $1/N$ if $j\neq i$. Nevertheless, this smallness is compensated by the
number $N$ of interactions. Ideally, since each $u^{i}$ is also symmetric in
$\boldsymbol{x}^{-i}$ one may hope in the convergence of
$u^{i}(t,\boldsymbol{x})$ to $U(t,x^{i},m)$, where $U$ is a function over
$[0,T]\times{\mathbb{R}}^{d}\times\mathcal{P}({\mathbb{R}}^{d})$ that should
be characterised by being the solution of the formal limit of (1.1). This
limit object is called the Master Equation, and it is a nonlinear transport
equation over the set of probability measures. The function $U(t,x,m)$ can be
regarded as the value function of a player with state $x$ at time $t$,
observing a population distribution $m$. This approach to the limit of
$N$-player symmetric games has been discussed for the first time by P.-L.
Lions during his lectures at the Collège de France [36]. Since then, the
derivation of the estimates needed to complete the program in a general
framework remained an open problem.

Such a convergence result is of primary importance. Not only it justifies the
use of the limit object (broadly speaking, the MFG theory) to approximate Nash
equilibria of the $N$-players problem, but it carries fundamental
implications. For instance, the optimal control in the limit for each agent
reads as $-D_{x}U(t,x^{i},m)$, so it has decentralized structure, or, in other
words, it is open-loop in nature. This fact reflects into the possibility of
characterising Nash equilibria in the limit by a simple (compared to the Nash
system (1.1)) backward-forward PDE system that goes under the name of MFG
system.

The convergence problem has been matter of extensive research, and the
fundamental contribution from the PDE perspective has been given by [14].
Interestingly, since the estimates for the Nash system (1.1) seemed to be out
of reach, the authors proposed a different approach: from the stability
properties of the MFG system, one may produce smooth solutions $U(t,x,m)$ to
the Master equation. These can be in turn projected over empirical measures;
the $v^{i}(\boldsymbol{x})=U(t,x^{i},m_{\boldsymbol{x}^{-i}})$ are then shown
to be “almost” solutions to the Nash system. Comparing the $v^{i}$ with the
solutions $u^{i}$, one observes a discrepancy that vanishes as $N\to\infty$.

This program has been carried out under the so-called Lasry-Lions monotonicity
assumption

(1.3)
$\int_{{\mathbb{R}}^{d}}\big{(}F(x,m_{1})-F(x,m_{2})\big{)}(m_{1}-m_{2})(\mathrm{d}x)\geqslant
0\qquad\forall m_{1},m_{2}\in\mathcal{P}({\mathbb{R}}^{d}),$

which is crucial to get uniqueness and stability at the level of the MFG
system. Later on, other conditions guaranteeing uniqueness of solutions have
been formulated [1, 5, 19, 24, 25, 39]. Among them, we focus our attention now
on the displacement monotonicity assumption

(1.4)
$\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\big{(}D_{x}F(x,m_{1})-D_{x}F(y,m_{2})\big{)}\cdot(x-y)\,\mu(\mathrm{d}x,\mathrm{d}y)\geqslant
0$

for all $m_{1},m_{2}$ and $\mu$ having $m_{1}$ and $m_{2}$ as first and second
marginal, respectively. This has been used successfully to obtain the well-
posedness of the Master Equation in [24], and later in [38] to deduce
uniqueness of solutions to the MFG system. To the best of our knowledge, the
convergence problem in this framework has not been addressed, at least in the
closed-loop case (for a recent result in the open-loop framework, see [29]).
An important observation is that the monotonicity assumptions (1.3) and (1.4)
both propagate at the level of the (limit) value function $U(t,x,m)$; that is,
(1.3) or (1.4) hold for $U(t,\cdot,\cdot)$ in place of $F(\cdot,\cdot)$ for
any $t\in[0,T]$.

The estimates. To attack the convergence problem directly from the Nash system
(that is, without employing smooth solutions of the Master Equation), one has
to look for estimates of the following form:

(1.5) $|D_{x_{i}}u^{i}|,|D_{x_{i}}(D_{x_{i}}u^{i})|\lesssim
1,\qquad|D_{x_{j}}u^{i}|,|D_{x_{j}}(D_{x_{i}}u^{i})|\lesssim\frac{1}{N}\qquad\forall\,i,j\neq
i.$

Roughly speaking, if such bounds are uniform in $N$, a compactness argument
allows to define (up to subsequences) a limit value function $U$ and a limit
drift $D_{x}U$, which is the optimal feedback of a typical player. These will
be Lipschitz continuous in the $(x,m)$ variables. Our strategy to derive these
estimates originated within the displacement monotone setting, and it has been
developed first in some simple linear-quadratic case [20, Section 3]. It can
be described as follows.

As we already mentioned, under the monotonicity condition (1.4) one knows that
$U(t,\cdot,\cdot)$ also satisfies (1.4) for any $t\in[0,T]$, see for example
[38, eq. (4.7)]. Hence, by evaluating such inequality on empirical measures,
and assuming that $u^{i}(t,\boldsymbol{x})$ is close to
$U(t,x^{i},m_{\boldsymbol{x}^{-i}})$, one observes that for all $t\in[0,T]$

(1.6) $\sum_{1\leqslant i\leqslant
N}\big{(}D_{i}u^{i}(t,\boldsymbol{x})-D_{i}u^{i}(t,\boldsymbol{y})\big{)}\cdot(x^{i}-y^{i})\geqslant-
c_{N}|\boldsymbol{x}-\boldsymbol{y}|^{2},$

where $c_{N}$ vanishes as $N\to\infty$ (see Remark 3.4). Recall that
$\alpha^{*,i}(t,\boldsymbol{x})=-D_{i}u^{i}(t,\boldsymbol{x})$ is the optimal
drift for the $i$-th player. Therefore, the previous inequality reads as a
one-sided Lipschitz condition (or dissipativity) on the global drift vector
$\boldsymbol{\alpha}^{*}$. This property turns out to be crucial in deriving
estimates on the derivatives of $u^{i}$. For example, one can deduce gradient
bounds that are independent of the dimension $N$ for $u^{i}$, which solves in
fact

$-\partial_{t}u^{i}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i})+\displaystyle\sum_{1\leqslant
j\leqslant N}(-\alpha^{*,j})\cdot
D_{j}u^{i}=f^{i}+\dfrac{1}{2}|D_{i}u^{i}|^{2},$

by using doubling of variables methods (see Lemma 4.3 below). This observation
on the structure of $\boldsymbol{\alpha}^{*}$ suggests to pursue the following
program:

  1. (1)

assume first that the vector $(D_{i}u^{i})$ satisfies the one-sided Lipschitz
condition (1.6) with some $c_{N}=M>0$ (so, uniformly in $N$), and, with this
condition in force, prove the desired estimates on the derivatives of $u^{i}$;

  2. (2)

employ the estimates (that will depend on the value of $M$, but not on $N$),
along with the full structure of the Nash system to show that in fact the
vector $(D_{i}u^{i})$ satisfies (1.6) with $c_{N}=M/2$, at least for $N$ large
enough.

This shows that the Nash system enforces one-sided Lipschitz estimates on the
optimal drift, or, in other words, (1.6) defines (with suitable $c_{N}=M$) an
invariant set for the equilibrium controls. Practically, it allows to show
that solutions $u^{i}$ satisfy the one-sided Lipschitz condition for some
constant that does not depend on $N$, implying the desired estimates on
$u^{i}$.

We will first show that this strategy works in a framework which resembles the
one of displacement monotonicity in the symmetric case. In particular, we will
prove that such an $M$ exists, and it has to be smaller and smaller as $T$
increases. We stress that we are not going to assume any symmetry on the data,
but only the fact that derivatives of $f^{i},g^{i}$ with respect to $x^{j}$
behave as in the MF (Mean Field) case. This is why we will use the terminology
Mean-Field-like costs, see (MF) below. Similarly, we will not assume (1.4),
but rather formulate the condition

$\sum_{1\leqslant i\leqslant
N}\,(D_{i}f^{i}(\boldsymbol{x})-D_{i}f^{i}(\boldsymbol{y}))\cdot(x^{i}-y^{i})\geqslant-C|\boldsymbol{x}-\boldsymbol{y}|^{2}\quad\forall\,\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N}$

which reads like (1.4) once it is specialized to the MF setting and $C=0$.
Since we are going to allow for a negative right-hand side, we will refer to
this case as the $\mathscr{D}$*semimonotone one, see (DS) below.

Let us now come back to the Lasry-Lions monotonicity assumption (1.3), and
discuss how the previous program can be adapted to this case. It is convenient
to look at the one-sided Lipschitz condition (1.6) via the pointwise
formulation (cf. Remark 3.3)

(1.7) $\sum_{1\leqslant i,j\leqslant
N}D^{2}_{ji}u^{i}(t,\boldsymbol{x})\xi^{i}\cdot\xi^{j}\geqslant-
c_{N}|\boldsymbol{\xi}|^{2}\qquad\forall\,\boldsymbol{\xi}\in({\mathbb{R}}^{d})^{N}.$

It turns out that, under Lasry-Lions monotonicity, the counterpart of (1.7) is
(as before by projecting onto empirical measures the Lasry-Lions monotonicity
of $U$)

$\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\ i\neq
j\end{subarray}}D^{2}_{ji}u^{i}(t,\boldsymbol{x})\xi^{i}\cdot\xi^{j}=\sum_{1\leqslant
i,j\leqslant
N}D^{2}_{ji}u^{i}(t,\boldsymbol{x})\xi^{i}\cdot\xi^{j}-\sum_{1\leqslant
i\leqslant N}D^{2}_{ii}u^{i}(t,\boldsymbol{x})\xi^{i}\cdot\xi^{i}\geqslant-
c_{N}|\boldsymbol{\xi}|^{2}.$

Therefore, we have now an off-diagonal one-sided Lipschitz information on the
drift. To get the full one-sided Lipschitz control, it is necessary to bound
from below the term
$\sum_{i}D^{2}_{ii}u^{i}(t,\boldsymbol{x})\xi^{i}\cdot\xi^{i}$, which encodes
a sort of “global” semiconvexity of the set of value functions $u^{i}$. This
introduces an additional step in our analysis, that is handled as before.
Under Lasry-Lions monotonicity, we show that the Nash system enforces not only
the one-sided Lipschitz estimates on the optimal drift
$\boldsymbol{\alpha}^{*}$, but also this kind of global semiconvexity. In
other words, in step (1) we assume that both
$\sum_{i,j}D_{ij}u^{i}(t,\boldsymbol{x})\xi^{i}\cdot\xi^{j}$ and
$\sum_{i}D^{2}_{ii}u^{i}(t,\boldsymbol{x})\xi^{i}\cdot\xi^{i}$ are bounded
below, and in step (2) we verify that these bounds are improved. To do so, as
before we do not require symmetry, but the Mean-Field like assumption (MF) and
the global (with respect to $i$) condition

$\sum_{1\leqslant i\leqslant
N}\big{(}f^{i}(\boldsymbol{x})-f^{i}(\boldsymbol{x}^{-i},y^{i})-f^{i}(\boldsymbol{y}^{-i},x^{i})+f^{i}(\boldsymbol{y})\big{)}\geqslant-\kappa|\boldsymbol{x}-\boldsymbol{y}|^{2}\quad\forall\,\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N},$

$(\boldsymbol{x}^{-i},y)=(x^{1},\dots,x^{i-1},y,x^{i+1},\dots,x^{N})$, as the
monotonicity condition. This inquality will be called the
$\mathscr{L}$*semimonotone assumption (see (LS) below), and becomes (1.3) once
it is specialized to the MF setting and $\kappa=0$.

Our main result on the estimates for $u^{i}$ is stated in Theorem 3.6:
assuming Mean-Field-like costs and either $\mathscr{D}$*semimonotonicity or
$\mathscr{L}$*semimonotonicity, the desired (a priori) estimates follow for
any fixed $T>0$, sufficiently small semimonotonicity constants and
sufficiently large $N\in\mathbb{N}$. Note that we are not exactly obtaining
(1.5); regarding second-order derivatives, we get a bound of the form
$\sup_{i}\sum_{j\neq i}|D_{j}D_{i}u^{i}|^{2}\lesssim 1/N$. This is still
enough, in the symmetric case described below, to produce in the limit
$N\to\infty$ a Lipschitz function (with respect to the Wasserstein $W_{2}$
distance).

A few comments on the two steps (1) and (2) are now in order. Step (2) is
achieved, in the $\mathscr{D}$*semimonotone case, in Section 5. We evaluate
the quantity
$\sum_{i}\big{(}D_{i}u^{i}(t,\boldsymbol{x})-D_{i}u^{i}(t,\boldsymbol{y})\big{)}\cdot(x^{i}-y^{i})$
along optimal trajectories, see Proposition 5.1. One observes some nice
properties (signs …) that appear also when operating at the limit, with an
additional term, which is proven to be of order $1/N$ provided that
derivatives of $u^{i}$ are suitably controlled up to the third order. A
similar strategy is developed in Section 6 for the $\mathscr{L}$*monotone
case, see in particular Proposition 6.3.

The estimates under the one-sided Lipschitz condition are the core of step (1)
presented above, which are developed in Section 4. Mainly two techniques are
employed there: the method of doubling variables and the Bernstein method.
Both are well suited to exploit the a priori one-sided assumption on
$D_{i}u^{i}$ (the former uses (1.6), while the latter works well in the form
(1.7)). To reach the third-order derivatives, several steps are necessary:
sometimes we address each equation separately (Lemma 4.3, Proposition 4.8),
sometimes we need to proceed by estimating derivatives of $u^{i}$ for all $i$
at the same time (Proposition 4.6, Proposition 4.7, Proposition 4.9,
Proposition 4.11).

The large population limit. With the estimates obtained in the first part of
the work, we address the convergence problem $N\to\infty$. While in the full
symmetric case, where all the players react in the same way with respect to
the empirical measure of the population, we are able to obtain the classical
MFG limit (see Proposition 7.8), the convergence problem in full generality
appears to be at this stage too complicated. We put ourselves in a sort of
intermediate setting: each player observes the empirical measure as in the
standard MFG theory, but players can react differently. More precisely, we
assume that the costs $f^{i}$ are taken from a pool of possible costs; that
is,

$f^{i}(\boldsymbol{x})=F(\lambda^{i},x^{i},m_{\boldsymbol{x}^{-i}})$

where the parameter $\lambda^{i}\in[0,1]$. If $\lambda^{i}$ varies in a finite
subset of $[0,1]$ for all $N$, one may think of a multi-population MFG as in
[8, 18], but $\lambda^{i}$ can actually vary in the continuum $[0,1]$. To
obtain a convergence result, we assume Lipschitz regularity of $F$ in the
$\lambda$ variable. This setting is somehow similar to the one of Graphon MFG
[3, 11, 33], with the difference that typically the cost functionals are
player-independent in Graphon games, but may depend on weighted empirical
measures, and weights are player-dependent. Here, costs are player-dependent,
but they see the standard empirical measure of the others.

Our convergence argument is in the spirit of [33]. If one thinks of any player
$i$ in terms of its label-state couple $(\lambda^{i},X^{i}_{t})$, it is
natural to work in the limit with probabilities $\mu$ over
$[0,1]\times{\mathbb{R}}^{d}$. More precisely, we are able to define a value
function $U(\lambda,t,x,\mu)$ from limits of the Nash system. Furthermore, $U$
generates a solution of a generalised MFG system, that is a continuum of
classical MFG systems that are parametrised by $\lambda$. This is the second
main result of this work, and it is stated in Theorem 3.10.

For simplicity, we proceed in this part without common noise; that is,
$\beta=0$. A compactness argument allows to define $U$ from limits of $u^{i}$
(Theorem 7.5), and the convergence is strong enough to pass to the limit the
notion of Nash equilibrium (via a propagation of chaos argument), see
Proposition 7.8 and Theorem 7.9; this in turn yields the MFG system. We first
show how to get convergence in the usual MFG setting; that is, for
$\lambda^{i}=\lambda^{N,i}=\lambda$ for all $N,i$. In the multi-population
(but finite) case, the argument is similar, even though it is technically more
involved. Finally, the general case is obtained by approximating via a finite
(and increasing) number of populations.

Note that, since compactness arguments are involved, convergence is obtained
up to subsequences. Clearly, under displacement or Lasry-Lions monotonicity
and in the classical homogeneous MFG setting one has that limit objects (MFG
equilibria, or solutions to the MFG system, or solutions $U(t,x,m)$ of the
Master Equation, …) are unique, hence convergence is a posteriori along the
full sequence. Nevertheless, we prefer not to stress this point here, as in
fact we are requiring the milder assumptions of semimonotonicity, and we allow
for a certain degree of heterogeneity among agents. While we are not aware of
uniqueness theorems in such generality, we believe that they should be true,
as consequence of the fact that $U(\lambda,t,x,\mu)$ and
$D_{x}U(\lambda,t,x,\mu)$ are Lipschitz continuous.

Further remarks.  In this work, we propose a new PDE approach to the
convergence problem in large population stochastic differential games, which
allows to treat in an unified way two scenarios of monotonicity. In addition,
by employing the notion of semimonotonicity, we can consider at the same time
the large time horizon $T$ case, where one needs almost monotonicity, and the
short time $T$ case, where (almost) no monotonicity is needed, as in [13].

Besides the aforementioned approach via the Master Equation, let us mention
that different probabilistic methods have been developed to tackle the
convergence problem in MFG, within the framework of closed-loop equilibria. In
[42], propagation of chaos for BSDEs arguments are employed. In [21], by
making use of the notion of measure-valued MFG equilibria, convergence is
studied for a general class of MFG of controls; remarkably, no uniqueness of
MF equilibria is assumed here, as in the important works [31, 32], where semi-
Markov equilibria are obtained by means of compactness arguments.

By developing further the analytic approach proposed here, we plan to extend
our convergence arguments in the following two directions, for which, to the
best of our knowledge, no results are yet available. First, the
$\mathscr{D}$*monotone setting appears to be strong enough to guarantee
estimates that are not only independent of $N$, but also on $T$, thus allowing
to study the infinite-horizon (or ergodic) problem. Second, we aim at
analysing the vanishing viscosity limit, where the idiosyncratic noise
$\sigma=\sigma(N)>0$ vanishes as $N\to\infty$. Note that the estimates
obtained here are stable with respect to the common noise strength $\beta$,
but $\sigma$ needs to be bounded away from zero uniformly in $N$ in the
present work. Furthermore, we suspect that the two concepts of
semimonotonicity explored here might be two special cases of a more general
framework.

Finally, what is developed here is restricted to quadratic
Lagrangians/Hamiltonians. We expect that this method can be adapted to more
general Hamiltonians $H(t,\boldsymbol{x},D_{i}u^{i})$ with suitable
semimonotone structure, though there are nontrivial technical issues that need
to be overcome. Most importantly, while one still verifies the one-sided
Lipschitz condition on $D_{i}u^{i}$ in step (2), this does not propagate
immediately to the optimal drift
$\alpha^{*,i}(t,\boldsymbol{x})=-D_{p}H(t,\boldsymbol{x},D_{i}u^{i})$, since a
composition of semimonotone maps needs not be semimonotone in general. This
introduces difficulties in closing the argument that links step (1) and step
(2).

