# Rates of Convergence in Periodic Homogenization of Nonlocal Hamilton-Jacobi-
Bellman Equations

Andrei Rodríguez-Paredes  Andrei Rodríguez-Paredes: Departamento de Matemática
y C.C., Universidad de Santiago de Chile, Casilla 307, Santiago, CHILE.
<EMAIL_ADDRESS>and  Erwin Topp  Erwin Topp: Departamento de
Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Santiago,
CHILE.
<EMAIL_ADDRESS>
###### Abstract.

In this paper we provide a rate of convergence for periodic homogenization of
Hamilton-Jacobi-Bellman equations with nonlocal diffusion. The result is based
on the regularity of the associated effective problem, where the convexity
plays a crucial role. Such regularity estimates are possible from the
available literature once we provide a representation formula for the
effective Hamiltonian, a result that has an independent interest.

Keywords: Rates of convergence, homogenization, nonlocal elliptic equations,
nonlinear equations, Hamilton-Jacobi equations, representation formulas.

AMS classification (2020): 35D40, 35J60, 35R09, 35B27.

##  1\. Introduction

In this paper we are interested in the periodic homogenization of stationary
Hamilton-Jacobi-Bellman equations of the form

$\displaystyle
u^{\epsilon}+H(x,\nicefrac{{x}}{{\epsilon}},Du^{\epsilon}(x),u^{\epsilon})=0\quad\mbox{for}\
x\in\mathbb{R}^{N},$ (1)

where $\epsilon\in(0,1)$. The Hamiltonian $H$ in the equation above has a
nonlocal dependence in the last variable, and for this reason we start this
note describing its structure.

Denote by $\mathbb{T}^{N}$ the $N$-dimensional flat torus, and let $\Theta$ be
a compact metric space. We consider continuous and bounded functions

$f:\mathbb{R}^{N}\times\mathbb{T}^{N}\times\Theta\to\mathbb{R}^{N};\quad
l:\mathbb{R}^{N}\times\mathbb{T}^{N}\times\Theta\to\mathbb{R},$

and for $x,p\in\mathbb{R}^{N},y\in\mathbb{T}^{N}$ and $\varphi\in
C^{2}_{b}(\mathbb{R}^{N})$, we write

$H(x,y,p,\varphi)=\sup_{\theta}\\{-L_{y}^{\theta}\varphi(x)-f^{\theta}(x,y)\cdot
p-l^{\theta}(x,y)\\},$ (2)

where we have adopted the notation $f^{\theta}(x,y)=f(x,y,\theta)$, and
similarly for $l$.

We fix $\sigma\in(1,2)$—the “order” of the operator $L$—and consider a
function $A:\mathbb{T}^{N}\times S^{N-1}\times\Theta\to\mathbb{S}^{N}$, where
$S^{N-1}$ is the unit sphere in $\mathbb{R}^{N}$ and $\mathbb{S}^{N}$ is the
set of $N\times N$ symmetric matrices. Thus, we have at disposal a family of
kernels $\\{K^{\theta}\\}_{\theta\in\Theta}$ defined as

$K^{\theta}(y,z)=|z^{t}A^{\theta}(y,\hat{z})z|^{-(N+\sigma)/2},\quad
y\in\mathbb{T}^{N},\ z\in\mathbb{R}^{N}\setminus\\{0\\},$ (3)

where $\hat{z}=\nicefrac{{z}}{{|z|}}$, with
$K^{\theta}(y,z)=K^{\theta}(y,-z)$, and through them, for each
$\theta\in\Theta$, $x,y\in\mathbb{R}^{N}$ and $\varphi\in
C_{b}^{2}(\mathbb{R}^{N})$, we define

$L_{y}^{\theta}\varphi(x)=\mathrm{P.V.}\int_{\mathbb{R}^{N}}[\varphi(x+z)-\varphi(x)]K^{\theta}(y,z)\,dz,$
(4)

where $P.V.$ stands for the Cauchy Principal Value. When
$A^{\theta}(y,\hat{z})=c_{\sigma}I_{N}$ for all $y\in\mathbb{T}^{N}$, $I_{N}$
is the identity matrix, and $c_{\sigma}>0$ is an adequate normalizing
constant, $L^{\theta}$ equals $\Delta^{\sigma/2}$, the nowadays well-known
fractional Laplacian of order $\sigma$. See [23] for a complete review of this
operator. Notice that the kernels $K$ defining $L$ are homogeneous of degree
$N+\sigma$, that is $K^{\theta}(y,\lambda
z)=\lambda^{-(N+\sigma)}K^{\theta}(y,z)$ for all $\theta,y$, $z\neq 0$ and
$\lambda>0$.

Having described the structure of (1), we are in position to reveal our main
concerns: establishing the convergence of the family of (viscosity) solutions
$\\{u^{\epsilon}\\}_{\epsilon}$ of (1) to a function $\bar{u}\in
C(\mathbb{R}^{N})$, and providing a rate for the convergence
$u^{\epsilon}\to\bar{u}$; i.e., an estimate of the form
$\|u^{\epsilon}-\bar{u}\|_{\infty}=\omega(\epsilon)$, where $\omega$ is an
explicit modulus of continuity. In the procedure, we are also interested in
the study of the limit function $\bar{u}$ as the solution of an “average”
fractional PDE, the so-called effective problem.

Fractional PDEs have attracted the attention of the community in the last
decade because of its wide number of applications. For instance, problem (1)
can be regarded as the dual formulation of an optimal control problem of jump
diffusion processes. In the case $A$ is independent of $z$ in (3) and unitary,
denoting $\tilde{A}=\tilde{A}^{\theta}(y)$ such that
$A=\tilde{A}\tilde{A}^{t}$ (the “square root” of $A$), we have the nonlocal
operator $L$ takes the so-called Lévy-Ito form

$L^{\theta}_{y}u(x)=\mathrm{P.V.}\int_{\mathbb{R}^{N}}[u(x+\tilde{A}^{\theta}(y)z)-u(x)]|z|^{-(N+\sigma)}dz.$

Written in this way, $L$ arises in the Ito formula of a general Lévy process
as the jump component of its generator. Now, we consider the controlled SDE

$dX_{t}=f(X_{t},\epsilon^{-1}X_{t},\theta_{t})dt+\int_{\mathbb{R}^{N}}\tilde{A}(\epsilon^{-1}X_{s},\theta_{s})z\tilde{N}(dt,dz),$

where for a given filtered probability space, $\tilde{N}$ is the compensated
Poisson random measure of the isotropic $\sigma$-stable Lévy process (whose
infinitesimal generator is the fractional Laplacian of order $\sigma$), and
where $\theta_{t}$ is an adapted process with values in $\Theta$. Then, for
each $x\in\mathbb{R}^{N}$, the value function

$u^{\epsilon}(x)=\inf_{\theta(\cdot)}\mathbb{E}\int_{0}^{\infty}e^{-s}\ell(X_{s},\epsilon^{-1}X_{s},\theta_{s})ds,$

is a viscosity solution to (1), see [37, 38]. Thus, the homogenization
phenomena observed as $\epsilon\to 0$ in (1) are related to the study of the
average behavior of this highly oscillatory stochastic optimal control
problem. Other applications related to nonlocal problems can be found in the
literature, such as dislocation dynamics [31] and front propagation of
reaction-diffusion equations [44].

Periodic homogenization results for first- and second-order degenerate
elliptic Hamilton-Jacobi equations have a long history and have constituted an
active research field since the appearance of the pioneering work of Lions,
Papanicolaou and Varadhan [34]. A (non-exhaustive) list of notable
contributions to the field developed since then includes [25, 26, 1, 21, 8].
We also highlight recent works on stochastic homogenization (see, e.g., [43,
35, 15, 4]). For an introduction to homogenization results in elliptic PDEs
with variational structure, where a different set of tools is available, we
refer the reader to the books [20, 46].

In the nonlocal setting, periodic homogenization results have been obtained by
working roughly within the framework outlined above, see e.g. [2, 3, 40, 5,
19, 33, 39]. However, to the best of our knowledge, the rate of convergence
for the homogenization of nonlocal equations has not yet been addressed. For
expository reasons, we proceed to explain the main ideas behind this pursuit,
beginning with some well-known facts, and refer the reader to Section 2 for
precise assumptions on the data in (1) and the definition of some auxiliary
concepts.