Acknowledgements.  The authors are partially supported by the Gruppo Nazionale
per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of
the Istituto Nazionale di Alta Matematica (INdAM), Italy. Cirant partially
supported by the King Abdullah University of Science and Technology (KAUST)
project CRG2021-4674 “Mean-Field Games: models, theory, and computational
aspects”, and by the project funded by the EuropeanUnion–NextGenerationEU
under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2
Investment 1.1 - Call PRIN 2022 No. 104 of February 2, 2022 of Italian
Ministry of University and Research; Project 2022W58BJ5 (subject area: PE -
Physical Sciences and Engineering) “PDEs and optimal control methods in mean
field games, population dynamics and multi-agent models”. Redaelli partially
supported by Fondazione Cassa di Risparmio di Padova e Rovigo, Italy.

##  2\. Recurring general notation

Before proceeding, we believe it is useful to fix some notation, a part of
which has already been explained in the Introduction.

  * •

Vectors and matrices.

    * $\circ$

Given $m\in\mathbb{N}$, the _all-ones vector_ is denoted $1_{m}$.

    * $\circ$

The (Euclidean) _inner product_ is denoted by $\cdot$ .

    * $\circ$

Given $\boldsymbol{x}\in({\mathbb{R}}^{n})^{N}$ and
$\boldsymbol{y}\in({\mathbb{R}}^{m})^{N}$, we define

$(\boldsymbol{x}:\boldsymbol{y})\vcentcolon=((x^{j},y^{j}))_{1\leqslant
j\leqslant N}\in({\mathbb{R}}^{n}\times{\mathbb{R}}^{m})^{N}.$

    * $\circ$

The transpose of a matrix $\mathsf{M}$ is denoted by
$\mathsf{M}^{\mathsf{T}}$.

    * $\circ$

Given a vector $\boldsymbol{\xi}\in{\mathbb{R}}^{m}$, ${\rm
diag}\,\boldsymbol{\xi}$ is the diagonal $m\times m$ matrix having $\xi^{i}$
as the $(i,i)$-entry .

    * $\circ$

Given a real $m\times m$ matrix $\mathsf{M}$, we write $\mathsf{M}\geqslant 0$
(even if $\mathsf{M}$ is not symmetric) whenever $\sum_{1\leqslant
i,j\leqslant m}\mathsf{M}_{i,j}\xi^{i}\xi^{j}\geqslant 0$ for all
$\boldsymbol{\xi}\in{\mathbb{R}}^{m}$.

    * $\circ$

The space of all (real) _symmetric $m\times m$ matrices_ is $\mathscr{S}(m)$.
The _identity_ matrix therein is $\mathsf{I}_{m}$, while the _matrix of ones_
is $\mathsf{J}_{m}$.

    * $\circ$

The operator $\otimes$ can be both the _Kronecker product_ (between matrices)
and the _outer product_ (between vectors). Then, for instance,
$\mathsf{J}_{m}=1_{m}\otimes 1_{m}$ and
$\mathsf{I}_{Nd}=\mathsf{I}_{N}\otimes\mathsf{I}_{d}$.

    * $\circ$

An _ad hoc_ notation is used for the two matrices appearing in the Nash
system: we write $\boldsymbol{\mathsf{I}}\vcentcolon=\mathsf{I}_{Nd}$ and
$\boldsymbol{\mathsf{J}}\vcentcolon=\mathsf{J}_{N}\otimes\mathsf{I}_{d}$.

  * •

Vectors with suppressed or replaced coordinates.

    * $\circ$

Given a set $\mathcal{S}$, $\boldsymbol{x}\in\mathcal{S}^{N}$ and
$i\in\\{1,\dots,N\\}$, we denote by $\boldsymbol{x}^{-i}\in S^{N-1}$ the
vector $\boldsymbol{x}^{-i}=(x^{1},\dots,x^{i-1},x^{i+1},\dots,x^{N})$

    * $\circ$

Given $y\in\mathcal{S}$, the symbol $(\boldsymbol{x}^{-i},y)$ will be
understood as the vector $\boldsymbol{z}\in\mathcal{S}^{N}$ with $z^{j}=x^{j}$
if $j\neq i$ and $z^{i}=y$; that is, the vector $\boldsymbol{x}$ with the
$i$-th coordinate replaced by $y$.
The order will be important in the sense that we will specify first the vector
$\boldsymbol{x}$ and then the replacement $y$, so we point out that both the
following _not equivalent_ notations will occur: given a function $h$ defined
on $\mathcal{S}^{N}$,

$h(\boldsymbol{x}^{-i},y)=h(x^{1},\dots,x^{i-1},y,x^{i+1},\dots,x^{N}),$

but

$h(y,\boldsymbol{x}^{-i})=h(y,x^{1},\dots,x^{i-1},x^{i+1},\dots,x^{N}).$

  * •

Measures and spaces of measures.

    * $\circ$

Given two measurable spaces $\mathcal{X}$ and $\mathcal{Y}$, a measure $\mu$
on $\mathcal{X}$ and a measurable map $\psi\colon\mathcal{X}\to\mathcal{Y}$,
the _push-forward_ (or _image_) _measure_ of $\mu$ via $\psi$ is the measure
$\nu$ on $\mathcal{Y}$ given by
$\psi{}_{\sharp}\mu\vcentcolon=\mu\circ\psi^{-1}$.

    * $\circ$

Given a Polish space $\mathcal{X}$ and $p\in[1,\infty)$, we will denote by
$\mathcal{P}_{p}(\mathcal{X})$ the space of all probability measures $\nu$ on
$\mathcal{X}$ such that
$\int_{\mathcal{X}}|\mspace{1.8mu}\cdot\mspace{1.8mu}|^{p}\,\mathrm{d}\nu$ is
finite, endowed with the $p$-Wasserstein distance $W_{p}$. Recall that $W_{p}$
is given by

$W_{p}(\mu,\nu)^{p}=\inf_{\pi\in\Pi(\mu,\nu)}\int_{\mathcal{X}\times\mathcal{X}}|x-y|^{p}\,\varpi(\mathrm{d}x,\mathrm{d}y)\quad\forall\,\mu,\nu\in\mathcal{P}_{p}(\mathcal{X}),$

where $\Pi(\mu,\nu)$ is the set of all couplings of $(\mu,\nu)$; that is, of
all measures in $\mathcal{P}_{p}(\mathcal{X}\times\mathcal{X})$ with $\mu$ and
$\nu$ as their marginals (that is, if
$\pi_{j}\colon\mathcal{X}\times\mathcal{X}\to\mathcal{X}$ is the projection
onto the $j$-th copy of $\mathcal{X}$, $\pi_{1}{}_{\sharp}\varpi=\mu$ and
$\pi_{2}{}_{\sharp}\varpi=\nu$).

    * $\circ$

Given $N\in\mathbb{N}$ and
$\boldsymbol{x}=(x^{1},\dots,x^{N})\in\mathcal{X}^{N}$, we will use the
notation $m_{\boldsymbol{x}}\vcentcolon=\frac{1}{N}\sum_{1\leqslant i\leqslant
N}\delta_{x^{j}}$; note that
$m_{\boldsymbol{x}}\in\mathcal{P}_{p}(\mathcal{X})$ for all $p$ as above.

  * •

Derivatives. Most functions we deal with are defined on
$[0,T]\times({\mathbb{R}}^{d})^{N}$, with typical element
$(t,\boldsymbol{x})$, where $\boldsymbol{x}=(x^{1},\dots,x^{N})$,
$x^{i}\in{\mathbb{R}}^{d}$ for all $i\in\\{1,\dots,N\\}$. The coordinated of
$x^{i}$ will be in turn written as $x^{ij}$ with $j\in\\{1,\dots,d\\}$. We
will refer to $t$ as _time_ (temporal coordinate) and on $\boldsymbol{x}$,
$x^{i}$, $x^{ij}$ as _space_ (spatial coordinates).

    * $\circ$

The derivative with respect to time is denoted by $\partial_{t}$.

    * $\circ$

The gradient (or derivative) with respect to a spatial coordinate $y$ is
$D_{y}$; in particular, we will use the shorter notations
$D\vcentcolon=D_{\boldsymbol{x}}$ and $D_{j}\vcentcolon=D_{x^{j}}$. The same
applies to higher order derivatives, so that, for instance, $D^{2}$ is the
Hessian with respect to $\boldsymbol{x}$ and
$D^{2}_{ij}=D^{2}_{x^{i}x^{j}}=D_{x^{i}}D_{x^{j}}$.
(Note that, when seeing $D$ as a vector of $N$ vectors of $d$ differential
operators, and $D_{j}$ as a vector of $d$ differential operators, one has that
$D_{j}$ is the $j$-th coordinate of $D$. Analogously, when seeing $D^{2}$ as
an $N\times N$ matrix of $d\times d$ matrices of differential operators, and
$D^{2}_{ij}$ as a $d\times d$ matrix of differential operators, one has
$(D^{2})_{ij}=D^{2}_{ij}$; that is, the entry $(i,j)$ of $D^{2}$ is
$D^{2}_{ij}$.)

    * $\circ$

The _multi-index_ notation is the usual one: for instance, given
$\alpha=(\alpha_{1},\dots,\alpha_{N})\in\mathbb{N}^{N}$ and $h$ differentiable
on $({\mathbb{R}}^{d})^{N}$, we have $D^{\alpha}h=D^{\alpha_{1}}_{1}\cdots
D^{\alpha_{N}}_{N}h$.

    * $\circ$

For functions depending on measures, the _flat derivative_ and the _intrinsic
derivative_ with respect to a probability measure (according to the
definitions given in [14, Section 2.2.1]) are $\frac{\delta}{\delta m}$ and
$D_{m}$, respectively.

  * •

Norms. Unless otherwise specified:

    * $\circ$

the norm $|\mspace{1.8mu}\cdot\mspace{1.8mu}|$ is understood to be the
_Euclidean_ one when applied to a vector, the _Frobenius norm_ for matrices
and, more generally, the _full tensor contraction_ for tensors (and according
to this, inner products between tensors will be understood as tensor
contractions); for example,

$|D^{3}_{ijk}u^{j}|^{2}=\sum_{1\leqslant h_{i},h_{j},h_{k}\leqslant
d}\big{|}D^{3}_{x^{ih_{i}}x^{jh_{j}}x^{kh_{k}}}u^{j}\big{|}^{2};$

    * $\circ$

given $h:({\mathbb{R}}^{d})^{N}\to{\mathbb{R}}$, the _supremum norms_ of $h$
and $Dh$ are given by

$\|h\|_{\infty}=\sup_{\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}}|h(\boldsymbol{x})|,\qquad\|Dh\|_{\infty}=\||Dh|\|_{\infty}=\bigg{\|}\bigg{(}\sum_{j=1}^{N}|D_{j}h|^{2}\bigg{)}^{\frac{1}{2}}\bigg{\|}_{\infty};$

    * $\circ$

the supremum norm of a function which depends on more variables (for example,
also on time) is understood with respect to all of them;

    * $\circ$

the _$i$ -th weighted norm_ is defined by

$\lVert\boldsymbol{x}\rVert^{i}\vcentcolon=\Bigl{(}|x^{i}|^{2}+{\frac{1}{N}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}|x^{j}|^{2}\Bigr{)}^{\frac{1}{2}},\quad\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}.$

  * •

Particular symbols. For $\mathcal{D}$ and $Q$, see (3.2) and below; for
$\mathcal{L}$, see (3.4); for $\boldsymbol{\triangle}$, see (3.5).

  * •

Assumptions. The main assumptions in this paper are (MF), (DS) and (LS), (S),
and (L). They are given at pages (MF), (DS), (S), and (LP), respectively.

##  3\. Summary of main results

###  3.1. Two notions of (semi)monotonicity

Along with the hypothesis of _Mean-Field-like_ interactions (which will be
presented later in Section 3.2), our main structural assumption on the system
will be of _semimonotonicity_ of the data, according to either one of the
following definitions.

######  Definition 3.1.

Let $h\in C^{1}(({\mathbb{R}}^{d})^{N};{\mathbb{R}}^{N})$. For $M\geqslant 0$,
we say that $h$ is _$M$ -$\mathscr{D}$*semimonotone_ if

(3.1) $\sum_{1\leqslant i\leqslant
N}\,(D_{i}h^{i}(\boldsymbol{x})-D_{i}h^{i}(\boldsymbol{y}))\cdot(x^{i}-y^{i})\geqslant-M|\boldsymbol{x}-\boldsymbol{y}|^{2}\quad\forall\,\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N}$

and $M$ is called the _$\mathscr{D}$ *semimonotonicity constant_. We simply
say that $h$ is $\mathscr{D}$*semimonotone if (3.1) holds for some $M\geqslant
0$. If (3.1) holds with $M=0$, we say that $h$ is $\mathscr{D}$*monotone.

In order to abridge the notation in what follows we shall write
$h|_{\boldsymbol{y}}^{\boldsymbol{x}}=h(\boldsymbol{x})-h(\boldsymbol{y})$ and

(3.2)
$\mathcal{D}[h](\boldsymbol{x},\boldsymbol{y})\vcentcolon=\sum_{1\leqslant
i\leqslant
N}D_{i}h^{i}\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i}),$

so that, inequality (3.1) will appear as $\mathcal{D}[h]\geqslant-MQ$ on
$({\mathbb{R}}^{d})^{2N}\simeq({\mathbb{R}}^{d})^{N}\times({\mathbb{R}}^{d})^{N}$,
where $Q=Q_{\boldsymbol{\mathsf{A}}}$ is the quadratic form induced on
$({\mathbb{R}}^{d})^{2N}$ by the matrix
$\boldsymbol{\mathsf{A}}\vcentcolon=\begin{pmatrix}1&-1\\\
-1&1\end{pmatrix}\otimes\boldsymbol{\mathsf{I}}\in\mathscr{S}(2Nd)$.

######  Definition 3.2.

Let $h\colon({\mathbb{R}}^{d})^{N}\to{\mathbb{R}}^{N}$. For $\kappa\geqslant
0$, we say that $h$ is $\kappa$-$\mathscr{L}$*semimonotone if

(3.3) $\sum_{1\leqslant i\leqslant
N}\big{(}h^{i}(\boldsymbol{x})-h^{i}(\boldsymbol{x}^{-i},y^{i})-h^{i}(\boldsymbol{y}^{-i},x^{i})+h^{i}(\boldsymbol{y})\big{)}\geqslant-\kappa|\boldsymbol{x}-\boldsymbol{y}|^{2}\quad\forall\,\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N}$

and $\kappa$ is called the _$\mathscr{L}$ *semimonotonicity constant_. We
simply say that $h$ is $\mathscr{L}$*semimonotone if (3.3) holds for some
$\kappa\geqslant 0$. If (3.3) holds with $\kappa=0$, we also say that $f$ is
$\mathscr{L}$*monotone.

For the sake of brevity, we define

(3.4)
$\mathcal{L}[h](\boldsymbol{x},\boldsymbol{y})\vcentcolon=\sum_{1\leqslant
i\leqslant
N}\big{(}h^{i}(\boldsymbol{x})-h^{i}(\boldsymbol{x}^{-i},y^{i})-h^{i}(\boldsymbol{y}^{-i},x^{i})+h^{i}(\boldsymbol{y})\big{)},$

so that inequality (3.3) will appear as $\mathcal{L}[h]\geqslant-\kappa Q$ on
$({\mathbb{R}}^{d})^{2N}$.

######  Remark 3.3 (Pointwise characterisation of semimonotonicity and
comparison).

If $h$ is of class $C^{2}$, then exploiting the mean value theorem and the
fundamental theorem of calculus one easily shows that

$\text{$h$ is $M$-$\mathscr{D}$\Hyphdash*semimonotone (that is,
$\mathcal{D}[h]\geqslant-MQ$ on
$({\mathbb{R}}^{d})^{2N}$)}\quad\iff\quad(D^{2}_{ij}h^{i})_{1\leqslant
i,j\leqslant N}\geqslant-M\boldsymbol{\mathsf{I}}.$

Analogously, applying the mean value theorem twice to (3.3),

$\mathcal{L}[h](\boldsymbol{x},\boldsymbol{y})=\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D^{2}_{ij}h^{i}(\boldsymbol{z}^{-i}_{s^{\prime}},z^{i}_{s})(x^{j}-y^{j})\cdot(x^{i}-y^{i})\,\mathrm{d}s\,\mathrm{d}s^{\prime},$

where we have set
$\boldsymbol{z}_{s}\vcentcolon=s\boldsymbol{x}+(1-s)\boldsymbol{y}$; thus, one
easily shows that

$\begin{multlined}\text{$h$ is $\kappa$-$\mathscr{L}$\Hyphdash*semimonotone
(that is, $\mathcal{L}[h]\geqslant-\kappa Q$ on $({\mathbb{R}}^{d})^{2N}$)}\\\
\iff\quad(D^{2}_{ij}h^{i})_{1\leqslant i,j\leqslant
N}-\mathrm{diag}\,(D^{2}_{ii}h^{i})_{1\leqslant i\leqslant
N}\geqslant-\kappa\boldsymbol{\mathsf{I}}.\end{multlined}\text{$h$ is
$\kappa$-$\mathscr{L}$\Hyphdash*semimonotone (that is,
$\mathcal{L}[h]\geqslant-\kappa Q$ on $({\mathbb{R}}^{d})^{2N}$)}\\\
\iff\quad(D^{2}_{ij}h^{i})_{1\leqslant i,j\leqslant
N}-\mathrm{diag}\,(D^{2}_{ii}h^{i})_{1\leqslant i\leqslant
N}\geqslant-\kappa\boldsymbol{\mathsf{I}}.$

At the level of second-order derivatives, one then observes that the two
notions of semimonotonicity differ by the diagonal term

(3.5)
$\boldsymbol{\triangle}_{h}\vcentcolon=\mathrm{diag}\,(D_{ii}^{2}h^{i})_{1\leqslant
i\leqslant N}.$

Therefore, under some unilateral control of such diagonal, they can be somehow
compared: if $D_{ii}^{2}h^{i}\geqslant-\gamma\mathsf{I}_{d}$ on
${\mathbb{R}}^{d}$, for some constant $\gamma>0$ and all $i$,
$\mathscr{L}$*semimonotonicity implies $\mathscr{D}$*semimonotonicity

$\begin{cases}\mathcal{L}[h]\geqslant-\kappa Q&\text{on}\
({\mathbb{R}}^{d})^{2N}\\\
\boldsymbol{\triangle}_{h}\geqslant-\gamma\boldsymbol{\mathsf{I}}&\text{on}\
({\mathbb{R}}^{d})^{N}\end{cases}\quad\implies\quad\mathcal{D}[h]\geqslant-(\kappa+\gamma)Q\
\ \text{on}\ ({\mathbb{R}}^{d})^{2N}.$

On the other hand, if $D_{ii}^{2}h^{i}\leqslant\gamma\mathsf{I}_{d}$ on
$({\mathbb{R}}^{d})^{N}$, $\mathscr{D}$*semimonotonicity implies
$\mathscr{L}$*semimonotonicity

$\begin{cases}\mathcal{D}[h]\geqslant-MQ&\text{on}\ ({\mathbb{R}}^{d})^{2N}\\\
\boldsymbol{\triangle}_{h}\leqslant\gamma\boldsymbol{\mathsf{I}}&\text{on}\
({\mathbb{R}}^{d})^{N}\end{cases}\quad\implies\quad\mathcal{L}[h]\geqslant-(M+\gamma)Q\
\ \text{on}\ ({\mathbb{R}}^{d})^{2N}.$

Nevertheless, for a given $h$, it is clear that the two constants involved in
the definitions may differ substantially, hence one should not regard the two
notions as comparable at all, especially when $\kappa$ or $M$ are small, that
will be likely in our analysis. In the limit case that $f$ be
$\mathscr{D}$*monotone, it will be false in general that $f$ is also
$\mathscr{L}$*monotone, and vice versa.

######  Remark 3.4 (Relationship with monotonicity in the Mean-Field setting).

Consider the case $f^{i}(\boldsymbol{x})=F(x^{i},m_{\boldsymbol{x}})$ for all
$i\in\\{1,\dots,N\\}$, where
$F:{\mathbb{R}}^{d}\times\mathcal{P}({\mathbb{R}}^{d})\to{\mathbb{R}}$ is
smooth enough for the following computations to be admissible.

$\bullet$ Suppose that $F$ is displacement monotone; that is,

(3.6)
$\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\big{(}D_{x}F(x,m_{1})-D_{x}F(y,m_{2})\big{)}\cdot(x-y)\,\mu(\mathrm{d}x,\mathrm{d}y)\geqslant
0$

for every $m_{1},m_{2}\in\mathcal{P}({\mathbb{R}}^{d})$ and
$\mu\in\mathcal{P}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})$ having $m_{1}$ and
$m_{2}$ as first and second marginal, respectively. We show that $f$ is
“almost” $\mathscr{D}$*monotone. Indeed, for every $i$,

$D_{i}f^{i}(\boldsymbol{x})=D_{x}F(x^{i},m_{\boldsymbol{x}})+\frac{1}{N}\,D_{m}F(x^{i},m_{\boldsymbol{x}},x^{i}),$

hence

$\begin{multlined}\sum_{1\leqslant i\leqslant
N}\,(D_{i}f^{i}(\boldsymbol{x})-D_{i}f^{i}(\boldsymbol{y}))\cdot(x^{i}-y^{i})\\\
\begin{multlined}=N\int_{({\mathbb{R}}^{d})^{2}}(D_{x}F(x,m_{\boldsymbol{x}})-D_{x}F(y,m_{\boldsymbol{y}}))\cdot(x-y)\,\frac{1}{N}\\!\sum_{1\leqslant
i\leqslant N}\delta_{(x^{i},y^{i})}(\mathrm{d}x,\mathrm{d}y)\\\
+\frac{1}{N}\sum_{1\leqslant i\leqslant
N}(D_{m}F(x^{i},m_{\boldsymbol{x}},x^{i})-D_{m}F(y^{i},m_{\boldsymbol{y}},y^{i}))\cdot(x^{i}-y^{i}).\end{multlined}=N\int_{({\mathbb{R}}^{d})^{2}}(D_{x}F(x,m_{\boldsymbol{x}})-D_{x}F(y,m_{\boldsymbol{y}}))\cdot(x-y)\,\frac{1}{N}\\!\sum_{1\leqslant
i\leqslant N}\delta_{(x^{i},y^{i})}(\mathrm{d}x,\mathrm{d}y)\\\
+\frac{1}{N}\sum_{1\leqslant i\leqslant
N}(D_{m}F(x^{i},m_{\boldsymbol{x}},x^{i})-D_{m}F(y^{i},m_{\boldsymbol{y}},y^{i}))\cdot(x^{i}-y^{i}).\end{multlined}\sum_{1\leqslant
i\leqslant
N}\,(D_{i}f^{i}(\boldsymbol{x})-D_{i}f^{i}(\boldsymbol{y}))\cdot(x^{i}-y^{i})\\\
\begin{multlined}=N\int_{({\mathbb{R}}^{d})^{2}}(D_{x}F(x,m_{\boldsymbol{x}})-D_{x}F(y,m_{\boldsymbol{y}}))\cdot(x-y)\,\frac{1}{N}\\!\sum_{1\leqslant
i\leqslant N}\delta_{(x^{i},y^{i})}(\mathrm{d}x,\mathrm{d}y)\\\
+\frac{1}{N}\sum_{1\leqslant i\leqslant
N}(D_{m}F(x^{i},m_{\boldsymbol{x}},x^{i})-D_{m}F(y^{i},m_{\boldsymbol{y}},y^{i}))\cdot(x^{i}-y^{i}).\end{multlined}=N\int_{({\mathbb{R}}^{d})^{2}}(D_{x}F(x,m_{\boldsymbol{x}})-D_{x}F(y,m_{\boldsymbol{y}}))\cdot(x-y)\,\frac{1}{N}\\!\sum_{1\leqslant
i\leqslant N}\delta_{(x^{i},y^{i})}(\mathrm{d}x,\mathrm{d}y)\\\
+\frac{1}{N}\sum_{1\leqslant i\leqslant
N}(D_{m}F(x^{i},m_{\boldsymbol{x}},x^{i})-D_{m}F(y^{i},m_{\boldsymbol{y}},y^{i}))\cdot(x^{i}-y^{i}).$

The former term on the right-hand side is nonnegative by the displacement
monotonicity of $F$. The latter one can be estimated in absolute value,
assuming that $D_{m}F$ is globally $L$-Lipschitz with respect to the
($|\cdot|,W_{1},|\cdot|)$ distance, by

$\frac{L}{N}\,W_{1}(m_{\boldsymbol{x}},m_{\boldsymbol{y}})\sum_{1\leqslant
i\leqslant N}|x^{i}-y^{i}|+\frac{2L}{N}\sum_{1\leqslant i\leqslant
N}|x^{i}-y^{i}|^{2}\leqslant\frac{3L}{N}\sum_{1\leqslant i\leqslant
N}|x^{i}-y^{i}|^{2}.$

Therefore, $f$ is $\frac{3L}{N}$-$\mathscr{D}$*semimonotone. More generally,
if the right-hand side of (3.6) is not zero, but
$-M\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|x-y|^{2}\,\mu(\mathrm{d}x,\mathrm{d}y)$,
then $F$ is just displacement semimonotone, as in [24, Definition 2.7]. In
such case, $f$ becomes
$\big{(}\frac{3L}{N}+M\big{)}$-$\mathscr{D}$*semimonotone.

$\bullet$ Suppose now that $F$ is monotone in the Lasry–Lions sense; that is,
for all $m_{1},m_{2}\in\mathcal{P}({\mathbb{R}}^{d})$,

(3.7)
$\int_{{\mathbb{R}}^{d}}\big{(}F(z,m_{1})-F(z,m_{2})\big{)}(m_{1}-m_{2})(\mathrm{d}z)\geqslant
0.$

Then,

$\begin{multlined}\sum_{1\leqslant i\leqslant
N}\bigl{(}f^{i}(\boldsymbol{x})-f^{i}(\boldsymbol{x}^{-i},y^{i})-f^{i}(\boldsymbol{y}^{-i},x^{i})+f^{i}(\boldsymbol{y})\bigr{)}\\\
\begin{multlined}=N\int_{{\mathbb{R}}^{d}}\big{(}F(z,m_{\boldsymbol{x}})-F(z,m_{\boldsymbol{y}})\big{)}(m_{\boldsymbol{x}}-m_{\boldsymbol{y}})(\mathrm{d}z)\\\
+\sum_{1\leqslant i\leqslant
N}\biggl{(}F(y^{i},m_{\boldsymbol{x}})-F\Bigl{(}y^{i},m_{\boldsymbol{x}}+\frac{\delta_{y^{i}}-\delta_{x^{i}}}{N}\Bigr{)}-F\Bigl{(}x^{i},m_{\boldsymbol{y}}+\frac{\delta_{x^{i}}-\delta_{y^{i}}}{N}\Bigr{)}+F(x^{i},m_{\boldsymbol{y}})\biggr{)}.\end{multlined}=N\int_{{\mathbb{R}}^{d}}\big{(}F(z,m_{\boldsymbol{x}})-F(z,m_{\boldsymbol{y}})\big{)}(m_{\boldsymbol{x}}-m_{\boldsymbol{y}})(\mathrm{d}z)\\\
+\sum_{1\leqslant i\leqslant
N}\biggl{(}F(y^{i},m_{\boldsymbol{x}})-F\Bigl{(}y^{i},m_{\boldsymbol{x}}+\frac{\delta_{y^{i}}-\delta_{x^{i}}}{N}\Bigr{)}-F\Bigl{(}x^{i},m_{\boldsymbol{y}}+\frac{\delta_{x^{i}}-\delta_{y^{i}}}{N}\Bigr{)}+F(x^{i},m_{\boldsymbol{y}})\biggr{)}.\end{multlined}\sum_{1\leqslant
i\leqslant
N}\bigl{(}f^{i}(\boldsymbol{x})-f^{i}(\boldsymbol{x}^{-i},y^{i})-f^{i}(\boldsymbol{y}^{-i},x^{i})+f^{i}(\boldsymbol{y})\bigr{)}\\\
\begin{multlined}=N\int_{{\mathbb{R}}^{d}}\big{(}F(z,m_{\boldsymbol{x}})-F(z,m_{\boldsymbol{y}})\big{)}(m_{\boldsymbol{x}}-m_{\boldsymbol{y}})(\mathrm{d}z)\\\
+\sum_{1\leqslant i\leqslant
N}\biggl{(}F(y^{i},m_{\boldsymbol{x}})-F\Bigl{(}y^{i},m_{\boldsymbol{x}}+\frac{\delta_{y^{i}}-\delta_{x^{i}}}{N}\Bigr{)}-F\Bigl{(}x^{i},m_{\boldsymbol{y}}+\frac{\delta_{x^{i}}-\delta_{y^{i}}}{N}\Bigr{)}+F(x^{i},m_{\boldsymbol{y}})\biggr{)}.\end{multlined}=N\int_{{\mathbb{R}}^{d}}\big{(}F(z,m_{\boldsymbol{x}})-F(z,m_{\boldsymbol{y}})\big{)}(m_{\boldsymbol{x}}-m_{\boldsymbol{y}})(\mathrm{d}z)\\\
+\sum_{1\leqslant i\leqslant
N}\biggl{(}F(y^{i},m_{\boldsymbol{x}})-F\Bigl{(}y^{i},m_{\boldsymbol{x}}+\frac{\delta_{y^{i}}-\delta_{x^{i}}}{N}\Bigr{)}-F\Bigl{(}x^{i},m_{\boldsymbol{y}}+\frac{\delta_{x^{i}}-\delta_{y^{i}}}{N}\Bigr{)}+F(x^{i},m_{\boldsymbol{y}})\biggr{)}.$

While the former term on the right-hand side is nonnegative by the
monotonicity of $F$, the latter one can be written as

$\begin{multlined}\frac{1}{N}\sum_{1\leqslant i\leqslant
N}\int_{0}^{1}\Bigl{(}\frac{\delta F}{\delta
m}(y^{i},m_{s},x^{i})-\frac{\delta F}{\delta
m}(y^{i},m_{s},y^{i})-\frac{\delta F}{\delta
m}(x^{i},m_{s},x^{i})+\frac{\delta F}{\delta
m}(x^{i},m_{s},y^{i})\Bigr{)}\mathrm{d}s\\\ +\frac{1}{N}\sum_{1\leqslant
i\leqslant N}\int_{0}^{1}\Bigl{(}\frac{\delta F}{\delta
m}(x^{i},m_{s},x^{i})-\frac{\delta F}{\delta
m}(x^{i},m_{s},y^{i})-\frac{\delta F}{\delta
m}(x^{i},\hat{m}_{s},x^{i})+\frac{\delta F}{\delta
m}(x^{i},\hat{m}_{s},y^{i})\Bigr{)}\mathrm{d}s\end{multlined}\frac{1}{N}\sum_{1\leqslant
i\leqslant N}\int_{0}^{1}\Bigl{(}\frac{\delta F}{\delta
m}(y^{i},m_{s},x^{i})-\frac{\delta F}{\delta
m}(y^{i},m_{s},y^{i})-\frac{\delta F}{\delta
m}(x^{i},m_{s},x^{i})+\frac{\delta F}{\delta
m}(x^{i},m_{s},y^{i})\Bigr{)}\mathrm{d}s\\\ +\frac{1}{N}\sum_{1\leqslant
i\leqslant N}\int_{0}^{1}\Bigl{(}\frac{\delta F}{\delta
m}(x^{i},m_{s},x^{i})-\frac{\delta F}{\delta
m}(x^{i},m_{s},y^{i})-\frac{\delta F}{\delta
m}(x^{i},\hat{m}_{s},x^{i})+\frac{\delta F}{\delta
m}(x^{i},\hat{m}_{s},y^{i})\Bigr{)}\mathrm{d}s$

where $\frac{\delta}{\delta m}$ denotes the flat derivative (see, e.g., [14,
Definition 2.2.1]) and we set
$m_{s}=m_{\boldsymbol{x}}+\frac{1-s}{N}(\delta_{y^{i}}-\delta_{x^{i}})$ and
$\hat{m}_{s}=m_{\boldsymbol{y}}+\frac{1-s}{N}(\delta_{x^{i}}-\delta_{y^{i}})$.
Using repeatedly the mean value theorem, these two sums can be bounded,
repsectively, by

$\frac{\|D_{x}D_{m}F\|_{\infty}}{N}\sum_{1\leqslant i\leqslant
N}|x^{i}-y^{i}|^{2}$

and

$\begin{multlined}\Biggl{\lvert}\frac{1}{N}\sum_{1\leqslant i\leqslant
N}\biggl{(}\,\int_{0}^{1}\\!\int_{0}^{1}\\!\int_{0}^{1}\
\\!\int_{{\mathbb{R}}^{d}}\frac{\delta(D_{m}F)}{\delta
m}(\boldsymbol{\ast})\,(m_{s}-\hat{m}_{s})(\mathrm{d}z)\,\mathrm{d}s^{\prime\prime}\mathrm{d}s^{\prime}\mathrm{d}s\biggr{)}\cdot(x^{i}-y^{i})\Biggr{\rvert}\\\
\leqslant\|D^{2}_{mm}F\|_{\infty}\biggl{(}\biggl{(}\frac{1}{N}\sum_{1\leqslant
i\leqslant N}|x^{i}-y^{i}|\biggr{)}^{2}+\frac{2}{N^{2}}\sum_{1\leqslant
i\leqslant
N}|x^{i}-y^{i}|^{2}\biggr{)},\end{multlined}\Biggl{\lvert}\frac{1}{N}\sum_{1\leqslant
i\leqslant N}\biggl{(}\,\int_{0}^{1}\\!\int_{0}^{1}\\!\int_{0}^{1}\
\\!\int_{{\mathbb{R}}^{d}}\frac{\delta(D_{m}F)}{\delta
m}(\boldsymbol{\ast})\,(m_{s}-\hat{m}_{s})(\mathrm{d}z)\,\mathrm{d}s^{\prime\prime}\mathrm{d}s^{\prime}\mathrm{d}s\biggr{)}\cdot(x^{i}-y^{i})\Biggr{\rvert}\\\
\leqslant\|D^{2}_{mm}F\|_{\infty}\biggl{(}\biggl{(}\frac{1}{N}\sum_{1\leqslant
i\leqslant N}|x^{i}-y^{i}|\biggr{)}^{2}+\frac{2}{N^{2}}\sum_{1\leqslant
i\leqslant N}|x^{i}-y^{i}|^{2}\biggr{)},$

where
$\boldsymbol{\ast}=(x^{i},s^{\prime}m_{s}+(1-s^{\prime})\hat{m}_{s},s^{\prime\prime}x^{i}+(1-s^{\prime\prime})y^{i},z)$.
Therefore, $f$ is $\kappa$-$\mathscr{L}$*semionotone with $\kappa=O(N^{-1})$.
More generally, if the right-hand side of (3.7) is not zero, but
$-MW_{2}(m_{1},m_{2})^{2}$, then $f$ becomes
$\big{(}M+O(N^{-1})\big{)}$-$\mathscr{L}$*semimonotone.

###  3.2. Estimates on the Nash system

We suppose that all data $f^{i}$ and $g^{i}$ are sufficiently smooth (locally
$C^{2}$ and $C^{4}$, respectively) with bounded derivatives. Solutions to the
Nash system are assumed to be classical, locally $C^{1}$ in time and $C^{4}$
in space, with bounded derivatives. Moreover, we require second order
derivatives in space to be uniformly continuous. Note that we are not assuming
that $f^{i},g^{i}$ are globally bounded here: they may have linear growth.

As prefigured, we work in a regime of _Mean-Field-like interactions_ ; that
is, we assume the following:

  * (MF)

for $h\in\\{f,g\\}$, there exists $L_{h}>0$, independent of $N$, such that

$\displaystyle\sup_{1\leqslant i\leqslant
N}\big{(}\|D_{i}h^{i}\|_{\infty}+\|D^{2}_{ii}h^{i}\|_{\infty}\big{)}\leqslant
L_{h},$ $\displaystyle\sup_{1\leqslant i\leqslant
N}\Big{(}\sup_{\begin{subarray}{c}1\leqslant k\leqslant N\\\ k\neq
i\end{subarray}}\|D_{k}h^{i}\|_{\infty}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant N\\\ k\neq
i\end{subarray}}\|D^{2}_{ik}h^{k}\|^{2}_{\infty}\Big{)}\leqslant\frac{L_{h}^{2}}{N},$

and in addition

$\displaystyle\sup_{1\leqslant k\leqslant N}\sum_{1\leqslant i\leqslant
N}\lVert D(D_{k}g^{i})\rVert^{2}_{\infty}+\sum_{\begin{subarray}{c}1\leqslant
k,i\leqslant N\\\ i\neq k\end{subarray}}\lVert
D(D^{2}_{ki}g^{i})\rVert_{\infty}^{2}\leqslant L^{2}_{g},$
$\displaystyle\sup_{1\leqslant i\leqslant
N}\sum_{\begin{subarray}{c}1\leqslant k\leqslant N\\\ k\neq
i\end{subarray}}\lVert
D(D_{k}g^{i})\rVert_{\infty}^{2}\leqslant\frac{L^{2}_{g}}{N}.$

We stress the fact that these bounds are of Mean-Field type in that they
provide the weight with respect to $N$ which one expects in the Mean Field
setting, as described in the remark below. Yet, we are not asking for any
symmetry with respect to the variables of $h$; that is, we are essentially
dropping the classic indistinguishability (or exchangeability) hypothesis of
Mean Field Games.