Previous results for fractional equations show that the study of the cell
problem associated to (1) is possible following the scheme of Lions,
Papanicolaou and Varadhan. This leads to the existence of an effective
Hamiltonian $\bar{H}$, which in this case can be regarded as a function
$\bar{H}:\mathbb{R}^{N}\times\mathbb{R}^{N}\times
C_{b}^{2}(\mathbb{R}^{N})\to\mathbb{R}$. This function is defined through an
eigenvalue problem that we describe next: for each $x,p\in\mathbb{R}^{N}$ and
$\phi\in C_{b}^{2}(\mathbb{R}^{N})$, there exists a unique constant
$c\in\mathbb{R}$ such that

$\sup_{\theta}\\{-L^{\theta}_{y}w(y)-L^{\theta}_{y}\phi(x)-f^{\theta}(x,y)\cdot
p-l^{\theta}(x,y)\\}=c\quad\text{for}\ y\in\mathbb{T}^{N},$ (5)

has a viscosity solution $w\in C(\mathbb{T}^{N})$. This equation is referred
as the cell problem associated to $H$, and $c$ is the ergodic constant, see
for instance [10] for the solvability of such a problem. This allows us to
define $\bar{H}(x,p,\phi):=c$.

In spite of not having closed formulae for $\bar{H}$ in most cases, it is
possible to prove that this function _inherits_ various properties from $H$,
such as continuity and/or convexity. This is obtained following ideas already
in [26], and in the nonlocal setting it is done in e.g. [19]. Another property
satisfied by $\bar{H}$ in the present nonlocal context is the _degenerate
ellipticity_ , intended in the following sense: given $x,p\in\mathbb{R}^{N}$
and $\varphi_{1},\varphi_{2}\in C^{2}_{b}(\mathbb{R}^{N})$ such that
$\varphi_{1}(x)=\varphi_{2}(x)$ and $\varphi_{1}\geq\varphi_{2}$ in
$\mathbb{R}^{N}$, then

$\bar{H}(x,p,\varphi_{1})\leq\bar{H}(x,p,\varphi_{2}).$ (6)

Following the classical perturbed test function method of Evans [25, 26], it
is possible to prove that the sequence of solutions $u^{\epsilon}$ to (1)
converges as $\epsilon\to 0$ (up to subsequence) to the solution of the
effective problem

$\bar{u}+\bar{H}(x,D\bar{u}(x),\bar{u})=0\quad\mbox{for}\ x\in\mathbb{R}^{N}.$
(7)

Thus, it is a well-known fact that uniqueness for the limit problem (7)
implies the full convergence of the sequence $(u^{\epsilon})$, leading to
homogenization.

In general—for instance, when the Hamiltonian is not translation-invariant—the
monotonicity condition (6) is not sufficient to compare discontinuous
viscosity sub- and supersolutions of (7). This impedes the use of the half-
relaxed limits method of Barles and Perthame [7], an efficient tool used in
the first and second-order setting to obtain homogenization. Though partial
results on the comparison principle for non-translation-invariant fractional
equations exist, they are either related to Lévy-Ito type operators, or
require some regularity of the sub- or supersolution being compared; see e.g.
[11, 27, 5, 36]. Nevertheless, condition (6) by itself is sufficient to
compare classical solutions of (7). This motivates the study of
$C^{\sigma+\alpha}$-estimates for the effective equation.

Once $C^{\sigma+\alpha}$-estimates for (7) are at hand, they can be employed
to obtain the result on rate of convergence, following the approach presented
by Camilli and Marchi in [16] for second-order convex, fully nonlinear
equations. The argument relies on the study of the difference
$u^{\epsilon}-\bar{u}$ using the classical doubling variables method in the
viscosity solution’s theory. Since $u^{\epsilon}$ and $\bar{u}$ solve
different problems, the usual penalization procedure requires the
consideration of the corrector function $w$ solving (5) to modulate the
difference among the original problem and the effective one. Actually, we need
to consider the solution $w_{\lambda}$ of the approximating problem (21) below
(the so-called vanishing discount problem), which satisfies $w_{\lambda}\to w$
as $\lambda\to 0$, using the last convergence in an adequate regime depending
on $\epsilon$. This is the content of our main Theorem 2.2 below, from which
homogenization is a byproduct of such a rate of convergence.

The mentioned regularity estimates are made possible in our setting by
representation formulas for the effective Hamiltonian $\bar{H}$ that have an
interest in their own right. These formulas are contained in Theorem 2.1,
which in turn is a direct consequence of Proposition 3.1. The proof of this
last proposition follows the lines of Ishii, Mitake and Tran [32], which
characterizes the ergodic constant associated to second-order Hamilton-Jacobi-
Bellman problems. This allows us to conclude an “average” formula for
$\bar{H}$ in analogy to the characterization of the (multiplicative) principal
eigenvalue in the celebrated paper of Donsker and Varadhan [24] for linear
second-order equations; see also Armstrong [2] for the fully nonlinear second-
order case.

The problem of obtaining such a representation formula for an operator like
$\bar{H}$ dates back to the work of Courrège [22], in which linear operators
satisfying a _global_ comparison principle are characterized of being of Lèvy
type. More recently, this study has been extended to the nonlinear setting in
[29, 30], in even greater generality than is necessary for the problem at hand
(see also [40, 28]). Nevertheless, these formulas are not sufficient to
conclude comparison principles and/or regularity estimates for equation (7),
at least in the most general setting presented in this paper.

We finish mentioning that our representation formula is valid in greater
generality, for instance when the nonlocal operator also depends on the slow
variable $x$. However, the conclusion regarding regularity for the effective
problem and its subsequent application on the rate of convergence in the
homogenization procedure requires further analysis in this case, and we do not
pursue in this direction here. See the discussion about this issue in
Subsection 3.1.

The paper is organized as follows. In Section 2 we provide the assumptions and
present the main results of the paper. In Section 3 we prove the
representation formula for the effective Hamiltonian. In Section 4 we provide
relevant estimates of the discount problem, the approximating equation that
defines the effective Hamiltonian. Finally, in Section 5 we provide the rate
of convergence for our homogenization problem.

##  2\. Assumptions and Main results

Throughout this note, $H$ will have the form (2). We assume $f\in
C(\mathbb{R}^{N}\times\mathbb{T}^{N}\times\Theta;\mathbb{R}^{N})$ and $l\in
C(\mathbb{R}^{N}\times\mathbb{T}^{N}\times\Theta;\mathbb{R})$ and there exists
a constant $C$ such that

$|Df^{\theta}|,|Dl^{\theta}|\ \leq C,\quad\textrm{for all }\theta\in\Theta,$
(8)

for some $C>0$, in the viscosity sense; in other words, both functions are
Lipschitz continuous with respect to both variables, uniformly in $\theta$.

The nonlocal operator $L$ has the form (4), with $A\in C(\mathbb{T}^{N}\times
S^{N-1}\times\Theta;\mathbb{S}^{N})$ satisfying

$\|A^{\theta}\|_{C^{\alpha}(\mathbb{T}^{N}\times S^{N-1})}\leq C_{0}$ (9)

for some $C_{0}>0$, uniformly with respect to $\theta$, and is uniformly
elliptic in the sense that there exists constants $0<\gamma\leq\Gamma$ such
that

$\Gamma^{-2/(N+\sigma)}\leq A^{\theta}\leq\gamma^{-2/(N+\sigma)}$ (10)

in the sense of matrices.

In particular, the family of kernels $\\{K^{\theta}\\}$ satisfies

$\frac{\gamma}{|z|^{N+\sigma}}\leq
K^{\theta}(y,z)\leq\frac{\Gamma}{|z|^{N+\sigma}}\quad\textrm{for all
}y\in\mathbb{T}^{N},z\in\mathbb{R}^{N}\backslash\\{0\\},$ (11)

in addition to being $\mathbb{T}^{N}$-periodic in $y$.