######  Remark 3.5.

Assume that $f^{i}(\boldsymbol{x})=F^{i}(x^{i},m_{\boldsymbol{x}^{-i}})$;
recall that $m_{\boldsymbol{x}^{-i}}=\frac{1}{N-1}\sum_{j\neq
i}\delta_{x^{j}}$. Suppose that
$F^{i}\colon{\mathbb{R}}^{d}\times\mathcal{P}_{1}({\mathbb{R}}^{d})\to{\mathbb{R}}$
is Lipschitz continuous with Lipschitz constant $L>0$. Since for any
$v\in{\mathbb{R}}^{d}$ and $k\neq i$

$\begin{multlined}|f^{i}(x^{1},\ldots,x^{k-1},x^{k}+v,x^{k+1},\ldots)-f^{i}(\boldsymbol{x})|\\\
\begin{split}&\leqslant
L\,W_{1}\Big{(}\frac{1}{N-1}\sum_{j\neq\\{i,k\\}}\delta_{x^{j}}+\frac{1}{N-1}\delta_{x^{k}+v},\frac{1}{N-1}\sum_{j\neq
i}\delta_{x^{j}}\Big{)}\leqslant\frac{L}{N+1}\,|v|,\end{split}\end{multlined}|f^{i}(x^{1},\ldots,x^{k-1},x^{k}+v,x^{k+1},\ldots)-f^{i}(\boldsymbol{x})|\\\
\begin{split}&\leqslant
L\,W_{1}\Big{(}\frac{1}{N-1}\sum_{j\neq\\{i,k\\}}\delta_{x^{j}}+\frac{1}{N-1}\delta_{x^{k}+v},\frac{1}{N-1}\sum_{j\neq
i}\delta_{x^{j}}\Big{)}\leqslant\frac{L}{N+1}\,|v|,\end{split}$

one easily checks that the assumptions on $f^{i}$ in (MF) hold, independently
of $N$, provided that

(3.8) $\text{$F^{i}$ and $D_{x}F^{i}$ are Lipschitz continuous with respect to
the $(|\cdot|,W_{1})$ distance},$

since in such case $\|D_{k}F^{i}\|_{\infty}$ and
$\|D^{2}_{ki}F^{i}\|_{\infty}$ remain bounded proportionally to $1/N$
uniformly with respect to $k\neq i$ and $N$.

On the other hand, the assumptions on $g^{i}$ in (MF) are a bit more
restrictive. This is because several estimates below exploit the
regularization effect of the diffusion, hence bounds on $D^{\alpha}u^{i}$ will
depend on $D^{\alpha-1}f^{i}$ and on $D^{\alpha}g^{i}$. If
$g^{i}(\boldsymbol{x})=G^{i}(x^{i},\frac{1}{N-1}\sum_{j\neq
i}\delta_{x^{j}})$, where $G^{i}\in
C^{1}({\mathbb{R}}^{d}\times\mathcal{P}_{1}({\mathbb{R}}^{d}))$, then,
whenever $j\neq i$,

$D_{j}g^{i}(\boldsymbol{x})=\frac{1}{N-1}D_{m}G^{i}\Big{(}x^{i},\frac{1}{N-1}\sum_{j\neq
i}\delta_{x^{j}},x^{j}\Big{)},$

see [14, Remark 6.1.2]. Therefore, for (MF) to hold one should require (3.8)
with $G$ in lieu of $F$ and that

$D_{m}G^{i}$ and $D_{x}D_{m}G^{i}$ are Lipschitz continuous with respect to
the $(|\cdot|,W_{1},|\cdot|)$ distance.

A priori estimates, useful in studying the limit $N\to\infty$, are derived on
the $N$-dimensional Nash system under one of the following two additional
structural assumptions on $h\in\\{f,g\\}$ of _semimonotonicity_ :

  * (DS)

there exists a constant $M_{h}>0$, independent of $N$, such that $h$ is
$M_{h}$-$\mathscr{D}$*semimonotone;

  * (LS)

there exists a constant $\kappa_{h}\geqslant 0$, independent of $N$, such that
$h$ is $\kappa_{h}$-$\mathscr{L}$*semimonotone.

Our first main result is the following, which consists in the merging of
Theorems 5.2 and 6.5.

######  Theorem 3.6 (Estimates on the Nash system with semimonotone data).

Assume Mean-Field-like and semimonotone interactions; that is, assume (MF) and
either (DS) or (LS). For any $T>0$, if the semimonotonicity constants are
sufficiently small and $N\in\mathbb{N}$ is sufficiently large, any solution
$u$ to the Nash system (1.1) on $[0,T]\times({\mathbb{R}}^{d})^{N}$ satisfies

$\sup_{1\leqslant i\leqslant N}\bigg{(}\sup_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D_{j}u^{i}\rVert_{\infty}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}+\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D(D_{j}u^{i})\rVert_{\infty}^{2}\bigg{)}\lesssim\frac{1}{N}$

and

$\sup_{1\leqslant i\leqslant N}\lVert Du^{i}\rVert_{\infty}+\sup_{1\leqslant
i\leqslant N}\sum_{1\leqslant j\leqslant N}\lVert
D(D_{i}u^{j})\rVert_{\infty}^{2}+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D(D_{ij}u^{j})\rVert_{\infty}^{2}\lesssim 1,$

where the implied constants are independent of $N$. In addition, $u$ shares
the same type of semimonotonicity of the data.

As discussed in the introduction, we proceed as follows. First, if the
solution $u$ to the Nash system is semimonotone, then the desired estimates on
the derivatives hold; then, the semimonotonicity of the data provide
semimonotonicity $u$ near the final time $T$ in such a way that, as a
consequence of the aforementioned estimates, the semimonotonicity of $u$
“propagates” up to time $0$, provided that $N$ is large enough.

This is done for $\mathscr{D}$*semimonotone data first. Then, making use of
the relationship highlighted in Remark 3.3 between the two notions of
semimonotonicity, with little additional effort the case of
$\mathscr{L}$*semimonotone data is covered as well.

###  3.3. Nonsymmetric Mean-Field Games

In the second part of the work, we show that the estimates contained in
Theorem 3.6 allow to attack the convergence problem in the $N\to\infty$ limit.
To be sure that, for any fixed $N$, solutions to the Nash system exist, we
assume that $f^{i}$ and $g^{i}$ are of class
$C^{2+\alpha}(({\mathbb{R}}^{d})^{N})$ and
$C^{4+\alpha}(({\mathbb{R}}^{d})^{N})$ respectively, for every $i$ and $N$. By
Theorem A.1, solutions $u^{i}$ indeed exists and are unique, and belong to the
class $C^{2+\alpha/2,4+\alpha}$. We have then enough regularity to apply the a
priori estimates obtained in first part.

We work, for the sake of simplicity, without common noise; that is, we set
$\beta=0$ in (1.2) and thus in (1.1). Then we set $\sigma=1$

Besides this reduction, we let $\Lambda\vcentcolon=[0,1]$ and we suppose that,
for $h\in\\{f,g\\}$, the following symmetry assumption is fulfilled:

  * (S)

in the $N$-dimensional Nash system,
$h^{i}(\boldsymbol{x})=h_{N}(\lambda_{N}^{i},x^{i},\boldsymbol{x}^{-i})$, for
some _bounded_ sequence of maps
$h_{N}\colon\Lambda\times{\mathbb{R}}^{d}\times({\mathbb{R}}^{d})^{N-1}\to{\mathbb{R}}$,
$N\in\mathbb{N}$, that are _symmetric_ on $({\mathbb{R}}^{d})^{N-1}$, and for
some $\lambda_{N}^{i}\in\Lambda$;

more explicitly, the assumption of symmetry means that for any $\sigma\in
S_{N-1}$ (that is, $\sigma$ which is a permutation on
$\\{1,\dots,N-1\\}$)222Note that we are using now $\sigma$ to denote both the
diffusion coefficient in the Nash system and a permutation. Nevertheless, we
believe this should not cause any confusion, also because, as we have said, in
this part of the work the former is set to be $1$ (and thus it will not even
explicitly appear). one has

$h_{N}(\lambda,y,\boldsymbol{z}^{\sigma})=h_{N}(\lambda,y,\boldsymbol{z}),\qquad\boldsymbol{z}^{\sigma}\vcentcolon=(z^{\sigma(1)},\dots,z^{\sigma(N-1)}),$

for all $\lambda\in\Lambda$, $y\in{\mathbb{R}}^{d}$ and
$\boldsymbol{z}\in({\mathbb{R}}^{d})^{N-1}$.

######  Remark 3.7.

Letting $\lambda_{N}^{i}=\lambda\in\Lambda$ for all $N,i$, one obtains
standard symmetric MFG interactions, so that
$u^{i}(\boldsymbol{x})=u^{1}(x^{i},\boldsymbol{x}^{-i})$.

######  Remark 3.8.

A bounded and continuous function
$h_{N}\colon({\mathbb{R}}^{d})^{N}\to{\mathbb{R}}$ that is symmetric on its
domain can be seen as a “finite-dimensional projection” of the function
$\mathfrak{h}_{N}\colon\mathcal{P}_{p}({\mathbb{R}}^{d})\to{\mathbb{R}}$
defined by

$\mathfrak{h}_{N}(m)\vcentcolon=\inf_{\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}}\\{h(\boldsymbol{x})+W_{p}(m_{\boldsymbol{x}},m)\\}.$

Indeed, it is known (cf. [40, Theorem 2.1]) that there exists $\sigma\in
S_{N}$ such that

$W_{p}(m_{\boldsymbol{x}},m_{\boldsymbol{y}})=N^{-\frac{1}{p}}\bigg{(}\sum_{1\leqslant
i,j\leqslant
N}\big{|}x^{i}-y^{\sigma(i)}\big{|}^{p}\bigg{)}^{\frac{1}{p}}=N^{-\frac{1}{p}}|\boldsymbol{x}-\boldsymbol{y}^{\sigma}|_{p}\qquad\forall\,\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N},$

so that a symmetric $h$ is continuous with respect to the Euclidean distance
if and only if it is continuous with respect to the above Wasserstein
distance; then one readily checks that
$h_{N}(\boldsymbol{x})=\mathfrak{h}_{N}(m_{\boldsymbol{x}})$.

With this additional structure on the data, we require _Lipschitz continuous_
dependence on the _parameter_ $\lambda$:

  * (LP)

there exist a $L_{\Lambda}>0$, independent of $N$, such that

$\lVert
D^{\alpha}h_{N}(\lambda,\mspace{1.8mu}\cdot\mspace{1.8mu})-D^{\alpha}h_{N}(\lambda^{\prime},\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}\leqslant
L_{\Lambda}|\lambda-\lambda^{\prime}|\qquad\forall\,\lambda,\lambda^{\prime}\in\Lambda,\
i\in\\{1,\dots,N\\},\ |\alpha|\in\\{0,1\\}.$

In order to specify the choice of $\boldsymbol{\lambda}\in\Lambda^{N}$ with
which a Nash system is built, we use the notation $u_{\boldsymbol{\lambda}}$
to denote the solution to (1.1) on $[0,T]\times({\mathbb{R}}^{d})^{N}$ where
$h^{i}=h_{N}(\lambda_{i},\mspace{1.8mu}\cdot\mspace{1.8mu})$ for
$h\in\\{f,g\\}$, $i\in\\{1,\dots,N\\}$. Then, we focus on the map

$\mathfrak{u}_{N}\colon(\boldsymbol{\lambda},t,\boldsymbol{x})\mapsto
u_{\boldsymbol{\lambda}}^{1}(t,\boldsymbol{x}),$

which is representative of any solution of an $N$-dimensional Nash system of
the form we are considering, as the following remark shows.

######  Remark 3.9.

A solution $u_{\boldsymbol{\lambda}}$ is recovered by noticing that
$u^{i}_{\boldsymbol{\lambda}}(t,\boldsymbol{x})=\mathfrak{u}_{N}(\boldsymbol{\lambda}^{\sigma},t,\boldsymbol{x}^{\sigma})$
for any $\sigma\in S_{N}$ such that $\sigma(1)=i$, as by construction
$\mathfrak{u}_{N}$ is symmetric on $(\Lambda\times({\mathbb{R}}^{d}))^{N-1}$,
in the sense that it is invariant under permutations of the pairs
$(\lambda^{j},x^{j})$, $j>1$. Similarly, the optimal controls $D_{i}u^{i}$ are
recovered from $D_{1}\mathfrak{u}$. Heuristically, this means that switching
two players does not affect the value functions, but the two corresponding
labels need to be switched correspondingly in the Nash system.

As the first coordinate of $\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}$ and of
$\boldsymbol{\lambda}\in\Lambda^{N}$ have a distinct role for $\mathfrak{u}$
with respect to the other coordinates, we use the notations
$\boldsymbol{x}=(x,\hat{\boldsymbol{x}})\in{\mathbb{R}}^{d}\times({\mathbb{R}}^{d})^{N-1}$
and
$\boldsymbol{\lambda}=(\lambda,\hat{\boldsymbol{\lambda}})\in\Lambda\times\Lambda^{N-1}$.
Also, we write $\partial_{x}=D_{1}$ for the derivative with respect to
$x\in{\mathbb{R}}^{d}$ and $D_{j}$ for the one with respect to
$\hat{x}^{j}\in{\mathbb{R}}^{d}$.

Our main contribution contained in this part consists in identifying a limit
function and showing that it satisfies a certain representation formula. This
in turn implies that a solution to a generalised Mean-Field system can be
constructed. These results are contained in Theorems 7.5, 7.9 and 7.13 and we
summarize them here, omitting the more technical integral characterisation
provided by the second theorem.

######  Theorem 3.10.

Let assumptions (MF), (S) and (LP) be in force. Assume also that one between
(DS) and (LS) holds, and that the corresponding semimonotonicity constant is
sufficiently small, according to Theorem 3.6.

Then, there exists a Lipschitz continuous map
$U\colon\Lambda\times[0,T]\times{\mathbb{R}}^{d}\times\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})\to{\mathbb{R}}$
with bounded derivative $\partial_{x}U$ which is Lipschitz continuous on
$\Lambda\times{\mathbb{R}}^{d}\times\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$
and $\frac{1}{3}$-Hölder continuous on $[0,T]$, such that, up to a
subsequence,

$\sup|\partial_{x}^{k}\mathfrak{u}_{N}({\boldsymbol{\lambda}},t,\boldsymbol{x})-\partial_{x}^{k}U(\lambda,t,x,m_{(\hat{\boldsymbol{\lambda}}:\hat{\boldsymbol{x}})})|\,\xrightarrow[N\to\infty]{}\,0,\quad
k\in\\{0,1\\},$

whenever the supremum is taken over a set of the form

$\bigl{\\{}(\boldsymbol{\lambda},t,\boldsymbol{x})\in\Lambda^{N}\times[0,T]\times({\mathbb{R}}^{d})^{N}:\
|x|\leqslant R,\ m_{\hat{\boldsymbol{x}}}\in\mathcal{K}\bigr{\\}}$

for some $R>0$ and some compact set
$\mathcal{K}\subset\mathcal{P}_{2}({\mathbb{R}}^{d})$.

Furthermore, given
$\boldsymbol{\mu}\in\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$ with
continuous disintegration with respect to the projection onto $\Lambda$, the
function $u^{\lambda}(t,x)\vcentcolon=U^{\lambda}(t,x,\mu_{t})$ solves the
following (generalised) Mean-Field system on
$(0,T)\times\Lambda\times{\mathbb{R}}^{d}$:

$\begin{dcases}-\partial_{t}u^{\lambda}-\Delta_{x}u^{\lambda}+\frac{1}{2}|D_{x}u^{\lambda}|^{2}=f^{\lambda}(x,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{t})\\\
\partial_{t}\mu-\Delta_{x}\mu-\mathrm{div}_{x}(Du^{\lambda}\mu)=0\\\
u^{\lambda}(T,\cdot)=g^{\lambda}(\cdot,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{T}),\quad\mu_{0}=\boldsymbol{\mu},\end{dcases}$

where both the Hamilton–Jacobi and the Fokker–Planck equations are satisfied
in the classical sense.

######  Example 3.11.

Let
$F,G:\Lambda\times{\mathbb{R}}^{d}\times\mathcal{P}_{2}({\mathbb{R}}^{d})\to{\mathbb{R}}$
be bounded. Consider
$f_{N}(\lambda,y,\boldsymbol{z})=F(\lambda,y,m_{\boldsymbol{z}})$ and
$g_{N}(\lambda,y,\boldsymbol{z})=G(\lambda,y,m_{\boldsymbol{z}})$, so that

$f^{i}(\boldsymbol{x})=F(\lambda_{N}^{i},x^{i},m_{\boldsymbol{x}^{-i}}),\qquad
g^{i}(\boldsymbol{x})=G(\lambda_{N}^{i},x^{i},m_{\boldsymbol{x}^{-i}});$

that is, players’ costs are taken from a pool of admissible symmetric costs,
according to some label $\lambda_{N}^{i}$.

While (S) clearly holds and so does (MF) provided that the properties
highlighted in Remark 3.5 are satisfied for any fixed $\lambda$ (uniformly),
(LP) requires Lipschitz continuity with respect to $\lambda$ uniformly in the
other variables. Summing up, $F$ needs to satisfy

$F$ and $D_{x}F$ are Lipschitz continuous with respect to the
$(|\cdot|,|\cdot|,W_{1})$ distance

and so does $G$, for which we also require that $D_{m}G(\lambda,\cdot)$ and
$D_{x}D_{m}G(\lambda,\cdot)$ are Lipschitz continuous with respect to the
$(|\cdot|,W_{1},|\cdot|)$ distance, uniformly in $\lambda$.

Finally, one can check the semimonotonicity assumption (DS) or (LS) by
revising Remark 3.4, as follows. For any fixed $T>0$, Theorem 3.6 gives some
smallness condition on the semimonotonicity constant $M_{0}$.

$\bullet$ Suppose that, for any
$\nu\in\mathcal{P}(\Lambda\times({\mathbb{R}}^{d})^{2})$,

$\int_{\Lambda}\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\big{(}D_{x}F(\lambda,x,\pi^{(1)}_{{\mathbb{R}}^{d}}{}_{\sharp}\nu)-D_{x}F(\lambda,y,\pi^{(2)}_{{\mathbb{R}}^{d}}{}_{\sharp}\nu)+M(x-y)\big{)}\cdot(x-y)\,\nu(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}y)\geqslant
0,$

where
$\pi^{(i)}_{{\mathbb{R}}^{d}}\colon\Lambda\times({\mathbb{R}}^{d})^{2}\to{\mathbb{R}}^{d}$
is the projection onto the $i$-th copy of ${\mathbb{R}}^{d}$ ($i=1,2$). If
$D_{m}F(\lambda,\cdot)$ is $L$-Lipschitz with respect to the
$(|\mspace{1.8mu}\cdot\mspace{1.8mu}|,W_{1},|\mspace{1.8mu}\cdot\mspace{1.8mu}|)$
distance, uniformly in $\lambda$, then $f_{N}$ is
$(M+3L/N)$-$\mathscr{D}$*semimonotone. Therefore, if $M<M_{0}$, Theorem 3.10
applies (by requiring also the same inequality on $G$ of course).

$\bullet$ Suppose that, for any
$\mu_{1},\mu_{2}\in\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$,

$\int_{\Lambda\times{\mathbb{R}}^{d}}\bigl{(}F(\lambda,z,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{1})-F(\lambda,z,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{2})\bigr{)}(\mu_{1}-\mu_{2})(\mathrm{d}\lambda,\mathrm{d}z)\geqslant-
MW_{2}(\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{1},\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{2})^{2}.$

Then, arguing as in Remark 3.4, $f_{N}$ is
$(M+O(N^{-1}))$-$\mathscr{L}$*semimonotone. Hence, if $D_{x}D_{m}F$,
$D^{2}_{mm}F$ are bounded uniformly in $\lambda$ and $M<M_{0}$, Theorem 3.10
applies.

######  Remark 3.12.

The above result says, within the setting of the previous example, that the
Mean-Field system arises from a sequence of $N$-player games built by choosing
certain $\boldsymbol{\lambda}^{N}$, provided that the $\lambda^{N}_{i}$ are
“sufficiently well distributed”, in the sense that their empirical measures
weakly converge to some $\rho\in\mathcal{P}(\Lambda)$ as $N\to\infty$. Limit
properties of those games are then captured by the Mean-Field system when one
chooses $\boldsymbol{\mu}=\int_{\Lambda}m^{\lambda}\rho(\mathrm{d}\lambda)$,
being $m^{\lambda}$ the distribution of $\lambda$-labelled players.

On the other hand, by a theorem of Hedrlín (see [26]), for each Mean-Field
system of that kind there exists a suitable choice of
$\boldsymbol{\lambda}^{N}$ in order for it to arise from the respective
sequence of $N$-player games.

##  4\. Estimates under the one-sided Lipschitz condition on the drift

For the following computations, we assume that $u$ is of class $C^{1}$ in time
and of class $C^{4}$ in space. Throughout the section, we will always assume a
one-sided Lipschitz condition on the first-order (drift) term, as in (4.2)
below. Such term will be often the one appearing in (1.1), that is
$b^{i}=D_{i}u^{i}$, in which case the one-sided condition is equivalent to the
$\mathscr{D}$*semimonotonicity of $u$.

###  4.1. A Lipschitz estimate for linear equations

The first ingredient is a Lipschitz estimate for solutions of linear equations
with semimonotone drifts. Here, we implement the method of doubling variables
(also known as Ishii–Lions method, or coupling method in the probabilistic
community) as in [41]. The crucial features of the estimate are that the
Lipschitz constant is universal with respect to $N$, and its dependence on the
Lipschitz norm of the final condition, the supremum norm of the right-hand
side and $t$ is explicit. We note that the technique is well-known, but we are
aware of very few examples of statements that are dimensional independent
(e.g. [2]). Having a nondegenerate diffusion is crucial here, while the size
of $\beta$ does not play any role.

######  Lemma 4.1.

Let $v\colon[0,T]\times({\mathbb{R}}^{d})^{N}\to{\mathbb{R}}$ be a bounded
classical solution to

(4.1)
$\begin{cases}-\partial_{t}v-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}v)+\displaystyle\sum_{1\leqslant
j\leqslant N}b^{j}\cdot D_{j}v=F\\\ v|_{t=T}=G,\end{cases}$

where $F$ and $G$ are continuous and bounded, $G$ having bounded derivative,
$b\colon[0,T]\times({\mathbb{R}}^{d})^{N}\to({\mathbb{R}}^{d})^{N}$ is
continuous and bounded and for some $\tau\in[0,T]$ and $M\geqslant 0$,

(4.2) $\sum_{1\leqslant i\leqslant
N}\,(b^{i}(t,\boldsymbol{x})-b^{i}(t,\boldsymbol{y}))\cdot(x^{i}-y^{i})\geqslant-M|\boldsymbol{x}-\boldsymbol{y}|^{2}\quad\forall
t\in(\tau,T],\ \boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N}.$

Then

$\lVert
Dv(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}\leqslant\big{(}4\sigma^{-\frac{1}{2}}\lVert
F\rVert_{\infty}(T-t)^{\frac{1}{2}}e^{M(T-t)}+\lVert
DG\rVert_{\infty}\big{)}e^{M(T-t)}\quad\forall\,t\in[\tau,T].$

Note that the rate $\lVert F\rVert_{\infty}\sqrt{(T-t)/\sigma}$ is sharp and
coincides with the one of the heat equation $-(\partial_{t}+\sigma\Delta)v=F$,
that can be obtained, for instance, using Duhamel’s formula.

###### Proof.

Let
$w\colon[0,T]\times({\mathbb{R}}^{d})^{N}\times({\mathbb{R}}^{d})^{N}\to{\mathbb{R}}$
be given by
$w(t,\boldsymbol{x},\boldsymbol{y})\vcentcolon=v(t,\cdot)|_{\boldsymbol{y}}^{\boldsymbol{x}}-\psi(t,|\boldsymbol{x}-\boldsymbol{y}|)-\lVert
DG\rVert_{\infty}e^{M(T-t)}|\boldsymbol{x}-\boldsymbol{y}|$, where

$\psi(t,r)\vcentcolon=2{\sigma^{-\frac{1}{2}}}\lVert
F\rVert_{\infty}e^{2M(T-t)}\bigl{(}r\sqrt{T-t}+\sigma^{\frac{1}{2}}(T-t)(1-e^{-\frac{r}{\sqrt{\sigma(T-t)}}})\bigr{)},$

Computations show that for $\boldsymbol{x}\neq\boldsymbol{y}$

$\begin{split}D^{2}_{(\boldsymbol{x},\boldsymbol{y})}w(t,\boldsymbol{x},\boldsymbol{y})&=\mathrm{diag}\big{(}D^{2}v(t,\boldsymbol{x}),-D^{2}v(t,\boldsymbol{y})\big{)}-\psi_{rr}(t,|\boldsymbol{x}-\boldsymbol{y}|)\bigg{(}\begin{matrix}\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}&-\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}\\\
-\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}&\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}\end{matrix}\bigg{)}\\\
&\quad\
-\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\big{(}\psi_{r}(t,|\boldsymbol{x}-\boldsymbol{y}|)+\lVert
DG\rVert_{\infty}e^{M(T-t)}\big{)}\bigg{(}\begin{matrix}\boldsymbol{\mathsf{I}}-\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}&-\boldsymbol{\mathsf{I}}+\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}\\\
-\boldsymbol{\mathsf{I}}+\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}&\boldsymbol{\mathsf{I}}-\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}\end{matrix}\bigg{)},\end{split}$

where
$\boldsymbol{\mathsf{P}}_{\boldsymbol{e}}\vcentcolon=|\boldsymbol{e}|^{-2}\boldsymbol{e}\otimes\boldsymbol{e}$
for $\boldsymbol{e}\in({\mathbb{R}}^{d})^{N}\setminus\\{\boldsymbol{0}\\}$. We
now want to show that $w$ is a subsolution of the parabolic operator
$-\partial_{t}-L$ on
$(0,T)\times\big{(}({\mathbb{R}}^{d})^{N}\times({\mathbb{R}}^{d})^{N}\setminus\\{\boldsymbol{x}=\boldsymbol{y}\\}\big{)}$,
where

$L=\operatorname{tr}\\!\big{(}(\mathsf{J}_{2}\otimes(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}))D^{2}_{(\boldsymbol{x},\boldsymbol{y})}\big{)}-4\sigma\operatorname{tr}\\!\big{(}\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}D^{2}_{\boldsymbol{x}\boldsymbol{y}}\big{)}-b(t,\boldsymbol{x})\cdot
D_{\boldsymbol{x}}-b(t,\boldsymbol{y})\cdot D_{\boldsymbol{y}}.$

Note that the operator is indeed (degenerate) parabolic, since its diffusion
matrix

$\begin{pmatrix}\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}&\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}-2\sigma\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}\\\
\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}-2\sigma\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}&\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}\end{pmatrix}$

is nonnegative. We have

$\operatorname{tr}\\!\big{(}(\mathsf{J}_{2}\otimes(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}))D^{2}_{(\boldsymbol{x},\boldsymbol{y})}w(t,\boldsymbol{x},\boldsymbol{y})\big{)}=\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}v(t,\cdot))\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}$

and

$-4\sigma\operatorname{tr}\\!\big{(}\boldsymbol{\mathsf{P}}_{\boldsymbol{x}-\boldsymbol{y}}D^{2}_{\boldsymbol{x}\boldsymbol{y}}w(t,\boldsymbol{x},\boldsymbol{y})\big{)}=-4\sigma\psi_{rr}(t,|\boldsymbol{x}-\boldsymbol{y}|),$

where we used the notation $\psi_{r}=\frac{\partial}{\partial r}\psi$. Also,
we compute

$\displaystyle
D_{\boldsymbol{x}}w(t,\boldsymbol{x},\boldsymbol{y})=Dv(t,\boldsymbol{x})-\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\big{(}\psi_{r}(t,|\boldsymbol{x}-\boldsymbol{y}|)+\lVert
DG\rVert_{\infty}e^{M(T-t)}\big{)}(\boldsymbol{x}-\boldsymbol{y}),$
$\displaystyle
D_{\boldsymbol{y}}w(t,\boldsymbol{x},\boldsymbol{y})=-Dv(t,\boldsymbol{y})+\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\big{(}\psi_{r}(t,|\boldsymbol{x}-\boldsymbol{y}|)+\lVert
DG\rVert_{\infty}e^{M(T-t)}\big{)}(\boldsymbol{x}-\boldsymbol{y})$

and, according to (1.1),

$\begin{split}\partial_{t}w(t,\boldsymbol{x},\boldsymbol{y})=\,&-\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}v(t,\cdot)\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\big{)}+\sum_{1\leqslant
j\leqslant
N}b^{j}(t,\cdot)D_{j}v(t,\cdot)\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}-F|_{\boldsymbol{y}}^{\boldsymbol{x}}\\\
&-\psi_{t}(t,|\boldsymbol{x}-\boldsymbol{y}|)+M\lVert
DG\rVert_{\infty}e^{M(T-t)}|\boldsymbol{x}-\boldsymbol{y}|,\end{split}$

where $\psi_{t}=\frac{\partial}{\partial t}\psi$. It follows that

$\begin{multlined}(\partial_{t}+L)w(t,\boldsymbol{x},\boldsymbol{y})\\\
\begin{multlined}=-4\sigma\psi_{rr}(t,|\boldsymbol{x}-\boldsymbol{y}|)-F|_{\boldsymbol{y}}^{\boldsymbol{x}}-\psi_{t}(t,|\boldsymbol{x}-\boldsymbol{y}|)+M\lVert
DG\rVert_{\infty}e^{M(T-t)}|\boldsymbol{x}-\boldsymbol{y}|\\\
+\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\big{(}\psi_{r}(t,|\boldsymbol{x}-\boldsymbol{y}|)(|\boldsymbol{x}-\boldsymbol{y}|)+\lVert
DG\rVert_{\infty}e^{M(T-t)}\big{)}\sum_{1\leqslant j\leqslant
N}b^{j}(t,\cdot)\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{j}-y^{j});\end{multlined}=-4\sigma\psi_{rr}(t,|\boldsymbol{x}-\boldsymbol{y}|)-F|_{\boldsymbol{y}}^{\boldsymbol{x}}-\psi_{t}(t,|\boldsymbol{x}-\boldsymbol{y}|)+M\lVert
DG\rVert_{\infty}e^{M(T-t)}|\boldsymbol{x}-\boldsymbol{y}|\\\
+\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\big{(}\psi_{r}(t,|\boldsymbol{x}-\boldsymbol{y}|)(|\boldsymbol{x}-\boldsymbol{y}|)+\lVert
DG\rVert_{\infty}e^{M(T-t)}\big{)}\sum_{1\leqslant j\leqslant
N}b^{j}(t,\cdot)\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{j}-y^{j});\end{multlined}(\partial_{t}+L)w(t,\boldsymbol{x},\boldsymbol{y})\\\
\begin{multlined}=-4\sigma\psi_{rr}(t,|\boldsymbol{x}-\boldsymbol{y}|)-F|_{\boldsymbol{y}}^{\boldsymbol{x}}-\psi_{t}(t,|\boldsymbol{x}-\boldsymbol{y}|)+M\lVert
DG\rVert_{\infty}e^{M(T-t)}|\boldsymbol{x}-\boldsymbol{y}|\\\
+\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\big{(}\psi_{r}(t,|\boldsymbol{x}-\boldsymbol{y}|)(|\boldsymbol{x}-\boldsymbol{y}|)+\lVert
DG\rVert_{\infty}e^{M(T-t)}\big{)}\sum_{1\leqslant j\leqslant
N}b^{j}(t,\cdot)\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{j}-y^{j});\end{multlined}=-4\sigma\psi_{rr}(t,|\boldsymbol{x}-\boldsymbol{y}|)-F|_{\boldsymbol{y}}^{\boldsymbol{x}}-\psi_{t}(t,|\boldsymbol{x}-\boldsymbol{y}|)+M\lVert
DG\rVert_{\infty}e^{M(T-t)}|\boldsymbol{x}-\boldsymbol{y}|\\\
+\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|}\big{(}\psi_{r}(t,|\boldsymbol{x}-\boldsymbol{y}|)(|\boldsymbol{x}-\boldsymbol{y}|)+\lVert
DG\rVert_{\infty}e^{M(T-t)}\big{)}\sum_{1\leqslant j\leqslant
N}b^{j}(t,\cdot)\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{j}-y^{j});$

hence by the hypothesis (4.2) on $b$,

$(\partial_{t}+L)w(t,\boldsymbol{x},\boldsymbol{y})\geqslant-4\sigma\psi_{rr}(t,|\boldsymbol{x}-\boldsymbol{y}|)-M\psi_{r}(t,|\boldsymbol{x}-\boldsymbol{y}|)|\boldsymbol{x}-\boldsymbol{y}|-\psi_{t}(t,|\boldsymbol{x}-\boldsymbol{y}|)-2\lVert
F\rVert_{\infty}.$

Now, direct computations show that, with $r=|\boldsymbol{x}-\boldsymbol{y}|$,
$s=T-t$ and $y=r/\sqrt{T-t}$,

$\begin{multlined}-4\sigma\psi_{rr}(t,r)-M\psi_{r}(t,r)r-\psi_{t}(t,r)\\\
\begin{split}&=2{\sigma^{-\frac{1}{2}}}\lVert
F\rVert_{\infty}e^{2Ms}\Bigl{(}\sigma^{\frac{1}{2}}+3\sigma^{\frac{1}{2}}e^{-y\sigma^{-\frac{1}{2}}}+2Ms\sigma^{\frac{1}{2}}(1-e^{-y\sigma^{-\frac{1}{2}}})+\frac{y}{2}(1+2Ms)(1-e^{-y\sigma^{-\frac{1}{2}}})\Bigr{)}\\\
&>2\lVert
F\rVert_{\infty}.\end{split}\end{multlined}-4\sigma\psi_{rr}(t,r)-M\psi_{r}(t,r)r-\psi_{t}(t,r)\\\
\begin{split}&=2{\sigma^{-\frac{1}{2}}}\lVert
F\rVert_{\infty}e^{2Ms}\Bigl{(}\sigma^{\frac{1}{2}}+3\sigma^{\frac{1}{2}}e^{-y\sigma^{-\frac{1}{2}}}+2Ms\sigma^{\frac{1}{2}}(1-e^{-y\sigma^{-\frac{1}{2}}})+\frac{y}{2}(1+2Ms)(1-e^{-y\sigma^{-\frac{1}{2}}})\Bigr{)}\\\
&>2\lVert F\rVert_{\infty}.\end{split}$

Since $w(t,\cdot)|_{\boldsymbol{x}=\boldsymbol{y}}=0$ for all $t$, by the
maximum principle333Here one needs to apply the maximum principle for
classical subsolutions of parabolic equations on unbounded domains. The proof
of this is rather classical, and it is based on the fact that interior maxima
of $w$ cannot exist. Since $w$ may not have a maximum on the unbounded set
$({\mathbb{R}}^{d})^{N}\times({\mathbb{R}}^{d})^{N}$, it should be additively
perturbed for instance by
$-\varepsilon(e^{K(T-t)}(1+|\boldsymbol{x}|^{2}+|\boldsymbol{y}|^{2})+(T-\tau)^{-1})$,
with $K$ large enough. Being now $w$ bounded (as well as $b$), then the
conclusion follows by letting $\varepsilon\to 0$. We do not provide here
further details, but mention that identical computations appear in the proof
of the next Lemma 4.3.

$w\leqslant\left(\max_{({\mathbb{R}}^{d})^{N}\times({\mathbb{R}}^{d})^{N}}w(T,\cdot)\right)_{\\!+}=0\quad\text{on
$[\tau,T]\times({\mathbb{R}}^{d})^{N}\times({\mathbb{R}}^{d})^{N}$.}$

Being the choice of $t$, $\boldsymbol{x}$ and $\boldsymbol{y}$ arbitrary, the
desired conclusion follows by observing that $\psi(t,r)\leqslant
4\sigma^{-\frac{1}{2}}\lVert F\rVert_{\infty}e^{2M(T-t)}r\sqrt{T-t}$. ∎

######  Remark 4.2.

The previous estimate can be proven also for viscosity solutions, using
standard methods. Unbounded solutions (with controlled growth) can be also
considered, but we are not going to address these generalizations here for the
sake of simplicity.

The following estimates will make use of Lemma 4.1 with $b^{j}=D_{j}u^{j}$.
They will be all derived under the assumption that $u$ be
$\mathscr{D}$*semimonotone on $({\mathbb{R}}^{d})^{2N}$ uniformly with respect
to $t\in(\tau,T]$; that is, according to the introduced notation,
$\mathcal{D}[u]\geqslant-MQ$ on $(\tau,T]\times({\mathbb{R}}^{d})^{2N}$.

###  4.2. First-order derivatives of the value functions

The next result is a Lipschitz estimate for the value function, it is obtained
by doubling variables. We show that the Hamilton–Jacobi equations preserve
some weighted Lipschitz seminorm on $({\mathbb{R}}^{d})^{N}$. Note that,
contrarily to the previous lemma, possible regularization effects from the
diffusion are not exploited below.

Recall that, for any $\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}$, we have the
_$i$ -th weighted norm_

$\lVert\boldsymbol{x}\rVert^{i}\vcentcolon=\Bigl{(}|x^{i}|^{2}+{\frac{1}{N}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}|x^{j}|^{2}\Bigr{)}^{\frac{1}{2}}.$

######  Lemma 4.3 (Weighted Lipschitz continuity of the value functions).

Let $\tau\in[0,T)$. Suppose that $\mathcal{D}[u]\geqslant-MQ$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$ for some $M>0$ and that, for
$h\in\\{f,g\\}$, there exist $\widetilde{L}_{h}>0$ such that

(4.3)
$|h^{i}(\boldsymbol{x})-h^{i}(\boldsymbol{y})|\leqslant\widetilde{L}_{h}\lVert\boldsymbol{x}-\boldsymbol{y}\rVert^{i}\qquad\forall\,\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N},\
i\in\\{1,\dots,N\\}.$

Then

(4.4) $|u^{i}(t,\boldsymbol{x})-u^{i}(t,\boldsymbol{y})|\leqslant
c_{1}\lVert\boldsymbol{x}-\boldsymbol{y}\rVert^{i}\qquad\forall\,t\in[\tau,T],\
\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N},$

where the constant $c_{1}$ depends only on $T$, $M$, $\widetilde{L}_{g}$ and
$\widetilde{L}_{f}$.