Now we present the first main result of this paper, which is a representation
formula for $\bar{H}$. Given $L$ as above, define the set

$\mathcal{G}_{0}=\\{\phi\in C(\mathbb{T}^{N}\times\Theta)\ :\ \exists\ u\
\mbox{s.t.}\
\sup_{\theta\in\Theta}\\{-L^{\theta}_{y}u(y)-\phi_{\theta}(y)\\}\leq 0\
\mbox{for}\ y\in\mathbb{T}^{N}\\},$ (12)

where the equation satisfied by $u$ is understood in the viscosity sense. As
it can be seen in Lemma 2.8 in [32], $\mathcal{G}_{0}$ is a nonempty convex
cone with vertex at the origin. We also denote $\mathcal{P}$ the space of
probability measures on $\mathbb{T}^{N}\times\Theta$, and consider the polar
cone $\mathcal{G}_{0}^{\prime}$ given by

$\mathcal{G}_{0}^{\prime}:=\left\\{\mu\in\mathcal{P}\ :\ \int\phi d\mu\geq 0\
\mbox{for all}\ \phi\in\mathcal{G}_{0}\right\\}.$ (13)

######  Theorem 2.1.

Assume $\\{K\\}$ has the form (3), and $f,l$ are continuous with respect to
all its variables, and that (8), (9) and (10) hold. Let
$\mathcal{G}_{0}^{\prime}$ be defined in (13). Then, for all
$x,p\in\mathbb{R}^{N}$ and $\phi\in C^{\sigma+\iota}$, the function $\bar{H}$
defined in (5) has the form

$\overline{H}(x,p,\phi)=\sup_{\mu\in\mathcal{P}\cap\mathcal{G}_{0}^{\prime}}\left\\{-\overline{L}^{\mu}\phi(x)-\bar{f}^{\mu}(x)\cdot
p-\bar{l}^{\mu}(x)\right\\},$

where, for each $\mu\in\mathcal{P}$ we have denoted

$\displaystyle\overline{L}^{\mu}\phi(x)$
$\displaystyle:=\int_{\mathbb{R}^{N}}[\phi(x+z)-\phi(x)]\overline{K}^{\mu}(z)\,dz;\quad\bar{K}^{\mu}(z)=\int_{\mathbb{T}^{N}\times\Theta}K^{\theta}(y,z)d\mu,$
$\displaystyle\bar{f}^{\mu}(x)$
$\displaystyle:=\int_{\mathbb{T}^{N}\times\Theta}f(x,y,\theta)d\mu,$
$\displaystyle\bar{l}^{\mu}(x)$
$\displaystyle:=\int_{\mathbb{T}^{N}\times\Theta}l(x,y,\theta)d\mu.$

In spite of $\bar{H}$ not being translation-invariant, the representation
formula obtained in Theorem 2.1 allows us to establish comparison –and thus,
homogenization– through a superposition of available regularity results which
allows us to ensure $C^{\sigma+\alpha}$ estimates for the effective problem,
see Corollary 3.2.

As mentioned earlier, the regularity estimates for the effective problem which
stem from Theorem 2.1 lead to the main result of this paper.

######  Theorem 2.2.

Under the assumptions of Theorem 2.1, there exist constants $C>0$ and
$\bar{\alpha}\in(0,1)$, depending on the data and parameters of equation (1),
but not on $\epsilon$, such that

$|u^{\epsilon}(x)-u(x)|\leq C\epsilon^{\bar{\alpha}}\quad\text{for all
}x\in\mathbb{R}^{N},\epsilon\in(0,1).$

where $u^{\epsilon}$ and $u$ are the solutions of (1) and (7), respectively.

For completeness, we include also a (much easier) result on the rates of
convergence for the homogenization of (1) when dependence on the slow
variable—and possibly the gradient of the solution—is dropped. The
corresponding results in the first- and second-order setting are contained in
[17] and [16], respectively.

######  Theorem 2.3.

Under the assumptions of Theorem 2.2,

  1. (i)

if $f^{\theta}\equiv 0$ and $l^{\theta}$ depends only on $y\in\mathbb{T}^{N}$
(i.e., $l^{\theta}(x,y)=l^{\theta}(x))$ for all $\theta\in\Theta$, then there
exists a constant $C>0$ such that

$|u(x)-u^{\epsilon}(x)|\leq C\epsilon^{\sigma}\quad\textrm{for all
}x\in\mathbb{R}^{N},\ \epsilon\in(0,1).$

  2. (ii)

if $f^{\theta}$ and $l^{\theta}$ depend only on $y\in\mathbb{T}^{N}$ (i.e.,
$f^{\theta}(x,y)=f^{\theta}(x)$, $l^{\theta}(x,y)=l^{\theta}(x))$ for all
$\theta\in\Theta$, then there exists a constant $C>0$ such that

$|u(x)-u^{\epsilon}(x)|\leq C\epsilon^{\sigma-1}\quad\textrm{for all
}x\in\mathbb{R}^{N},\ \epsilon\in(0,1).$

##  3\. Representation formula: Proof of Theorem 2.1

Theorem 2.1 will be obtained as a consequence of Proposition 3.1 below, which
provides two characterizations of the ergodic constant $c$ in (15). We present
the result in a slightly more general setting: we consider a measurable
function $K:\mathbb{T}^{N}\times\mathbb{R}^{N}\times\Theta\to\mathbb{R}$ such
that $K^{\theta}(y,z)=K^{\theta}(y,-z)$ for all
$y\in\mathbb{T}^{N},z\in\mathbb{R}^{N},\theta\in\Theta$, and satisfying the
ellipticity condition (11). We consider

$L_{y}^{\theta}u(y)=\mathrm{P.V.}\int_{\mathbb{R}^{N}}[u(y+z)-u(y)]K^{\theta}(y,z)dz.$
(14)

Let $\ell\in C(\mathbb{T}^{N}\times\Theta)$ and consider the following ergodic
problem: find a pair $(\psi,c)\in C(\mathbb{T}^{N})\times\mathbb{R}$ solving
the equation

$F(\psi,y):=\sup_{\theta}\\{-L^{\theta}_{y}\psi(y)-\ell^{\theta}(y)\\}=c\quad\mbox{in}\
\mathbb{T}^{N},$ (15)

in the viscosity sense. Under mild assumptions on $K$, it is possible to prove
the existence of a unique constant $c\in\mathbb{R}$ for which problem (15) has
a viscosity solution, and this solution is in $C^{\sigma+\alpha}$ for some
$\alpha\in(0,1)$, see [14, 41]. These assumptions are related with the
continuity on the data and ellipticity conditions that ensure comparison
principles and regularity, see for instance [10, 12].

######  Proposition 3.1.

Assume $\ell\in C(\mathbb{T}^{N}\times\Theta)$, $L$ has the form (14) with $K$
symmetric in $z$, satisfying (11) and such that the ergodic problem (15) has a
solution $(\psi,c)$. Let $\mathcal{G}_{0}^{\prime}$ defined as in (12) with
these operators $L$. Then, $c$ can be characterized as follows

$\begin{split}c&=-\inf_{\mu\in\mathcal{P}}\sup_{\psi\in
C^{\sigma+\iota}}\int_{\mathbb{T}^{N}\times\Theta}\\{L^{\theta}_{y}\psi(y)+\ell^{\theta}(y)\\}d\mu\\\
&=-\inf_{\mu\in\mathcal{G}_{0}^{\prime}}\int_{\mathbb{T}^{N}\times\Theta}\ell^{\theta}(y)d\mu.\end{split}$
(16)

###### Proof.

Using the comparison principle for the associated parabolic problem (see
Proposition 3.2 in [45]), which is possible in our case due to the smoothness
of the solution to (15), it is possible to prove that $c$ can be characterized
as

$c=\inf\\{\tilde{c}\in\mathbb{R}:\exists\ \psi\ \mbox{s.t.}\
F(\psi,y)\leq\tilde{c}\ \mbox{in}\ \mathbb{T}^{N}\\},$ (17)

where the inequality inside the $\inf$ is understood in the viscosity sense.
From here, we claim that $c$ meets the value

$\inf_{\psi\in C^{\sigma+\iota}}\sup_{y\in\mathbb{T}^{N}}F(\psi,y).$

Indeed, let us denote the latter value by $\bar{c}$. Since there exists a
smooth solution to (15), we clearly have $\bar{c}\leq c$. On the other hand,
if $\bar{c}<c$, there exists $\epsilon>0$ small and $\psi$ smooth such that

$F(\psi,y)\leq\bar{c}+\epsilon<c\quad\mbox{for all}\ y\in\mathbb{T}^{N},$

which contradicts the characterization (17).

Using this last formula for $c$ and the continuity of the map

$(y,\theta)\mapsto-L^{\theta}_{y}\psi(y)-\ell^{\theta}(y)$

when $\psi$ is smooth, we conclude that

$c=-\sup_{\psi\in
C^{\sigma+\iota}}\inf_{\mu\in\mathcal{P}}\int_{\mathbb{T}^{N}\times\Theta}\\{L^{\theta}_{y}\psi(y)+\ell^{\theta}(y)\\}d\mu.$

Note that for each $\psi$ smooth, the map

$\mu\mapsto\int_{\mathbb{T}^{N}\times\Theta}\\{L^{\theta}_{y}\psi(y)+\ell^{\theta}(y)\\}d\mu$

is a bounded linear map when the $*$-weak topology is considered on
$\mathcal{P}$. On the other hand, given $\mu\in\mathcal{P}$, the map

$\psi\mapsto\int_{\mathbb{T}^{N}\times\Theta}\\{L^{\theta}_{y}\psi(y)+\ell^{\theta}(y)\\}d\mu$

is affine (hence, concave). Then, using Sion’s Theorem [42], we get that $c$
can be written as

$-c=\inf_{\mu\in\mathcal{P}}\sup_{\psi\in
C^{\sigma+\iota}}\int_{\mathbb{T}^{N}\times\Theta}\\{L^{\theta}_{y}\psi(y)+\ell^{\theta}(y)\\}d\mu,$
(18)

which is the first characterization in (16).

For the second, we denote

$\bar{c}=\inf_{\mu\in\mathcal{G}_{0}^{\prime}}\int_{\mathbb{T}^{N}\times\Theta}\ell^{\theta}(y)d\mu.$

Note that if $\psi\in C^{\sigma+\iota}(\mathbb{T}^{N})$, the function
$(y,\theta)\mapsto-L^{\theta}_{y}\psi(y)$ is in $\mathcal{G}_{0}(x)$. Then, by
(18) we immediately see that $-c\leq\bar{c}$.

On the other hand, using that $\mathcal{G}_{0}$ is a cone with vertex at the
origin, for each $\mu\notin\mathcal{G}_{0}^{\prime}$ we have
$\inf_{\phi\in\mathcal{G}_{0}}\int\phi d\mu=-\infty$; whereas, if
$\mu\in\mathcal{G}_{0}^{\prime}$ we have
$\inf_{\phi\in\mathcal{G}_{0}}\int\phi d\mu=0$. Thus, we can write

$\displaystyle\bar{c}={}$
$\displaystyle\inf_{\mu\in\mathcal{P}}\left\\{\int_{\mathbb{T}^{N}\times\Theta}\ell^{\theta}(y)d\mu-\inf_{\phi\in\mathcal{G}_{0}}\int_{\mathbb{T}^{N}\times\Theta}\phi^{\theta}(y)d\mu\right\\}$
$\displaystyle={}$
$\displaystyle\inf_{\mu\in\mathcal{P}}\sup_{\phi\in\mathcal{G}_{0}}\left\\{\int_{\mathbb{T}^{N}\times\Theta}\ell^{\theta}-\phi^{\theta}d\mu\right\\}.$

Invoking again Sion’s Theorem, we conclude that

$\bar{c}=\sup_{\phi\in\mathcal{G}_{0}}\inf_{\mu\in\mathcal{P}}\left\\{\int_{\mathbb{T}^{N}\times\Theta}\ell^{\theta}-\phi^{\theta}d\mu\right\\}.$

Then, if by contradiction we assume that $-c<\bar{c}$, there exists
$\epsilon>0$ small and $\phi\in\mathcal{G}_{0}$ such that

$-c+\epsilon<\int_{\mathbb{T}^{N}\times\Theta}\ell^{\theta}-\phi^{\theta}d\mu\quad\mbox{for
all}\ \mu\in\mathcal{P}.$

Since the _Dirac deltas_ are in $\mathcal{P}$, in particular we have

$-c+\epsilon<\ell^{\theta}(y)-\phi^{\theta}(y)\quad\mbox{for all}\
(y,\theta)\in\mathbb{T}^{N}\times\Theta.$

Thus, by definition of $\mathcal{G}_{0}$, there exists $u$ such that

$-L^{\theta}_{y}u(y)-\ell^{\theta}(y)<c-\epsilon\quad\mbox{for all}\
(y,\theta)\in\mathbb{T}^{N}\times\Theta,$

but this contradicts the characterization of $c$ in (17). This concludes the
second characterization in (16). ∎

###### Proof of Theorem 2.1.

We again assume that $K^{\theta}$ has the form (3)—i.e., neither $K^{\theta}$
nor the sets $\mathcal{G}_{0},\mathcal{G}_{0}^{\prime}$ depend on the the slow
variable $x$.

As described in the introduction, $\bar{H}$ is defined as the ergodic constant
in (5). Therefore, for given $x,p\in\mathbb{R}^{N}$ and $\phi\in
C_{b}^{2}(\mathbb{R}^{N})$, we apply the second characterization in (16) with
$\ell^{\theta}(y)=L^{\theta}_{y}\phi(x)+f^{\theta}(x,y)\cdot
p+l^{\theta}(x,y)$, and conclude by integrating over
$\mathbb{T}^{N}\times\Theta$. ∎

As we mentioned in the Introduction, we have the following regularity result
for the solution of the effective problem (7).

######  Corollary 3.2.

Assume hypotheses of Corollary (2.1) holds, and assume further that $f,l$ are
Hölder continuous in the slow variable, uniformly with respect to the rest of
the others, Then, every continuous solution $u$ of (7) is in
$C^{\sigma+\alpha}$, and there exist $C>0$ and $\alpha>0$, depending only on
universal constants, such that $\|u\|_{\sigma+\alpha}\leq C$. Moreover, this
solution is unique in the class of smooth functions.

The proof follows by available regularity results. First, $C^{\alpha}$
estimates for (1) which are uniform in $\epsilon$ (see Chang-Lara and Dávila
[18]), and the perturbed test function method, allows us to prove that each
accumulation point of the sequence is a continuous viscosity solution to (7).
Such a solution becomes Lipschitz continuous using the results of Barles,
Chasseigne, Ciomaga and Imbert [9], and from here we get it is $C^{1,\alpha}$
by Caffarelli and Silvestre [13]. Finally, the $C^{\sigma+\alpha}$ estimates
in [14] or Serra [41] imply the desired higher-order regularity. This
procedure is the content of the regularity estimates presented in Corollary
3.2. We remark that such a procedure cannot be directly applied to (1) to
obtain $C^{\sigma+\alpha}$ estimates independent of $\epsilon$, due to the
high oscillatory phenomena as $\epsilon\to 0$ which deteriorates the
continuity of the data in the equation.

###  3.1. A discussion on the dependency of $\bar{H}$ on the slow variable.

We notice here that the arguments of Proposition 3.1 and Theorem 2.1 apply
equally to the more complicated case of $x$-dependent kernels, for instance,
for kernels with the form

$K^{\theta}(x,y,z)=|z^{t}A(x,y,\hat{z})z|^{-(N+\sigma)/2},$ (19)

with certain continuity assumptions on the slow variable $x$. In that case we
still produce a representation formula, but in that case the new index set
depends upon $x$, namely

$\overline{H}(x,p,\phi)=\sup_{\mu\in\mathcal{G}_{0}^{\prime}(x)}\left\\{-\overline{L}^{\mu}\phi(x)-\bar{f}^{\mu}(x)\cdot
p-\bar{l}^{\mu}(x)\right\\}.$ (20)

However, we notice that the presence of $\mathcal{G}_{0}^{\prime}(x)$ in (20)
does not allow us to use the available regularity results to conclude an
analogue of Corollary 3.2.

As the method to obtain rates of convergence in the following section relies
on such estimates, we are unable to reach a conclusion in the setting imposed
by (19).