###### Proof.

For $\varepsilon>0$, let $w^{i}$ be the function
$w^{i}(t,\boldsymbol{x},\boldsymbol{y})\vcentcolon=u^{i}(t,\cdot)|_{\boldsymbol{y}}^{\boldsymbol{x}}-\psi(t)\lVert\boldsymbol{x}-\boldsymbol{y}\rVert^{i}-\varepsilon\varphi(t,\boldsymbol{x},\boldsymbol{y})-\frac{\varepsilon}{t-\tau}$,
defined for $t\in(\tau,T]$,
$\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N}$, where
$\psi(t)\vcentcolon=e^{M(T-t)}\widetilde{L}_{g}+\frac{e^{M(T-t)}-1}{M}\,\widetilde{L}_{f}$
and
$\varphi(t,\boldsymbol{x},\boldsymbol{y})=e^{K(T-t)}(1+|\boldsymbol{x}|^{2}+|\boldsymbol{y}|^{2})$
for some constant $K\geqslant 0$ to be determined. Since $u^{i}$ is bounded,
$w^{i}$ attains its maximum at some point
$(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}})\in(\tau,T]\times({\mathbb{R}}^{d})^{N}\times({\mathbb{R}}^{d})^{N}$.
Suppose that $\overline{t}\neq T$ and
$\overline{\boldsymbol{x}}\neq\overline{\boldsymbol{y}}$. Computations show
that for $\boldsymbol{x}\neq\boldsymbol{y}$ (for simplicity, one may follow
the computations below with $\varepsilon=0$, being the corresponding terms
merely necessary to guarantee the existence of a maximum for $w$, and
therefore of perturbative nature)

$\begin{split}D^{2}_{(\boldsymbol{x},\boldsymbol{y})}w^{i}(t,\boldsymbol{x},\boldsymbol{y})&=\biggl{(}\begin{matrix}D^{2}u^{i}(t,\boldsymbol{x})-2\varepsilon
e^{K(T-t)}\boldsymbol{\mathsf{I}}&\boldsymbol{\mathsf{0}}\\\
\boldsymbol{\mathsf{0}}&-D^{2}u^{i}(t,\boldsymbol{y})-2\varepsilon
e^{K(T-t)}\boldsymbol{\mathsf{I}}\end{matrix}\,\biggr{)}\\\ &\quad\
-\frac{\psi(t)}{\lVert\boldsymbol{x}-\boldsymbol{y}\rVert^{i}}\bigg{(}\begin{matrix}\boldsymbol{\mathsf{I}}^{i}-\boldsymbol{\mathsf{P}}^{i}_{\boldsymbol{x}-\boldsymbol{y}}&-\boldsymbol{\mathsf{I}}^{i}+\boldsymbol{\mathsf{P}}^{i}_{\boldsymbol{x}-\boldsymbol{y}}\\\
-\boldsymbol{\mathsf{I}}^{i}+\boldsymbol{\mathsf{P}}^{i}_{\boldsymbol{x}-\boldsymbol{y}}&\boldsymbol{\mathsf{I}}^{i}-\boldsymbol{\mathsf{P}}^{i}_{\boldsymbol{x}-\boldsymbol{y}}\end{matrix}\bigg{)},\end{split}$

where $\boldsymbol{\mathsf{I}}^{i}$ is the block diagonal matrix given by

$(\boldsymbol{\mathsf{I}}^{i})_{jj}=\begin{cases}\mathsf{I}_{d}&\text{if}\
j=i\\\ N^{-1}\mathsf{I}_{d}&\text{if}\ j\neq i\end{cases}$

and we have set, for
$\boldsymbol{e}\in({\mathbb{R}}^{d})^{N}\setminus\\{\boldsymbol{0}\\}$,

$\boldsymbol{\mathsf{P}}^{i}_{\boldsymbol{e}}\vcentcolon=\frac{\mathsf{N}_{i}\boldsymbol{e}}{\lVert\boldsymbol{e}\rVert^{i}}\otimes\frac{\mathsf{N}_{i}\boldsymbol{e}}{\lVert\boldsymbol{e}\rVert^{i}},\qquad(\mathsf{N}_{i}\boldsymbol{e})^{j}=\begin{cases}e^{i}&\text{if}\
j=i\\\ N^{-1}e^{j}&\text{if}\ j\neq i.\end{cases}$

Then, as
$\mathsf{J}_{2}\otimes(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})\geqslant
0$ and
$\operatorname{tr}(D^{2}w^{i}(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}}))\leqslant
0$, we have

(4.5)
$\begin{split}0&\geqslant\operatorname{tr}\\!\big{(}(\mathsf{J}_{2}\otimes(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}))D^{2}_{(\boldsymbol{x},\boldsymbol{y})}w^{i}(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}})\big{)}\\\
&=\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu}))\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}-{4}(\sigma+\beta)Nd\varepsilon
e^{K(T-t)}.\end{split}$

Also, since
$(D_{\boldsymbol{x}},D_{\boldsymbol{y}},\partial_{t})w^{i}(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}})=0$,
we have

(4.6) $\displaystyle
0=Du^{i}(\overline{t},\overline{\boldsymbol{x}})-\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}\,\mathsf{N}_{i}(\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}})-2\varepsilon
e^{K(T-t)}\overline{\boldsymbol{x}},$ (4.7) $\displaystyle
0=-Du^{i}(\overline{t},\overline{\boldsymbol{y}})+\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}\,\mathsf{N}_{i}(\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}})-2\varepsilon
e^{K(T-t)}\overline{\boldsymbol{y}}$

and, using (1.1) when computing $\partial_{t}w^{i}$,

(4.8)
$\begin{split}0&=-\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}\big{)}+\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{i}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}\\\
&\quad\
+\frac{1}{2}|D_{i}u^{i}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})|^{2}\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}-f^{i}|_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}-\psi^{\prime}(\overline{t})\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}+\varepsilon
Ke^{K(T-t)}(1+|\overline{\boldsymbol{x}}|^{2}+|\overline{\boldsymbol{y}}|^{2})+\frac{\varepsilon}{(\overline{t}-\tau)^{2}}.\end{split}$

Exploiting relations (4.6) and (4.7) one sees that

(4.9) $\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{i}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}+\frac{1}{2}|D_{i}u^{i}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})|^{2}\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}\\\
=\frac{1}{N}\,\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}\cdot(\overline{x}^{j}-\overline{y}^{j})+4\varepsilon
e^{K(T-\overline{t})}\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}(\overline{x}^{i}-\overline{y}^{i})\cdot(\overline{x}^{i}+\overline{y}^{i})\\\
+4\varepsilon^{2}e^{2K(T-\overline{t})}(|\overline{x}^{i}|^{2}-|\overline{y}^{i}|^{2})+2\varepsilon
e^{K(T-\overline{t})}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D_{j}u^{j}(\overline{t},\overline{\boldsymbol{x}})\cdot\overline{x}^{j}+D_{j}u^{j}(\overline{t},\overline{\boldsymbol{y}})\cdot\overline{y}^{i}\big{)},$

where the terms can be estimated as follows: since
$\mathcal{D}[u](\overline{t})\geqslant-MQ$ and
$\sqrt{N}\,\lVert\boldsymbol{x}-\boldsymbol{y}\rVert^{i}\geqslant|\boldsymbol{x}-\boldsymbol{y}|$,
using relations (4.6) and (4.7) along with the Cauchy–Schwarz inequality,

(4.10)
$\frac{1}{N}\,\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}\cdot(\overline{x}^{j}-\overline{y}^{j})\\\
\begin{split}&=\frac{1}{N}\,\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}\sum_{1\leqslant
j\leqslant
N}D_{j}u^{j}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}\cdot(\overline{x}^{j}-\overline{y}^{j})-\frac{1}{N}\,\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}D_{i}u^{i}(\overline{t},\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}\cdot(\overline{x}^{i}-\overline{y}^{i})\\\
&\geqslant-M\psi(\overline{t})\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}-\frac{1}{N}\,\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}\,2\varepsilon
e^{K(T-\overline{t})}(\overline{x}^{i}+\overline{y}^{i})\cdot(\overline{x}^{i}-\overline{y}^{i})\\\
&\geqslant-M\psi(\overline{t})\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}-\frac{2\varepsilon}{\sqrt{N}}\,\psi(\overline{t})\varphi(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}});\end{split}$

since
$\lVert\boldsymbol{x}-\boldsymbol{y}\rVert^{i}\leqslant|\boldsymbol{x}-\boldsymbol{y}|$,
by the Cauchy–Schwarz inequality

(4.11) $\begin{multlined}4\varepsilon
e^{K(T-\overline{t})}\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}(\overline{x}^{i}-\overline{y}^{i})\cdot(\overline{x}^{i}+\overline{y}^{i})+4\varepsilon^{2}e^{2K(T-\overline{t})}(|\overline{x}^{i}|^{2}-|\overline{y}^{i}|^{2})\\\
\begin{split}&\geqslant-4\varepsilon
e^{K(T-\overline{t})}\psi(\overline{t})(|\overline{x}^{i}|+|\overline{y}^{i}|)-4\varepsilon^{2}e^{2K(T-\overline{t})}(|\overline{\boldsymbol{x}}|^{2}+|\overline{\boldsymbol{y}}|^{2})\\\
&\geqslant-4\varepsilon\Big{(}\psi(\overline{t})+\varepsilon
e^{K(T-\overline{t})}\Big{)}\varphi(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}});\end{split}\end{multlined}4\varepsilon
e^{K(T-\overline{t})}\frac{\psi(\overline{t})}{\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}}(\overline{x}^{i}-\overline{y}^{i})\cdot(\overline{x}^{i}+\overline{y}^{i})+4\varepsilon^{2}e^{2K(T-\overline{t})}(|\overline{x}^{i}|^{2}-|\overline{y}^{i}|^{2})\\\
\begin{split}&\geqslant-4\varepsilon
e^{K(T-\overline{t})}\psi(\overline{t})(|\overline{x}^{i}|+|\overline{y}^{i}|)-4\varepsilon^{2}e^{2K(T-\overline{t})}(|\overline{\boldsymbol{x}}|^{2}+|\overline{\boldsymbol{y}}|^{2})\\\
&\geqslant-4\varepsilon\Big{(}\psi(\overline{t})+\varepsilon
e^{K(T-\overline{t})}\Big{)}\varphi(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}});\end{split}$

finally, by the Cauchy–Schwarz and Young’s inequalities,

(4.12) $\begin{multlined}2\varepsilon
e^{K(T-\overline{t})}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D_{j}u^{j}(\overline{t},\overline{\boldsymbol{x}})\cdot\overline{x}^{j}+D_{j}u^{j}(\overline{t},\overline{\boldsymbol{y}})\cdot\overline{y}^{i}\big{)}\\\\[-15.0pt]
\geqslant-\varepsilon\Big{(}2N\sup_{1\leqslant j\leqslant N}\lVert
D_{j}u^{j}\rVert^{2}_{\infty}+1\Big{)}\varphi(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}}).\end{multlined}2\varepsilon
e^{K(T-\overline{t})}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D_{j}u^{j}(\overline{t},\overline{\boldsymbol{x}})\cdot\overline{x}^{j}+D_{j}u^{j}(\overline{t},\overline{\boldsymbol{y}})\cdot\overline{y}^{i}\big{)}\\\\[-15.0pt]
\geqslant-\varepsilon\Big{(}2N\sup_{1\leqslant j\leqslant N}\lVert
D_{j}u^{j}\rVert^{2}_{\infty}+1\Big{)}\varphi(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}}).$

Plugging now (4.9) into (4.8) and exploiting inequalities (4.5), (4.10),
(4.11) and (4.12) as well as (4.3), we get

(4.13)
$\begin{split}0<\big{(}&M\psi(\overline{t})+\widetilde{L}_{f}+\psi^{\prime}(\overline{t})\big{)}\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}\\\
&+\varepsilon\big{(}4(\sigma+\beta)Nd+4\psi(\overline{t})+2\varepsilon
e^{K(T-\overline{t})}+2NC_{N}+1-K\big{)}\varphi(\overline{t},\overline{\boldsymbol{x}},\overline{\boldsymbol{y}}),\end{split}$

where $C_{N}\vcentcolon=\sup_{1\leqslant j\leqslant N}\lVert
D_{j}u^{j}\rVert^{2}_{\infty}$. At this point, choose
$K=4(\sigma+\beta)Nd+4\psi(0)+2NC_{N}+3$; thus for all $\varepsilon<e^{-KT}$
one has $4(\sigma+\beta)Nd+4\psi(\overline{t})+2\varepsilon
e^{K(T-\overline{t})}+2NC_{N}+1-K<0$. On the other hand, by the definition of
$\psi$ we have $M\psi+\widetilde{L}_{f}+\psi^{\prime}=0$, thus the above
inequality (4.13) cannot hold. We conclude that one must have $\overline{t}=T$
or $\overline{\boldsymbol{x}}=\overline{\boldsymbol{y}}$, for any sufficiently
small $\varepsilon$. Letting $\varepsilon\to 0$, we deduce that
$u^{i}(t,\cdot)|_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}-\psi(t)\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i}\leqslant(g^{i}|_{\overline{\boldsymbol{y}}}^{\overline{\boldsymbol{x}}}-\widetilde{L}_{g}\lVert\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}\rVert^{i})_{+}=0$,
where we have used (4.3) to obtain the last equality. The desired conclusion
follows with $c_{1}=\psi(0)$. ∎

######  Remark 4.4.

The previous result can be obtained for solutions to Hamilton–Jacobi equations
where the transport term $\sum_{1\leqslant j\leqslant N,\,j\neq
i}D_{j}u^{j}\cdot D_{j}u^{i}$ is replaced by $\sum_{1\leqslant j\leqslant
N}b^{j}\cdot D_{j}u^{i}$, with $b$ satisfying (4.2).

The previous proposition immediately provides an estimate for the _skew_
first-order derivatives; that is, for those derivatives of the $i$-th value
functions with respect to $x^{j}$ with $j\neq i$.

######  Proposition 4.5 (Estimate on skew first-order derivatives).

Let $\tau\in[0,T)$. Suppose that $\mathcal{D}[u]\geqslant-MQ$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$. Then

$\sup_{1\leqslant i\leqslant N}\Big{\|}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}|D_{j}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})|^{2}\Big{\|}_{\infty}\leqslant\frac{c_{1}^{2}}{N}\quad\forall\,t\in[\tau,T],$

where $c_{1}$ is the constant given in Lemma 4.3 (that applies with
$\widetilde{L}_{f}=\sqrt{2}L_{f}$ and $\widetilde{L}_{g}=\sqrt{2}L_{g}$, where
$L_{f}$ and $L_{g}$ are the constants appearing in (MF)).

###### Proof.

Note first that by the mean value theorem and (MF) we have

$|f^{i}(\boldsymbol{x})-f^{i}(\boldsymbol{y})|\leqslant\Big{\|}\Big{(}|D_{i}f^{i}|^{2}+N\sum_{j\neq
i}|D_{j}f^{i}|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{\infty}\lVert\boldsymbol{x}-\boldsymbol{y}\rVert^{i}\leqslant\sqrt{2}\,L_{f}\lVert\boldsymbol{x}-\boldsymbol{y}\rVert^{i}.$

Let now $\boldsymbol{\xi}\in({\mathbb{R}}^{d})^{N}$ be such that
$|\boldsymbol{\xi}|=1$ and $\boldsymbol{\xi}^{i}=0$. Plug
$\boldsymbol{y}=\boldsymbol{x}+h\boldsymbol{\xi}$ into (4.4) and let $h\to 0$
to obtain $|Du^{i}(t,\boldsymbol{x})\cdot\boldsymbol{\xi}|\leqslant
c_{1}/\sqrt{N}$. Taking now the supremum over $\boldsymbol{\xi}$ yields
$\bigl{(}\sum_{j\neq
i}|D_{j}u^{i}(t,\boldsymbol{x})|^{2}\bigr{)}^{\frac{1}{2}}\leqslant{c_{1}}/{\sqrt{N}}$.
∎

###  4.3. Second-order derivatives of the value functions

Although all constants contained in the following statements will depend also
on the dimension $d$, we shall omit to specify it.

We now show that certain sums of second-order derivatives over the indices of
the players are controlled uniformly with respect to $N$. This is a
preliminary step to achieve more precise control on second-order derivatives.

######  Proposition 4.6.

Let $\tau\in[0,T)$. Suppose that $\mathcal{D}[u]\geqslant-MQ$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$. Then

$\sup_{1\leqslant k\leqslant N}\sum_{1\leqslant i\leqslant N}\lVert
D(D_{k}u^{i})(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}^{2}=\sup_{1\leqslant
k\leqslant N}\sum_{1\leqslant i\leqslant N}\bigg{\lVert}\sum_{1\leqslant
j\leqslant
N}\big{|}D^{2}_{jk}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant
C_{2}\quad\forall\,t\in[\tau,T],$

where the constant $C_{2}$ depends only on $\sigma$, $T$, $M$, $L_{g}$ and
$L_{f}$.

###### Proof.

Let $k\in\\{1,\dots,N\\}$, $\ell\in\\{1,\dots,d\\}$ and $s\in[0,T-\tau]$. Note
that $v=v^{k\ell}=D_{x^{k\ell}}u^{i}$ solves (4.1) on
$[T-s,T]\times({\mathbb{R}}^{d})^{N}$ with $b^{j}=D_{j}u^{j}$,
$F=D_{x^{k\ell}}f^{i}-\sum_{1\leqslant j\leqslant N,\,j\neq
i}D_{j}(D_{x^{k\ell}}u^{j})D_{j}u^{i}$ and $G=D_{x^{k\ell}}g^{i}$. By Lemma
4.1,

(4.14) $\begin{multlined}\lVert D_{x^{k\ell}}Du^{i}\rVert_{\infty}\\\
\leqslant C_{\sigma}\lVert
D_{x^{k\ell}}f^{i}\rVert_{\infty}s^{\frac{1}{2}}e^{2Ms}+\lVert
D(D_{x^{k\ell}}g^{i})\rVert_{\infty}e^{Ms}+C_{\sigma}s^{\frac{1}{2}}e^{2Ms}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty},\end{multlined}\lVert
D_{x^{k\ell}}Du^{i}\rVert_{\infty}\\\ \leqslant C_{\sigma}\lVert
D_{x^{k\ell}}f^{i}\rVert_{\infty}s^{\frac{1}{2}}e^{2Ms}+\lVert
D(D_{x^{k\ell}}g^{i})\rVert_{\infty}e^{Ms}+C_{\sigma}s^{\frac{1}{2}}e^{2Ms}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty},$

where the $L^{\infty}$-norm are understood to be computed on
$[T-s,T]\times({\mathbb{R}}^{d})^{N}$ or $({\mathbb{R}}^{d})^{N}$. By
Proposition 4.5 and the Cauchy–Schwarz inequality we have

$\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}^{2}\leqslant\frac{c_{1}^{2}}{N}\,\bigg{\lVert}\sum_{1\leqslant
j\leqslant N}\big{|}D^{2}_{jk}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty};$

therefore from (4.14) and Young’s inequality we get

(4.15) $\begin{multlined}\lVert D(D_{k}u^{i})\rVert_{\infty}^{2}\\\ \leqslant
3dC_{\sigma}^{2}\lVert D_{k}f^{i}\rVert_{\infty}^{2}e^{4Ms}s+3d\lVert
D(D_{k}g^{i})\rVert_{\infty}^{2}e^{2Ms}+\frac{3dC_{\sigma}^{2}c^{2}_{1}e^{4Ms}s}{N}\bigg{\lVert}\sum_{1\leqslant
j\leqslant
N}\big{|}D^{2}_{jk}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}.\end{multlined}\lVert
D(D_{k}u^{i})\rVert_{\infty}^{2}\\\ \leqslant 3dC_{\sigma}^{2}\lVert
D_{k}f^{i}\rVert_{\infty}^{2}e^{4Ms}s+3d\lVert
D(D_{k}g^{i})\rVert_{\infty}^{2}e^{2Ms}+\frac{3dC_{\sigma}^{2}c^{2}_{1}e^{4Ms}s}{N}\bigg{\lVert}\sum_{1\leqslant
j\leqslant N}\big{|}D^{2}_{jk}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}.$

Since by (MF) we have $\sup_{1\leqslant k\leqslant N}\sum_{1\leqslant
i\leqslant N}\lVert D_{k}f^{i}\rVert_{\infty}^{2}\leqslant 2L_{f}^{2}$,
summing (4.15) over $i$ one deduces that

$\sum_{1\leqslant i\leqslant N}\lVert
D(D_{k}u^{i})\rVert_{\infty}^{2}\leqslant
6dC_{\sigma}^{2}e^{4Ms}sL_{f}^{2}+3de^{2Ms}\sum_{1\leqslant i\leqslant
N}\lVert
D_{k}Dg^{i}\rVert_{\infty}^{2}+{3dC_{\sigma}^{2}c^{2}_{1}e^{4Ms}s}\sum_{1\leqslant
i\leqslant N}\lVert D(D_{k}u^{i})\rVert_{\infty}^{2}.$

If $3dC_{\sigma}^{2}c^{2}_{1}e^{4Ms}s\leqslant\frac{1}{2}$ this yields

(4.16) $\sum_{1\leqslant i\leqslant N}\lVert
D(D_{k}u^{i})\rVert_{\infty}^{2}\leqslant\frac{2L_{f}^{2}}{c_{1}^{2}}+6de^{2Ms}\sum_{1\leqslant
i\leqslant N}\lVert D(D_{k}g^{i})\rVert_{\infty}^{2}.$

Considering that $g^{i}=u^{i}(T,\cdot)$, by iterating estimate (4.16) on the
intervals $[T-\ell s,T-(\ell-1)s]$ with $u^{i}(T-(\ell-1)s,\cdot)$ in lieu of
$g^{i}$ for any positive integer
$\ell\leqslant\ell^{*}\vcentcolon=\lfloor(T-\tau)/s\rfloor$ and then on the
interval $[\tau,T-\ell^{*}s]$, one can prove by induction that

$\sum_{1\leqslant i\leqslant N}\lVert
D(D_{k}u^{i})\rVert_{\infty}^{2}\leqslant\sum_{1\leqslant\ell\leqslant\ell^{*}+1}\bigg{(}\frac{2L_{f}^{2}}{c_{1}^{2}}\sum_{0\leqslant
m\leqslant\ell-1}(6de^{2Ms})^{m}+(6de^{2Ms})^{\ell}\sum_{1\leqslant i\leqslant
N}\lVert D(D_{k}g^{i})\rVert_{\infty}^{2}\bigg{)},$

where the $L^{\infty}$-norms are understood to be computed on
$[\tau,T]\times({\mathbb{R}}^{d})^{N}$. We can estimate $s\ell^{*}\leqslant T$
and choose $s=(6dC_{\sigma}^{2}c_{1}^{2}e^{4MT})^{-1}\wedge(T-\tau)$, hence
the desired conclusion follows by noticing that (MF) includes
$\sup_{1\leqslant k\leqslant N}\sum_{1\leqslant i\leqslant N}\lVert
D(D_{k}g^{i})\rVert_{\infty}^{2}\leqslant L_{g}^{2}$. ∎

Exploiting the estimates deduced so far, Bernstein’s method yields the first
estimate on the skew second-order derivatives. It provides a control when
letting one derivative be “in the direction” of the player to which the value
function corresponds while the other derivative is skew and then summing the
squares of the second-order derivatives over the players. In this sense, as
the sum touches all value functions, we will refer to it as a _transversal_
estimate.

######  Proposition 4.7 (Transversal estimate on skew second-order
derivatives).

Let $\tau\in[0,T)$. Suppose that $\mathcal{D}[u]\geqslant-MQ$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$. Then

$\sup_{1\leqslant i\leqslant
N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant\frac{C_{3}}{N}\quad\forall\,t\in[\tau,T].$

where the constant $C_{3}$ depends only on $\sigma$, $T$, $M$, $L_{g}$ and
$L_{f}$.

###### Proof.

Let $w^{i}\vcentcolon=\frac{1}{2}\big{(}\sum_{1\leqslant\ell\leqslant
N,\,\ell\neq
i}|D^{2}_{i\ell}u^{\ell}|^{2}+\frac{1}{N}\,|D^{2}_{ii}u^{i}|^{2}\big{)}$.
Direct computations exploiting (1.1) show that

$\begin{split}&-\partial_{t}w^{i}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}w^{i})+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot D_{j}w^{i}\\\
&+\beta\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant N\\\ \ell\neq
i\end{subarray}}\bigg{|}\sum_{1\leqslant j\leqslant
N}D^{3}_{ij\ell}u^{\ell}\bigg{|}^{2}+\frac{\beta}{N}\bigg{|}\sum_{1\leqslant
j\leqslant
N}D^{3}_{iij}u^{i}\bigg{|}^{2}+\sigma\\!\sum_{\begin{subarray}{c}1\leqslant
j,\ell\leqslant N\\\ \ell\neq
i\end{subarray}}\big{|}D^{3}_{ij\ell}u^{\ell}\big{|}^{2}+\frac{\sigma}{N}\sum_{1\leqslant
j\leqslant N}\big{|}D^{3}_{iij}u^{i}\big{|}^{2}\\\
&+\sum_{\begin{subarray}{c}1\leqslant j,\ell\leqslant N\\\ \ell\neq
i\end{subarray}}D^{2}_{j\ell}u^{\ell}D^{2}_{ij}u^{j}D^{2}_{i\ell}u^{\ell}+\sum_{\begin{subarray}{c}1\leqslant
j,\ell\leqslant N\\\ j\neq\ell\neq
i\end{subarray}}D^{2}_{ij}u^{\ell}D^{2}_{j\ell}u^{j}D^{2}_{i\ell}u^{\ell}+\sum_{\begin{subarray}{c}1\leqslant
j,\ell\leqslant N\\\ j\neq\ell\neq
i\end{subarray}}D_{j}u^{\ell}D^{3}_{ij\ell}u^{j}D^{2}_{i\ell}u^{\ell}\\\
&+\frac{1}{N}\sum_{1\leqslant j\leqslant
N}D^{2}_{ij}u^{i}D^{2}_{ij}u^{j}D^{2}_{ii}u^{i}+\frac{1}{N}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D^{2}_{ij}u^{i}D^{2}_{ij}u^{j}D^{2}_{ii}u^{i}+\frac{1}{N}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{i}D^{3}_{iij}u^{j}D^{2}_{ii}u^{i}\\\ &\quad\
=\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant N\\\ \ell\neq
i\end{subarray}}D^{2}_{i\ell}f^{\ell}D^{2}_{i\ell}u^{\ell}+\frac{1}{N}\,D^{2}_{ii}f^{i}D^{2}_{ii}u^{i},\end{split}$

where according to the fact that the we are considering the Frobenius inner
product on ${\mathbb{R}}^{d}$, all products between tensors are to be
understood in the sense of tensor contraction over the coordinates of repeated
derivative indices; that is, for example,

$\displaystyle
D^{2}_{j\ell}u^{\ell}D^{2}_{ij}u^{j}D^{2}_{i\ell}u^{\ell}=\sum_{1\leqslant
k_{1},k_{2},k_{3}\leqslant d}D^{2}_{x^{jk_{2}}x^{\ell
k_{3}}}u^{\ell}D^{2}_{x^{ik_{1}}x^{jk_{2}}}u^{j}D^{2}_{x^{ik_{1}}x^{\ell
k_{3}}}u^{\ell}=\operatorname{tr}(D^{2}_{\ell
j}u^{\ell}D^{2}_{ji}u^{j}D^{2}_{i\ell}u^{\ell}),$ $\displaystyle
D_{j}u^{\ell}D^{3}_{ij\ell}u^{j}D^{2}_{i\ell}u^{\ell}=\sum_{1\leqslant
k_{1},k_{2},k_{3}\leqslant
d}D_{x^{jk_{2}}}u^{\ell}D^{3}_{x^{ik_{1}}x^{jk_{2}}x^{\ell
k_{3}}}u^{j}D^{2}_{x^{ik_{1}}x^{\ell k_{3}}}u^{\ell},$
$\displaystyle\bigg{|}\sum_{1\leqslant j\leqslant
N}D^{3}_{iij}u^{i}\bigg{|}^{2}=\sum_{1\leqslant k_{1},k_{2},k_{3}\leqslant
d}\bigg{|}\sum_{1\leqslant j\leqslant
N}D^{3}_{x^{ik_{1}}x^{ik_{2}}x^{jk_{3}}}u^{i}\bigg{|}^{2}.$

and analogously for all other products. We now estimate the sums above: since
$\mathcal{D}[u]\geqslant-MQ$ is equivalent to
$(D^{2}_{j\ell}h^{\ell})_{1\leqslant j,\ell\leqslant
N}\geqslant-M\boldsymbol{\mathsf{I}}$,

$\begin{split}\sum_{\begin{subarray}{c}1\leqslant j,\ell\leqslant N\\\
\ell\neq
i\end{subarray}}\\!\\!D^{2}_{j\ell}u^{\ell}D^{2}_{ij}u^{j}D^{2}_{i\ell}u^{\ell}&=\sum_{\begin{subarray}{c}1\leqslant
j,\ell\leqslant N\\\ j,\ell\neq
i\end{subarray}}\\!\\!D^{2}_{j\ell}u^{\ell}D^{2}_{ij}u^{j}D^{2}_{i\ell}u^{\ell}+\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant
N\\\ \ell\neq i\end{subarray}}\\!D^{2}_{ii}u^{i}|D^{2}_{i\ell}u^{\ell}|^{2}\\\
&\geqslant-2M\\!\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant N\\\ \ell\neq
i\end{subarray}}|D^{2}_{i\ell}u^{\ell}|^{2}\geqslant-4Mw^{i};\end{split}$

by the Cauchy–Schwarz inequality and Proposition 4.6,

$\sum_{\begin{subarray}{c}1\leqslant j,\ell\leqslant N\\\ j\neq\ell\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{\ell}D^{2}_{j\ell}u^{j}D^{2}_{i\ell}u^{\ell}\big{|}\leqslant\sup_{1\leqslant
i\leqslant N}\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant N\\\ \ell\neq
i\end{subarray}}|D^{2}_{i\ell}u^{\ell}|^{2}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant
j,\ell\leqslant N\\\ j\neq\ell\neq
i\end{subarray}}|D^{2}_{ij}u^{\ell}|^{2}\bigg{)}^{\frac{1}{2}}\leqslant
C_{2}^{\frac{1}{2}}\sup_{1\leqslant i\leqslant N}w^{i};$

by the Cauchy–Schwarz inequality, Proposition 4.5 and Young’s inequality,

$\begin{split}\sum_{\begin{subarray}{c}1\leqslant j,\ell\leqslant N\\\
j\neq\ell\neq
i\end{subarray}}\big{|}D_{j}u^{\ell}D^{3}_{ij\ell}u^{j}D^{2}_{i\ell}u^{\ell}\big{|}&\leqslant\bigg{(}\sup_{1\leqslant\ell\leqslant
N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\
j\neq\ell\end{subarray}}|D_{j}u^{\ell}|^{2}\bigg{)}^{\frac{1}{2}}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant
j,\ell\leqslant N\\\ j\neq\ell\neq
i\end{subarray}}|D^{3}_{ij\ell}u^{j}|^{2}\bigg{)}^{\frac{1}{2}}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant
N\\\ \ell\neq
i\end{subarray}}|D^{2}_{i\ell}u^{\ell}|^{2}\bigg{)}^{\frac{1}{2}}\\\
&\leqslant\frac{c_{1}}{\sqrt{N}}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant
j,\ell\leqslant N\\\ \ell\neq
i\end{subarray}}|D^{3}_{ij\ell}u^{\ell}|^{2}+\sum_{1\leqslant j\leqslant
N}|D^{3}_{iij}u^{i}|^{2}\bigg{)}^{\frac{1}{2}}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant
N\\\ \ell\neq
i\end{subarray}}|D^{2}_{i\ell}u^{\ell}|^{2}\bigg{)}^{\frac{1}{2}}\\\
&\leqslant\frac{\sigma}{2N}\sum_{\begin{subarray}{c}1\leqslant j,\ell\leqslant
N\\\ \ell\neq
i\end{subarray}}|D^{3}_{ij\ell}u^{\ell}|^{2}+\frac{\sigma}{2N}\sum_{1\leqslant
j\leqslant
N}|D^{3}_{iij}u^{i}|^{2}+\frac{c_{1}^{2}}{\sigma}\,w^{i};\end{split}$

by the Cauchy–Schwarz inequality, Young’s inequality and Proposition 4.6,

$\begin{split}\sum_{1\leqslant j\leqslant
N}\big{|}D^{2}_{ij}u^{i}D^{2}_{ij}u^{j}D^{2}_{ii}u^{i}\big{|}&\leqslant|D^{2}_{ii}u^{i}|\bigg{(}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}|D^{2}_{ij}u^{i}|^{2}\bigg{)}^{\frac{1}{2}}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}|D^{2}_{ij}u^{j}|^{2}\bigg{)}^{\frac{1}{2}}+|D^{2}_{ii}u^{i}|^{3}\\\
&\leqslant 2C_{2}^{\frac{3}{2}}+C_{2}^{\frac{1}{2}}w^{i};\end{split}$

by the Cauchy–Schwarz inequality, Young’s inequality and Proposition 4.5,

$\begin{split}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D_{j}u^{i}D^{3}_{iij}u^{j}D^{2}_{ii}u^{i}\big{|}&\leqslant|D^{2}_{ii}u^{i}|\bigg{(}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}|D_{j}u^{i}|^{2}\bigg{)}^{\frac{1}{2}}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}|D^{3}_{iij}u^{j}|^{2}\bigg{)}^{\frac{1}{2}}\\\
&\leqslant\frac{c_{1}^{2}}{\sigma}\,w^{i}+\frac{\sigma}{2}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}|D^{3}_{iij}u^{j}|^{2};\end{split}$

by Young’s inequality,

$\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant N\\\ \ell\neq
i\end{subarray}}D^{2}_{i\ell}f^{\ell}D^{2}_{i\ell}u^{\ell}+\frac{1}{N}\,D^{2}_{ii}f^{i}D^{2}_{ii}u^{i}\leqslant\frac{1}{2}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant
N\\\ \ell\neq
i\end{subarray}}|D^{2}_{i\ell}f^{\ell}|^{2}+\frac{1}{N}\,|D^{2}_{ii}f^{i}|^{2}\bigg{)}+w^{i}.$

Therefore, we get

$\begin{multlined}-\partial_{t}w^{i}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}w^{i})+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot D_{j}w^{i}\\\
\leqslant\Bigl{(}1+4M+\frac{2c_{1}^{2}}{\sigma}+C_{2}^{\frac{1}{2}}\Bigr{)}w^{i}+C_{2}^{\frac{1}{2}}\sup_{1\leqslant\ell\leqslant
N}w^{\ell}+\frac{2C_{2}^{\frac{3}{2}}}{N}+\frac{1}{2}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant
N\\\ \ell\neq
i\end{subarray}}|D^{2}_{i\ell}f^{\ell}|^{2}+\frac{1}{N}\,|D^{2}_{ii}f^{i}|^{2}\bigg{)}.\end{multlined}-\partial_{t}w^{i}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}w^{i})+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot D_{j}w^{i}\\\
\leqslant\Bigl{(}1+4M+\frac{2c_{1}^{2}}{\sigma}+C_{2}^{\frac{1}{2}}\Bigr{)}w^{i}+C_{2}^{\frac{1}{2}}\sup_{1\leqslant\ell\leqslant
N}w^{\ell}+\frac{2C_{2}^{\frac{3}{2}}}{N}+\frac{1}{2}\bigg{(}\sum_{\begin{subarray}{c}1\leqslant\ell\leqslant
N\\\ \ell\neq
i\end{subarray}}|D^{2}_{i\ell}f^{\ell}|^{2}+\frac{1}{N}\,|D^{2}_{ii}f^{i}|^{2}\bigg{)}.$

By (MF), $\sum_{1\leqslant\ell\leqslant N,\,\ell\neq
i}|D^{2}_{i\ell}f^{\ell}|^{2}+\frac{1}{N}|D^{2}_{ii}f^{i}|^{2}\leqslant{2L_{f}^{2}}/{N}$,
thus

$-\partial_{t}w^{i}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}w^{i})+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot D_{j}w^{i}\leqslant
C\Big{(}\sup_{1\leqslant\ell\leqslant N}w^{\ell}+\frac{1}{N}\Big{)},$

where the constant $C$ depends only on $\sigma$, $T$, $M$, $L_{g}$ and
$L_{f}$. By the maximum principle we get

$\lVert
w^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}\leqslant\lVert
w^{i}(T,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}+(T-t)C\Big{(}\sup_{1\leqslant\ell\leqslant
N}\lVert w^{\ell}\rVert_{\infty;[T-s,T]}+\frac{1}{N}\Big{)}.$

for any $s\in(0,T-\tau]$ and $t\in[T-s,T]$; here we used the notation
$\lVert\,\cdot\,\rVert_{\infty;I}$ for the standard norm of
$L^{\infty}(I\times({\mathbb{R}}^{d})^{N})$. For $s\leqslant(2C)^{-1}$ we
obtain

(4.17) $\sup_{1\leqslant i\leqslant N}\lVert
w^{i}\rVert_{\infty;[T-s,T]}\leqslant 2\sup_{1\leqslant i\leqslant N}\lVert
w^{i}(T,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}+\frac{1}{N};$

letting $s=(2C)^{-1}\wedge(T-\tau)$ and iterating estimate (4.17) on the
intervals $[T-\ell s,T-(\ell-1)s]$ for all positive integers
$\ell\leqslant\ell^{*}\vcentcolon=\lfloor(T-\tau)/s\rfloor$ and then on
$[\tau,T-\ell^{*}s]$, we get by induction

$\sup_{1\leqslant i\leqslant N}\lVert w^{i}\rVert_{\infty;[\tau,T]}\leqslant
2(2^{\ell^{*}+1}-1)\Big{(}2\sup_{1\leqslant i\leqslant N}\lVert
w^{i}(T,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}+\frac{1}{N}\Big{)}-\frac{\ell^{*}+1}{N}.$

The conclusion follows recalling that
$u^{i}(T,\mspace{1.8mu}\cdot\mspace{1.8mu})=g^{i}$, thus by (MF) we have
$\sup_{1\leqslant i\leqslant N}\lVert
w^{i}(T,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert\leqslant 2L_{g}^{2}/N$. ∎

In order to proceed, we need further control, for fixed $i$, on second-order
derivatives $D_{jk}u^{i}$ of the value function as $j,k$ varies. We will refer
to it as a _horizontal_ estimate. Differently from the one obtained in
Proposition 4.6, we sum over $k\neq i$, thus expecting a decay of order $1/N$.
To achieve this, we need more precise control on first-order derivatives: in
Proposition 4.5, a cumulative information on $Du^{i}$ was stated; here below,
we show _pointwise_ bounds, in the sense that no sum over the direction
$x^{j}$ is involved.