Nevertheless, we highlight that the structure of (20) is in accordance with
phenomena arising, for instance, in first-order equations. For Hamiltonians of
the form

$H(x,y,p)=\sup_{\theta\in\Theta}\\{-f^{\theta}(x,y)\cdot
p-l^{\theta}(x,y)\\},$

under certain controlability assumptions on the family of fluxes $f$, the
associated effective Hamiltonian takes the form

$\bar{H}(x,p)=\sup_{\mu\in Z(x)}\\{-f^{\mu}(x)\cdot p-l^{\mu}(x)\\},$

where for each $x\in\mathbb{R}^{N}$, $Z(x)$ is a subset of the space of
probability measures on $\mathbb{T}^{N}\times\Theta$—the set of “occupational
measures”, as described in [47] and references therein. In this setting, some
continuity on this set of indices $Z(x)$ (in the Hausdorff distance) is
required in the standard doubling-variables method to get comparison
principles for viscosity solutions. This makes possible to identify the
effective problem with a optimal control problem with average trajectories and
costs, see Bardi and Terrone in [6].

The case of nonlocal terms of order $\sigma\in(0,1]$ contains similar
difficulties related to those of the previous point. First, we observe that
the structure of the cell problem changes depending on the value of $\sigma$,
as it is shown in [5]. For the critical value $\sigma=1$—the case which is
studied in detail in [19]—the scaling of the problem makes the cell problem
(5) depends also on the gradient of the corrector $w$. Consequently, the
inequality defining $\mathcal{G}_{0}$ in (12) is now

$\sup_{\theta\in\Theta}\\{-L^{\theta}_{y}v(y)-f^{\theta}(x,y)\cdot
Dv(y)-\phi_{\theta}(y)\\}\leq 0\ \mbox{in}\ \mathbb{T}^{N}.$

This implies that the sets $\mathcal{G}_{0}$ and $\mathcal{G}_{0}^{\prime}$
both depend on the slow variable $x$, even if $K^{\theta}$ has the simpler
form (3), and we are again in the situation described in the previous remark.

##  4\. Estimates for the Discount Problem

In the study of the cell problem (5) it is convenient to consider the
approximation problem

$\lambda
w_{\lambda}+\sup_{\theta}\\{-L^{\theta}_{y}w_{\lambda}(y)-L^{\theta}_{y}\phi(x)-f^{\theta}(x,y)\cdot
p-l^{\theta}(x,y)\\}=0\quad\text{in }\mathbb{T}^{N},$ (21)

for $\lambda>0$, commonly referred to as the vanishing discount problem: it is
well-known fact that under the assumptions considered here, the solvability of
the eigenvalue problem (5) is obtained in the passage to the limit as
$\lambda\to 0^{+}$ in (21). This problem plays a key role in our main result
Theorem 2.2.

To stress the dependence of $w_{\lambda}$ on $x,p$ and $\phi$, we write
$w_{\lambda}=w_{\lambda}(y;x,p,\phi)$ for the solution of (21) and, similarly,
$w=w(y;x,p,\phi)$ for the solution of (5).

######  Lemma 4.1.

There exists a constant $C_{1}$ such that the solution of (21) satisfies the
following: for all $x,p\in\mathbb{R}^{N}$ and $\phi\in
C^{\sigma+\iota}(\mathbb{R}^{N})$

  1. (a)

$\|w_{\lambda}(\cdot;x,p,\phi)\|_{\infty}\leq\lambda^{-1}C_{1}\left(1+|p|+\|\phi\|_{\sigma+\iota}\right);$

  2. (b)

for some $\alpha\in(0,1)$,

$\|w_{\lambda}(\cdot;x,p,\phi)-w_{\lambda}(0;x,p,\phi)\|_{C^{\sigma+\alpha}(\mathbb{R}^{N})}\leq
C_{1}\left(1+|p|+\|\phi\|_{\sigma+\iota}\right)$

  3. (c)

$|D_{p}w_{\lambda}|\leq\lambda^{-1}C_{1}$,
$|D_{x}w_{\lambda}|\leq\lambda^{-1}C_{1}(1+|p|+\|\phi\|_{\sigma+\iota})$ (in
the viscosity sense); and if $\phi_{i}\in C^{\sigma+\iota}(\mathbb{R}^{N})$
for $i=1,2$, then

$\|w_{\lambda}(\cdot;x,p,\phi_{1})-w_{\lambda}(\cdot;x,p,\phi_{2})\|_{\infty}\leq\lambda^{-1}C_{1}\|\phi_{1}-\phi_{2}\|_{\sigma+\iota}$

  4. (d)

for all
$y\in\mathbb{T}^{N},\quad|w_{\lambda}(y;x,p,\phi)+\overline{H}(x,p,\phi)|\leq\lambda
C_{1}\left(1+|p|+\|\phi\|_{\sigma+\iota}\right)$.

###### Proof.

(a) From the structure of $H$, we infer that
$C=\pm\lambda^{-1}C_{1}\left(1+|p|+\|\phi\|_{C^{\sigma+\iota}(\mathbb{R}^{N})}\right)$
are respectively super- and subsolutions of (21). Indeed, it is immediate that

$L^{\theta}_{y}C\equiv 0,\quad|f^{\theta}(x,y)\cdot p|\leq
C_{1}|p|,\quad|l^{\theta}(x,y)|\leq C_{1},$

for sufficiently large $C_{1}$. For the principal part, we have

$|L^{\theta}_{y}\phi(x)|\leq|L^{\theta}_{y}[B]\phi(x)|+|L^{\theta}_{y}[\mathbb{R}^{N}\backslash
B]\phi(x)|=:I_{1}+I_{2}.$

Using the symmetry of $K^{\theta}$, we have

$\displaystyle I_{1}={}$
$\displaystyle\left|\int_{B}\left(\phi(x+z)-\phi(x)-D\phi(x)\cdot
z\right)K^{\theta}(y,z)\,dz\right|$ $\displaystyle={}$
$\displaystyle\left|\int_{B}\left(\int_{0}^{1}D\phi(x+tz)\cdot
z\,dt\right)-D\phi(x)\cdot z\ K^{\theta}(y,z)\,dz\right|$
$\displaystyle\leq{}$
$\displaystyle\int_{B}\left(\int_{0}^{1}|D\phi(x+tz)-D\phi(x)||z|\,dt\right)\
K^{\theta}(y,z)\,dz$ $\displaystyle\leq{}$
$\displaystyle\int_{0}^{1}[\phi]_{1,\beta}|z|^{1+\beta}\
K^{\theta}(y,z)\,dz\leq C\Gamma[\phi]_{1,\beta}.$

On the other hand, it is easy to see that $I_{2}\leq
C\Gamma\|\phi\|_{\infty}$, and by combining both estimates we conclude.

(b) We must first establish that for all $\lambda,x,p$ and $\phi$ as above,

$\|w_{\lambda}(\cdot;x,p,\phi)-w_{\lambda}(0;x,p,\phi)\|_{\infty}\leq
C_{1}\left(1+|p|+\|\phi\|_{\sigma+\iota}\right).$ (22)

Assume, on the contrary, that there exist sequences $(\lambda_{k})_{k}$ and
$\big{(}(x_{k},p_{k},\phi_{k})\big{)}_{k}$ such that $\lambda_{k}\to 0$ and
$w_{k}=w_{\lambda_{k}}(\cdot;x_{k},p_{k},\phi_{k})$ satisfies

$\|w_{k}-w_{k}(0)\|_{\infty}\geq
k\left(1+|p_{k}|+\|\phi_{k}\|_{\sigma+\iota}\right).$

For $\eta_{k}=\|w_{k}-w_{k}(0)\|_{\infty}^{-1}$, we define
$\tilde{w}_{k}=\eta_{k}(w_{k}-w_{k}(0))$ and note that $\tilde{w}_{k}$
satisfies $\tilde{w}_{k}(0)=0$, $\|\tilde{w}_{k}\|_{\infty}=1$ and

$\lambda_{k}\tilde{w}_{k}+\lambda_{k}\eta_{k}\tilde{w}_{k}(0)+\sup_{\theta}\\{-L^{\theta}_{y}\tilde{w}_{k}(y)-\tilde{l}^{\theta}_{k}(y)\\}=0,$
(23)

where
$\tilde{l}^{\theta}_{k}(y)=L^{\theta}_{y}\phi_{k}(x_{k})+f^{\theta}(x_{k},y)+l^{\theta}(x_{k},y)$.

By part (a)—again using (8)—we have that

$\eta_{k}\lambda_{k}|\tilde{w}_{k}(0)|+\eta_{k}|\tilde{l}^{\theta}_{k}|_{\infty}\leq\frac{C}{k}.$

for some $C>0$. We may thus apply the regularity results of [13] to (23) to
find that $(\tilde{w}_{k})$ is bounded in $C^{\bar{\iota}}$ for some
$\bar{\iota}>0$, uniformly with respect to $k$, and thus converges up to a
subsequence to some $\tilde{w}\in C^{\bar{\iota}}(\mathbb{T}^{N})$. Passing to
the limit in (23) in the viscosity sense, we find that $\tilde{w}$ is a
solution of

$\sup_{\theta}\left\\{-L^{\theta}_{y}\tilde{w}(y)\right\\}=0\quad\textrm{in
}\mathbb{T}^{N}.$

Since $\tilde{w}$ is periodic, it achieves its maximum at some point; hence,
it is constant by the strong maximum principle. This contradicts the fact
that, as $\tilde{w}$ is a limit of $\tilde{w}_{k}$, we have $\tilde{w}(0)=0$
and $\|\tilde{w}\|_{\infty}=1$, thus establishing (22).

We now note that $v_{\lambda}=w_{\lambda}-w_{\lambda}(0)$ satisfies

$\lambda
v_{\lambda}+\sup_{\theta}\\{-L^{\theta}_{y}v_{\lambda}(y)-\tilde{l}^{\theta}(y)\\}=0$
(24)

where
$\tilde{l}^{\theta}(y)=L^{\theta}_{y}\phi(x)+f^{\theta}(x,y)+l^{\theta}(x,y)$,
and using (22) and the results of [13] we conclude that
$\|v_{\lambda}\|_{C^{\alpha}}(\mathbb{T}^{N})\leq
C_{1}(1+|p|+\|\phi\|_{\sigma+\iota})$ for some $\alpha>0$, taking a larger
$C_{1}$ if necessary.

We wish to conclude by applying the results of [41] to (24). To this end, we
now show that $\tilde{l}^{\theta}$ as defined above is uniformly bounded in
$C^{\alpha}$.

Let $y_{i}\in\mathbb{T}^{N}$, $i=1,2$, $z\in\mathbb{R}^{N}\setminus\\{0\\}$.
Given the assumptions (8), it is immediate that

$\|f^{\theta}(x,\cdot)\cdot
p+l^{\theta}(x,\cdot)\|_{C^{\alpha}(\mathbb{T}^{N})}\leq C_{1}(1+|p|),$

in fact for any $\alpha\in(0,1]$. To bound the nonlocal term appearing in
$\tilde{l}^{\theta}$, we compute

$|K^{\theta}(y_{1},z)-K^{\theta}(y_{2},z)|\leq\frac{\Gamma^{2}}{|z|^{2(N+\sigma)}}\left|\left(z^{T}A^{\theta}(y_{1},\hat{z})z\right)^{\frac{N+\sigma}{2}}-\left(z^{T}A^{\theta}(y_{2},\hat{z})z\right)^{\frac{N+\sigma}{2}}\right|.$

Write $h(s):=s^{\frac{N+\sigma}{2}}$,
$s_{i}:=z^{T}A^{\theta}(y_{i},\hat{z})z$, $i=1,2$. Since $h$ is smooth and
convex, we have

$|h(s_{1})-h(s_{2})|\leq|s_{1}-s_{2}|\max\\{|h^{\prime}(s_{1})|,|h^{\prime}(s_{2})|\\}.$

Therefore,

$\displaystyle\left|\left(z^{T}A^{\theta}(y_{1},\hat{z})z\right)^{\frac{N+\sigma}{2}}-\left(z^{T}A^{\theta}(y_{2},\hat{z})z\right)^{\frac{N+\sigma}{2}}\right|$
$\displaystyle\quad\leq\left|z^{T}[A^{\theta}(y_{1},\hat{z})-A^{\theta}(y_{2},\hat{z})z]\right|\max\left\\{\left(z^{T}A^{\theta}(y_{1},\hat{z})z\right)^{\frac{N+\sigma-2}{2}},\left(z^{T}A^{\theta}(y_{2},\hat{z})z\right)^{\frac{N+\sigma-2}{2}}\right\\}$
$\displaystyle\quad\leq C|y_{1}-y_{2}|^{\alpha}|z|^{N+\sigma},$

where we have used (11) and (9) for the last inequality. Combining these
estimates we obtain

$\displaystyle|L^{\theta}_{y_{1}}\phi(x)-L^{\theta}_{y_{2}}\phi(x)|\leq\int_{\mathbb{R}^{N}}|\phi(x+z)-\phi(x)||K^{\theta}(y_{1},z)-K^{\theta}(y_{2},z)|\,dz$
$\displaystyle\quad\leq
C|y_{1}-y_{2}|^{\alpha}\int_{\mathbb{R}^{N}}|\phi(x+z)-\phi(x)||z|^{N+\sigma}\,dz$
$\displaystyle\quad\leq C|y_{1}-y_{2}|^{\alpha}\|\phi\|_{\sigma+\iota}.$

Here we have removed the principal value from each of the integrals defining
$L^{\theta}_{y_{1}}\phi$ and $L^{\theta}_{y_{2}}\phi$ by using that $\phi\in
C^{\sigma+\iota}$ and computing as in part (a) of the lemma. The third
inequality is similarly obtained.

Again using the assumption (9), we apply the results of [41] to equation (24),
and with this we conclude.

(c) Let $w_{i}=w_{i}(\cdot;x_{i},p_{1},\phi_{i})$ for
$x_{i},p_{1}\in\mathbb{R}^{N}$, $\phi_{i}\in
C^{\sigma+\iota}(\mathbb{R}^{N})$, $i=1,2$. Using the assumptions on the
structure of $H$, it is easy to see that
$w_{\pm}:=w_{1}\pm\lambda^{-1}\big{[}|x_{1}-x_{2}|(1+|p_{2}|+\|\phi_{1}\|_{\sigma+\iota})+|p_{1}-p_{2}|+\|\phi_{1}-\phi_{2}\|\big{]}$
are respectively a super- and a subsolution of (21) centered in
$(x_{2},p_{2},\phi_{2})$. The claim follows by comparison.

(d) Adapting the arguments of [16], [32], we consider fixed $(x,p,\phi)$ and
define

$\Gamma_{\lambda}=\lambda\sup_{y}w_{\lambda}(y;x,p,\phi).$

We first claim that

$\Gamma_{\lambda}\geq-\bar{H}(x,p,\phi).$ (25)

Indeed, by noting that $w_{\lambda}$ satisfies

$\Gamma_{\lambda}+\sup_{\theta}\left\\{-L^{\theta}_{y}w_{\lambda}(y)-L^{\theta}_{y}\phi(x)-f^{\theta}(x,y)\cdot
p-l^{\theta}(x,y)\right\\}\geq 0,$

the claim follows from applying the following characterization of the ergodic
constant: using the notation of Section 3,

$c=\sup\\{\tilde{c}\in\mathbb{R}:\exists\ \psi\ \mbox{s.t.}\
F(\psi,y)\geq\tilde{c}\ \mbox{in}\ \mathbb{T}^{N}\\}.$

This is in a sense dual to (17) and can be proved in exactly the same way.

We then note that part (b) of the lemma implies in particular that

$\lambda|w_{\lambda}(y_{1};x,p,\phi)-w_{\lambda}(y_{2};x,p,\phi)|\leq\lambda
C_{1}\sqrt{N}(1+|p|+\|\phi\|_{\sigma+\iota}),$

given that $\sqrt{N}$ is the diameter of the unit hypercube in
$\mathbb{R}^{N}$. Plugging (25) into the previous inequality, we find

$\lambda w_{\lambda}(y;x,p,\phi)\geq-\bar{H}(x,p,\phi)-\lambda
C_{1}\sqrt{N}(1+|p|+\|\phi\|_{\sigma+\iota})\quad\textrm{for all
}y\in\mathbb{T}^{N}.$

The proof of the corresponding upper bound is similarly obtained. ∎

In the proof of Theorem 2.2 we will make use of the following estimate.

######  Proposition 4.2.

Let $x_{1},x_{2},z\in\mathbb{R}^{N}$, $y\in\mathbb{T}^{N}$, $\phi\in
C^{\sigma+\alpha}(\mathbb{R}^{N})$. For any $\alpha^{\prime}<\alpha$, there
exists a $C>0$ such that

$|L^{\theta}_{y}\phi(x_{1})-L^{\theta}_{y}\phi(x_{2})|\leq
C|x_{1}-x_{2}|^{\alpha^{\prime}}\|\phi\|_{\sigma+\alpha}.$ (26)

###### Proof.

Write

$\delta:=|\phi(x_{1}+z)-\phi(x_{1})-(\phi(x_{2}+z)-\phi(x_{2}))|$

As (in particular) $\phi\in C^{1}$, we have

$\displaystyle\delta={}$
$\displaystyle\left|\int_{0}^{1}\left(D\phi(x_{1}+tz)-D\phi(x_{2}+tz)\right)\cdot
z\,dt\right|$ $\displaystyle\leq{}$
$\displaystyle\int_{0}^{1}|D\phi(x_{1}+tz)-D\phi(x_{2}+tz)||z|\,dt\leq\int_{0}^{1}[\phi]_{1,\beta}|x_{1}-x_{2}|^{\beta}|z|\,dt$
$\displaystyle\leq{}$
$\displaystyle[\phi]_{1,\beta}\,|x_{1}-x_{2}|^{\beta}|z|.$

Also, by rearranging the terms,

$\displaystyle\delta={}$
$\displaystyle|\phi(x_{1}+z)-\phi(x_{2}+z)-(\phi(x_{1})-\phi(x_{2}))|$
$\displaystyle\leq{}$
$\displaystyle\int_{0}^{1}|D\phi(x_{2}+z+t(x_{1}-x_{2}))-D\phi(x_{2}+t(x_{1}-x_{2}))||x_{1}-x_{2}|\,dt$
$\displaystyle\leq{}$
$\displaystyle[\phi]_{1,\beta}\,|z|^{\beta}|x_{1}-x_{2}|.$

Hence

$\delta=\delta^{s}\delta^{1-s}\leq[\phi]_{1,\beta}\,|x_{1}-x_{2}|^{s\beta+1-s}|z|^{s+\beta(1-s)}.$

Setting $s=\frac{1-\alpha}{1-\beta}+\nu$ for some $\nu>0$, we have

$\displaystyle|L^{\theta}_{y}\phi(x_{1})-L^{\theta}_{y}\phi(x_{2})|$
$\displaystyle\quad\leq\int_{\mathbb{R}^{N}}|\phi(x_{1}+z)-\phi(x_{1})-(\phi(x_{2}+z)-\phi(x_{2}))|K^{\theta}(y,z)\,dz$
$\displaystyle\quad\leq[\phi]_{1,\beta}\,|x_{1}-x_{2}|^{\alpha-\nu(1-\beta)}\int_{\mathbb{R}^{N}}|z|^{\sigma+\nu(1-\beta)}K^{\theta}(y,z)\,dz$
$\displaystyle\quad\leq C[\phi]_{1,\beta}|x_{1}-x_{2}|^{\alpha-\nu(1-\beta)}.$

We thus obtain (26) taking a sufficiently small $\nu$, noting $C$ depends on
$\alpha^{\prime}:=\alpha-\nu(1-\beta)$ and universal constants. ∎

##  5\. Rate of convergence: Proof of Theorem 2.2.

We now provide the proof of Theorem 2.2.

###### Proof.

The proof follows that of Theorem 2.1 in [16]. Crucially, we will employ the
following bound, consequence of Corollary 3.2:

$\|u\|_{C^{\sigma+\alpha}(\mathbb{R}^{N})}\leq M.$ (27)

Let $M_{0},b>0$ to be chosen, and $\psi\in C^{\infty}(\mathbb{R}^{N})$ a
radial function, nondecreasing with respect to $|x|$, such that

$\psi(x)=\left\\{\begin{array}[]{ll}|x|^{2},&\text{if }|x|\leq 1,\\\
M_{0},&\text{if }|x|\geq 2.\end{array}\right.$

We immediately note that

$|D\psi|,\ |L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}\psi|\leq M_{1}$
(28)

where $M_{1}$ is independent of $\epsilon$.

Writing $w_{\lambda}(\cdot;[u](x))=w_{\lambda}(\cdot;x,Du(x),u)$ for short, we
define

$\varphi(x)=u^{\epsilon}(x)-u(x)-\epsilon^{\sigma}w_{\lambda}(\nicefrac{{x}}{{\epsilon}};[u](x))-b\psi(x),$

and note that for sufficiently large $M_{0}$, $\varphi$ attains a global
maximum at some $\hat{x}\in B_{2}$.

For $c>0$, define

$\tilde{\varphi}(x)=u^{\epsilon}(x)-u(x)-\epsilon^{\sigma}w_{\lambda}(\nicefrac{{x}}{{\epsilon}};[u](\hat{x}))-b\psi(x)-c\psi(x-\hat{x}).$

Given $\tau>0$, we claim that $c$ can be chosen so that $\tilde{\varphi}$
attains a global maximum $\tilde{x}\in B_{\tau}(\hat{x})$. Indeed, set

$c=\frac{\epsilon^{\sigma}}{\lambda\tau^{2-\alpha^{\prime}}}2C(1+2M)$

for $C>C_{1}$, where $C_{1}$ and $\alpha^{\prime}$ are as in Lemma 4.1 and
Proposition 4.2, respectively, and let $x\in\mathbb{R}^{N}\backslash
B_{\tau}(\hat{x})$. By construction we have
$\tilde{\varphi}(\hat{x})=\varphi(\hat{x})\geq\varphi(x)$, hence

$\displaystyle\tilde{\varphi}(\hat{x})-\tilde{\varphi}(x)$
$\displaystyle{}\geq\varphi(x)-\tilde{\varphi}(x)=-\epsilon^{\sigma}\left[w_{\lambda}(\nicefrac{{x}}{{\epsilon}};[u](x))-w_{\lambda}(\nicefrac{{x}}{{\epsilon}};[u](\hat{x}))\right]+c\psi(x-\hat{x}).$

Assume $\tau<1$ (it will eventually be taken small). If
$\tau\leq|x-\hat{x}|<1$ we have

$c\psi(x-\hat{x})=c|x-\hat{x}|^{2}=\frac{\epsilon^{\sigma}}{\lambda\tau^{2-\alpha^{\prime}}}2C(1+2M)|x-\hat{x}|^{2}>\frac{\epsilon^{\sigma}}{\lambda}2C_{1}(1+2M)|x-\hat{x}|^{\alpha^{\prime}},$

while Lemma 4.1(c) and (26) gives that

$\displaystyle\left[w_{\lambda}(\nicefrac{{x}}{{\epsilon}};[u](x))-w_{\lambda}(\nicefrac{{x}}{{\epsilon}};[u](\hat{x}))\right]\leq\lambda^{-1}C_{1}(1+2M)|x-\hat{x}|^{\alpha^{\prime}}.$

Combining these inequalities we obtain that
$\tilde{\varphi}(\hat{x})-\tilde{\varphi}(x)>0$. If, on the contrary,
$|x-\hat{x}|\geq 1$, then $c\psi(x-\hat{x})\geq c$, and from Lemma 4.1(a),
again using $\tau<1$, we have

$\displaystyle\epsilon^{\sigma}\left[w_{\lambda}(\nicefrac{{x}}{{\epsilon}};[u](x))-w_{\lambda}(\nicefrac{{x}}{{\epsilon}};[u](\hat{x}))\right]\leq\epsilon^{\sigma}2\lambda^{-1}C_{1}(1+2M)<c.$

This again implies that $\tilde{\varphi}(\hat{x})-\tilde{\varphi}(x)>0$. Thus
$\tilde{\varphi}$ attains a global maximum in $B_{\tau}(\hat{x})$, which we
denote by $\tilde{x}$.

In particular, we have the following behavior at $\tilde{x}$:

$\displaystyle
Du^{\epsilon}(\tilde{x})=Du(\tilde{x})+\epsilon^{\sigma-1}D_{y}w_{\lambda}(\nicefrac{{\tilde{x}}}{{\epsilon}};[u](\hat{x}))-bD\psi(\tilde{x})-2c(\tilde{x}-\hat{x}),$
(29)

since $|\tilde{x}-\hat{x}|<\tau<1$ implies that
$\psi(x-\hat{x})=c|x-\hat{x}|^{2}$ in a neighborhood of $\tilde{x}$, and

$\displaystyle
L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}u^{\epsilon}(\tilde{x})$
$\displaystyle\quad\leq
L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}u(\tilde{x})+\epsilon^{\sigma}L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}w_{\lambda}(\nicefrac{{\cdot}}{{\epsilon}};[u](\hat{x}))(\tilde{x})+bL^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}\psi(\tilde{x})+cL^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}\psi(\cdot-\hat{x})(\tilde{x})$
$\displaystyle\quad=L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}u(\tilde{x})+L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}},w_{\lambda}(\cdot;[u](\hat{x}))(\nicefrac{{\tilde{x}}}{{\epsilon}})+bL^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}\psi(\tilde{x})+cL^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}\psi(\tilde{x}-\hat{x}),$
(30)

using the homogeneity and translation invariance of $L^{\theta}$.

For the following computation we write
$w_{\lambda}=w_{\lambda}(\cdot;[u](\hat{x}))$ to ease notation. Evaluating (1)
at $\tilde{x}$, we have

$u^{\epsilon}(\tilde{x})+\sup_{\theta}\left\\{-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}u^{\epsilon}(\tilde{x})-f^{\theta}(\tilde{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\cdot
Du^{\epsilon}(\tilde{x})-l^{\theta}(\tilde{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\right\\}=0,$

and continue estimating:

$\displaystyle\sup_{\theta}\left\\{-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}u^{\epsilon}(\tilde{x})-f^{\theta}(\tilde{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\cdot
Du^{\epsilon}(\tilde{x})-l^{\theta}(\tilde{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\right\\}$
$\displaystyle\quad\geq\sup_{\theta}\left\\{-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}u(\tilde{x})-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}w_{\lambda}(\nicefrac{{\tilde{x}}}{{\epsilon}})-bL^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}\psi(\tilde{x})-cL^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}\psi(\tilde{x}-\hat{x})\right.$
$\displaystyle\qquad\left.-f^{\theta}(\tilde{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\cdot\big{(}Du(\tilde{x})+\epsilon^{\sigma-1}D_{y}w_{\lambda}(\nicefrac{{\tilde{x}}}{{\epsilon}})-bD\psi(\tilde{x})-2c(\tilde{x}-\hat{x})\big{)}-l^{\theta}(\tilde{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\right\\}$
$\displaystyle\quad\geq\sup_{\theta}\left\\{-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}u(\tilde{x})-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}w_{\lambda}(\nicefrac{{\tilde{x}}}{{\epsilon}})-f^{\theta}(\tilde{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\cdot
Du(\tilde{x})-l^{\theta}(\tilde{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\right\\}$
$\displaystyle\qquad-M_{2}\big{(}\epsilon^{\sigma-1}+b+(2\tau+1)c\big{)}$
$\displaystyle\quad\geq\sup_{\theta}\left\\{-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}u(\hat{x})-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}w_{\lambda}(\nicefrac{{\tilde{x}}}{{\epsilon}})-f^{\theta}(\hat{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\cdot
Du(\hat{x})-l^{\theta}(\hat{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\right\\}$
$\displaystyle\qquad-
M_{2}\big{(}\tau^{\alpha^{\prime}}+\tau\big{)}-M_{2}\big{(}\epsilon^{\sigma-1}+b+(2\tau+1)c\big{)}.$

Here we have used (29) and (5) for the first inequality, the bounds (27) and
(28) from the second (and onward) together with Part (b), estimate (26) for
the third, and chosen a sufficiently large $M_{2}$—in particular, such that
$M_{2}\geq M_{0},M_{1}$, and independent of $\epsilon,\lambda,\tau,b$ and $c$.

We now use that $w_{\lambda}(\cdot;[u](\hat{x}))$ and $u$ are solutions of
(21) and (7), respectively, and Lemma 4.1 (d) to obtain

$\displaystyle\sup_{\theta}\left\\{-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}},u(\hat{x})-L^{\theta}_{\nicefrac{{\tilde{x}}}{{\epsilon}}}w_{\lambda}(\nicefrac{{\tilde{x}}}{{\epsilon}})-f^{\theta}(\hat{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\cdot
Du(\hat{x})-l^{\theta}(\hat{x},\nicefrac{{\tilde{x}}}{{\epsilon}})\right\\}$
$\displaystyle\quad=-\lambda
w_{\lambda}(\nicefrac{{\tilde{x}}}{{\epsilon}};[u](\hat{x}))\geq\overline{H}(\hat{x},Du(\hat{x}),u)-\lambda
C_{1}(1+2M)$ $\displaystyle\quad\geq-u(\hat{x})-\lambda C_{1}(1+2M).$

Combining this with the previous computation we have

$u^{\epsilon}(\tilde{x})-u(\hat{x})\leq
M_{2}\big{(}\lambda+\epsilon^{\sigma-1}+b+c+\tau^{\alpha^{\prime}}\big{)}.$

By construction, for all $x\in\mathbb{R}^{N}$ we have
$\varphi(x)\leq\varphi(\hat{x})=\tilde{\varphi}(\hat{x})\leq\tilde{\varphi}(\tilde{x})$,
hence

$\displaystyle u^{\epsilon}(x)-u(x)\leq{}$
$\displaystyle\big{(}u^{\epsilon}(\tilde{x})-u(\hat{x})\big{)}+\big{(}u(\hat{x})-u(\tilde{x})\big{)}$
$\displaystyle\quad+\epsilon^{\sigma}\left[w_{\lambda}\left(\nicefrac{{x}}{{\epsilon}};[u](x)\right)-w_{\lambda}\left(\nicefrac{{\tilde{x}}}{{\epsilon}};[u](\hat{x})\right)\right]+b[\psi(x)-\psi(\tilde{x})]$
$\displaystyle\quad\leq
M_{2}\big{(}\lambda+\epsilon^{\sigma-1}+b+c+\tau^{\alpha^{\prime}}\big{)}+M\tau+\frac{\epsilon^{\sigma}}{\lambda}2C_{1}(1+2M)+M_{0}b,$

by Lemma 4.1 (a), (28) and (27).

Sending $b\to 0$ and recalling the definition of $c$, this gives

$u^{\epsilon}(x)-u(x)\leq
M_{2}\left(\lambda+\epsilon^{\sigma-1}+\frac{\epsilon^{\sigma}}{\lambda\tau^{2-\alpha^{\prime}}}+\tau^{\alpha^{\prime}}\right),$

again by taking a larger value of $M_{2}$ if necessary. Thus, by symmetry, we
set

$\lambda=\epsilon^{\frac{\sigma\alpha^{\prime}}{2+\alpha^{\prime}}},\quad\tau=\epsilon^{\frac{\sigma}{2+\alpha^{\prime}}},$

and obtain

$u^{\epsilon}(x)-u(x)\leq
4M_{2}\epsilon^{\frac{\sigma\alpha^{\prime}}{2+\alpha^{\prime}}}.$

∎

######  Remark 5.1.

The proof on Theorem 2.2 does not rely on establishing the convergence
$u^{\epsilon}\to u$ as $\epsilon\to 0$ beforehand. Consequently, Theorem 2.2
trivially implies the convergence result (i.e., homogenization).

We proceed with the proof of the easier results on rates of convergence when
some of the dependencies of $H$ are dropped. We note that these results are
independent of Theorems 2.1 and 2.2.

###### Proof of Theorem 2.3.

(i) From the assumptions, (1) may be written as

$u^{\epsilon}+H(x,y,u^{\epsilon})=0\quad\textrm{in }\mathbb{R}^{N},$ (31)

where

$H(x,y,\varphi)=\sup_{\theta\in\Theta}\\{-L^{\theta}_{y}\varphi(x)-l^{\theta}(y)\\}.$

As a function in $\mathbb{R}^{N}\times\mathbb{T}^{N}\times
C^{2}_{b}(\mathbb{R}^{N})$, $H$ in fact independent of $x$, since $L^{\theta}$
depends on $x$ only through evaluating $u(x+z)-u(x)$ in the integrand of
$L^{\theta}$ and $K^{\theta}$ is translation invariant. It is however not
entirely correct to write $H(y,u^{\epsilon})$ instead of $H(x,y,u^{\epsilon})$
above. The same can be said for the effective problem,

$u+\bar{H}(x,u)=0\quad\textrm{in }\mathbb{R}^