######  Proposition 4.8 (Pointwise estimate on skew first-order derivatives).

Let $\tau\in[0,T)$. Suppose that $\mathcal{D}[u]\geqslant-MQ$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$. Then

$\sup_{\begin{subarray}{c}1\leqslant i,k\leqslant N\\\ k\neq
i\end{subarray}}\|D_{k}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\|_{\infty}\leqslant\frac{C_{1}}{N}\quad\forall\,t\in[\tau,T],$

where the constant $C_{1}$ depends only on $\sigma$, $T$, $M$, $L_{g}$ and
$L_{f}$.

###### Proof.

Let $i,k\in\\{1,\ldots,N\\}$, $i\neq k$, and $\ell\in\\{1,\ldots,d\\}$. Note
that $v=v^{k\ell}=D_{x^{k\ell}}u^{i}$ solves the linear problem

$\begin{cases}-\partial_{t}v-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}v)+\displaystyle\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot
D_{j}v=D_{x^{k\ell}}f^{i}-\displaystyle\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})\cdot
D_{j}u^{i}\vspace{-10pt}\\\ v|_{t=T}=D_{x^{k\ell}}g^{i}.\end{cases}$

Recall that by (MF), $\|D_{x^{k\ell}}f^{i}\|_{\infty}\leqslant L_{f}/N$. For
any $s\in(0,T-\tau]$ and $t\in[T-s,T]$, we control the second term of the
right-hand side of the previous equation using Proposition 4.5, Proposition
4.7 and Proposition 4.6 as follows

$\begin{multlined}\Biggr{|}\displaystyle\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ k\neq j\neq i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})\cdot
D_{j}u^{i}+D_{k}(D_{x^{k\ell}}u^{k})\cdot D_{k}u^{i}\Biggl{|}\\\\[-7.0pt]
\begin{split}&\leqslant\biggl{(}\,\sum_{j\neq
k}|D^{2}_{jk}u^{j}|^{2}\biggr{)}^{\frac{1}{2}}\biggl{(}\,\sum_{j\neq
i}|D_{j}u^{i}|^{2}\biggr{)}^{\frac{1}{2}}+|D^{2}_{kk}u^{k}|\,|D_{k}u^{i}|\\\
&\leqslant\frac{c_{1}C_{3}}{N}+C_{2}\|D_{k}u^{i}\|_{\infty;[T-s,T]}.\end{split}\end{multlined}\Biggr{|}\displaystyle\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ k\neq j\neq i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})\cdot
D_{j}u^{i}+D_{k}(D_{x^{k\ell}}u^{k})\cdot D_{k}u^{i}\Biggl{|}\\\\[-7.0pt]
\begin{split}&\leqslant\biggl{(}\,\sum_{j\neq
k}|D^{2}_{jk}u^{j}|^{2}\biggr{)}^{\frac{1}{2}}\biggl{(}\,\sum_{j\neq
i}|D_{j}u^{i}|^{2}\biggr{)}^{\frac{1}{2}}+|D^{2}_{kk}u^{k}|\,|D_{k}u^{i}|\\\
&\leqslant\frac{c_{1}C_{3}}{N}+C_{2}\|D_{k}u^{i}\|_{\infty;[T-s,T]}.\end{split}$

Therefore, by the maximum principle we have

$\|v\|_{\infty;[T-s,T]}\leqslant
s\,\frac{c_{1}C_{3}+L_{f}}{N}+sC_{2}\|D_{k}u^{i}\|_{\infty;[T-s,T]}+\|D_{k}g^{i}\|_{\infty}.$

Since the previous estimate holds for all $\ell\in\\{1,\ldots,d\\}$, for
$s\leqslant(2\sqrt{d}\,C_{2})^{-1}$ we conclude that

$\|D_{k}u^{i}\|_{\infty;[T-s,T]}\leqslant\frac{c_{1}C_{3}+L_{f}}{C_{2}N}+2\sqrt{d}\,\|D_{k}g^{i}\|_{\infty}.$

By iterating the previous inequality as in the previous proof, and using (MF)
again to control $\|D_{x^{k}}g^{i}\|_{\infty}\leqslant L_{g}/N$ we obtain the
assertion on the time interval $[T-\tau,T]$. ∎

We can now get the horizontal bounds on skew second-order derivatives.

######  Proposition 4.9 (Horizontal estimate on skew second-order
derivatives).

Let $\tau\in[0,T)$. Suppose that $\mathcal{D}[u]\geqslant-MQ$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$. Then

$\sup_{1\leqslant i\leqslant N}\sum_{\begin{subarray}{c}1\leqslant k\leqslant
N\\\ k\neq i\end{subarray}}\lVert
D(D_{k}u^{i})(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}^{2}=\sup_{1\leqslant
i\leqslant N}\sum_{\begin{subarray}{c}1\leqslant k\leqslant N\\\ k\neq
i\end{subarray}}\bigg{\lVert}\sum_{1\leqslant j\leqslant
N}\big{|}D^{2}_{jk}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant\frac{C_{4}}{N}\quad\forall\,t\in[\tau,T],$

where the constant $C_{4}$ depends only on $\sigma$, $T$, $M$, $L_{g}$ and
$L_{f}$.

###### Proof.

From inequality (4.14) we have

(4.18) $\begin{multlined}\lVert D_{x^{k\ell}}Du^{i}\rVert_{\infty}\leqslant
C_{\sigma}\lVert
D_{x^{k\ell}}f^{i}\rVert_{\infty}s^{\frac{1}{2}}e^{2Ms}+\lVert
D(D_{x^{k\ell}}g^{i})\rVert_{\infty}e^{Ms}\\\
+C_{\sigma}s^{\frac{1}{2}}e^{2Ms}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ k\neq j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}+C_{\sigma}s^{\frac{1}{2}}e^{2Ms}\lVert
D_{k}(D_{x^{k\ell}}u^{k})D_{k}u^{i}\rVert_{\infty}.\end{multlined}\lVert
D_{x^{k\ell}}Du^{i}\rVert_{\infty}\leqslant C_{\sigma}\lVert
D_{x^{k\ell}}f^{i}\rVert_{\infty}s^{\frac{1}{2}}e^{2Ms}+\lVert
D(D_{x^{k\ell}}g^{i})\rVert_{\infty}e^{Ms}\\\
+C_{\sigma}s^{\frac{1}{2}}e^{2Ms}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ k\neq j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}+C_{\sigma}s^{\frac{1}{2}}e^{2Ms}\lVert
D_{k}(D_{x^{k\ell}}u^{k})D_{k}u^{i}\rVert_{\infty}.$

By the Cauchy–Schwarz inequality, Proposition 4.5 and Proposition 4.7,

$\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ k\neq j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}^{2}\leqslant\frac{c_{1}^{2}}{N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
k\end{subarray}}\big{|}D^{2}_{jk}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant\frac{c_{1}^{2}C_{3}}{N^{2}};$

therefore from (4.18) and Young’s inequality we get

(4.19) $\begin{multlined}\lVert D(D_{k}u^{i})\rVert_{\infty}^{2}\\\ \lesssim
C_{\sigma}^{2}\lVert D_{k}f^{i}\rVert_{\infty}^{2}e^{4Ms}s+\lVert
D(D_{k}g^{i})\rVert_{\infty}^{2}e^{2Ms}+\frac{C_{\sigma}^{2}c^{2}_{1}C_{3}e^{4Ms}s}{N^{2}}+C_{\sigma}^{2}e^{4Ms}s\lVert
D^{2}_{kk}u^{k}\rVert_{\infty}^{2}\lVert
D_{k}u^{i}\rVert_{\infty}^{2},\end{multlined}\lVert
D(D_{k}u^{i})\rVert_{\infty}^{2}\\\ \lesssim C_{\sigma}^{2}\lVert
D_{k}f^{i}\rVert_{\infty}^{2}e^{4Ms}s+\lVert
D(D_{k}g^{i})\rVert_{\infty}^{2}e^{2Ms}+\frac{C_{\sigma}^{2}c^{2}_{1}C_{3}e^{4Ms}s}{N^{2}}+C_{\sigma}^{2}e^{4Ms}s\lVert
D^{2}_{kk}u^{k}\rVert_{\infty}^{2}\lVert D_{k}u^{i}\rVert_{\infty}^{2},$

where the implied constant depends only on $d$. By (MF), $\sum_{1\leqslant
k\leqslant N,\,k\neq i}\lVert
D_{k}f^{i}\rVert_{\infty}^{2}\leqslant{L_{f}^{2}}/{N}$ and $\sum_{1\leqslant
k\leqslant N,\,k\neq i}\lVert
D(D_{k}g^{i})\rVert_{\infty}^{2}\leqslant{L_{g}^{2}}/{N}$; by Proposition 4.6
and Proposition 4.8

(4.20) $\sum_{\begin{subarray}{c}1\leqslant k\leqslant N\\\ k\neq
i\end{subarray}}\lVert D^{2}_{kk}u^{k}\rVert_{\infty}^{2}\lVert
D_{k}u^{i}\rVert_{\infty}^{2}\leqslant\frac{C_{1}C_{2}}{N}.$

The desired conclusion now follows, summing (4.19) over $k\neq i$. ∎

######  Remark 4.10.

Even if Proposition 4.8 is clearly stronger than Proposition 4.5, we cited
them both in the previous proof in order to highlight when the weaker
Proposition 4.5 was sufficient and, consequently, that the stronger
Proposition 4.8 is only needed to get estimate (4.20).

###  4.4. Third-order derivatives of the value functions

Finally, we prove the most basic third-order version of the transversal
estimate on skew-second order derivatives.

######  Proposition 4.11 (Transversal estimate on third-order derivatives).

Let $\tau\in[0,T)$. Suppose that $\mathcal{D}[u]\geqslant-MQ$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$. Then

$\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\ j\neq
i\end{subarray}}\lVert
D(D_{ij}u^{j})(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}^{2}=\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq i\end{subarray}}\bigg{\lVert}\sum_{1\leqslant
k\leqslant
N}\big{|}D^{3}_{ijk}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant
C_{5}\quad\forall\,t\in[\tau,T],$

where the constant $C_{5}$ depends only on $\sigma$, $T$, $M$, $L_{g}$ and
$L_{f}$.

###### Proof.

Proceed as in the proof of Proposition 4.6. Let $h,k\in\\{1,\dots,N\\}$,
$\ell,m\in\\{1,\dots,d\\}$ and $s\in[0,T-\tau]$. Note that
$v=D_{x^{k\ell}x^{hm}}u^{i}$ solves problem (4.1) on
$[T-s,T]\times({\mathbb{R}}^{d})^{N}$ with $b^{j}=D_{j}u^{j}$,

$\begin{multlined}F=D^{2}_{x^{k\ell}x^{hm}}f^{i}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D^{2}_{x^{k\ell}x^{hm}}u^{j})D_{j}u^{i}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}(D_{x^{hm}}u^{i})\\\
-\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{hm}}u^{j})D_{j}(D_{x^{k\ell}}u^{i})-D_{i}(D_{x^{hm}}u^{i})D_{i}(D_{x^{k\ell}}u^{i})\end{multlined}F=D^{2}_{x^{k\ell}x^{hm}}f^{i}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D^{2}_{x^{k\ell}x^{hm}}u^{j})D_{j}u^{i}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}(D_{x^{hm}}u^{i})\\\
-\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{hm}}u^{j})D_{j}(D_{x^{k\ell}}u^{i})-D_{i}(D_{x^{hm}}u^{i})D_{i}(D_{x^{k\ell}}u^{i})$

and $G=D^{2}_{x^{k\ell}x^{hm}}g^{i}$. By Lemma 4.1,

(4.21) $\begin{multlined}\lVert
D(D^{2}_{x^{k\ell}x^{hm}}u^{i})\rVert_{\infty}\\\
\begin{multlined}\lesssim\,C_{\sigma}\biggl{(}\,\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D^{2}_{x^{k\ell}x^{hm}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}(D_{x^{hm}}u^{i})\bigg{\rVert}_{\infty}\\\
+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{hm}}u^{j})D_{j}(D_{x^{k\ell}}u^{i})\bigg{\rVert}_{\infty}+\lVert
D_{i}(D_{x^{hm}}u^{i})D_{i}(D_{x^{k\ell}}u^{i})\rVert_{\infty}s^{\frac{1}{2}}e^{2Ms}\\\\[-5.0pt]
+\lVert
D^{2}_{x^{k\ell}x^{hm}}f^{i}\rVert_{\infty}\biggr{)}s^{\frac{1}{2}}e^{2Ms}+\lVert
D^{2}_{x^{k\ell}x^{hm}}Dg^{i}\rVert_{\infty}e^{Ms},\end{multlined}\lesssim\,C_{\sigma}\biggl{(}\,\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D^{2}_{x^{k\ell}x^{hm}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}(D_{x^{hm}}u^{i})\bigg{\rVert}_{\infty}\\\
+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{hm}}u^{j})D_{j}(D_{x^{k\ell}}u^{i})\bigg{\rVert}_{\infty}+\lVert
D_{i}(D_{x^{hm}}u^{i})D_{i}(D_{x^{k\ell}}u^{i})\rVert_{\infty}s^{\frac{1}{2}}e^{2Ms}\\\\[-5.0pt]
+\lVert
D^{2}_{x^{k\ell}x^{hm}}f^{i}\rVert_{\infty}\biggr{)}s^{\frac{1}{2}}e^{2Ms}+\lVert
D^{2}_{x^{k\ell}x^{hm}}Dg^{i}\rVert_{\infty}e^{Ms},\end{multlined}\lVert
D(D^{2}_{x^{k\ell}x^{hm}}u^{i})\rVert_{\infty}\\\
\begin{multlined}\lesssim\,C_{\sigma}\biggl{(}\,\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D^{2}_{x^{k\ell}x^{hm}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}(D_{x^{hm}}u^{i})\bigg{\rVert}_{\infty}\\\
+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{hm}}u^{j})D_{j}(D_{x^{k\ell}}u^{i})\bigg{\rVert}_{\infty}+\lVert
D_{i}(D_{x^{hm}}u^{i})D_{i}(D_{x^{k\ell}}u^{i})\rVert_{\infty}s^{\frac{1}{2}}e^{2Ms}\\\\[-5.0pt]
+\lVert
D^{2}_{x^{k\ell}x^{hm}}f^{i}\rVert_{\infty}\biggr{)}s^{\frac{1}{2}}e^{2Ms}+\lVert
D^{2}_{x^{k\ell}x^{hm}}Dg^{i}\rVert_{\infty}e^{Ms},\end{multlined}\lesssim\,C_{\sigma}\biggl{(}\,\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D^{2}_{x^{k\ell}x^{hm}}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}}u^{j})D_{j}(D_{x^{hm}}u^{i})\bigg{\rVert}_{\infty}\\\
+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{hm}}u^{j})D_{j}(D_{x^{k\ell}}u^{i})\bigg{\rVert}_{\infty}+\lVert
D_{i}(D_{x^{hm}}u^{i})D_{i}(D_{x^{k\ell}}u^{i})\rVert_{\infty}s^{\frac{1}{2}}e^{2Ms}\\\\[-5.0pt]
+\lVert
D^{2}_{x^{k\ell}x^{hm}}f^{i}\rVert_{\infty}\biggr{)}s^{\frac{1}{2}}e^{2Ms}+\lVert
D^{2}_{x^{k\ell}x^{hm}}Dg^{i}\rVert_{\infty}e^{Ms},$

where the implied constant is a number and the $L^{\infty}$-norm are
understood to be computed on $[T-s,T]\times({\mathbb{R}}^{d})^{N}$ or
$({\mathbb{R}}^{d})^{N}$. By Proposition 4.5 and the Cauchy–Schwarz inequality
we have

$\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}(D_{x^{k\ell}x^{hm}}^{2}u^{j})D_{j}u^{i}\bigg{\rVert}_{\infty}^{2}\leqslant\frac{c_{1}^{2}}{N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{3}_{hjk}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty};$

therefore, choosing $k=i$ in (4.21), and applying Young’s inequality we get

(4.22) $\begin{multlined}\lVert D(D_{hi}^{2}u^{i})\rVert_{\infty}^{2}\\\
\lesssim\,\lVert D^{2}_{hi}f^{i}\rVert_{\infty}^{2}se^{4Ms}+\lVert
D^{2}_{hi}Dg^{i}\rVert_{\infty}^{2}e^{2Ms}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ji}u^{j}\big{|}^{2}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{hj}u^{i}\big{|}^{2}\bigg{\rVert}_{\infty}se^{4Ms}\\\
+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{hj}u^{j}\big{|}^{2}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ji}u^{i}\big{|}^{2}\bigg{\rVert}_{\infty}se^{4Ms}+\lVert
D_{hi}u^{i}\rVert_{\infty}^{2}\lVert
D_{ii}u^{i}\rVert_{\infty}^{2}se^{4Ms}\vspace{-5pt}\\\
+\frac{c^{2}_{1}se^{4Ms}}{N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant
N\\\ j\neq i\end{subarray}}\lVert
D^{3}_{hij}u^{j}\rVert_{\infty}^{2},\end{multlined}\lVert
D(D_{hi}^{2}u^{i})\rVert_{\infty}^{2}\\\ \lesssim\,\lVert
D^{2}_{hi}f^{i}\rVert_{\infty}^{2}se^{4Ms}+\lVert
D^{2}_{hi}Dg^{i}\rVert_{\infty}^{2}e^{2Ms}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ji}u^{j}\big{|}^{2}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{hj}u^{i}\big{|}^{2}\bigg{\rVert}_{\infty}se^{4Ms}\\\
+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{hj}u^{j}\big{|}^{2}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ji}u^{i}\big{|}^{2}\bigg{\rVert}_{\infty}se^{4Ms}+\lVert
D_{hi}u^{i}\rVert_{\infty}^{2}\lVert
D_{ii}u^{i}\rVert_{\infty}^{2}se^{4Ms}\vspace{-5pt}\\\
+\frac{c^{2}_{1}se^{4Ms}}{N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant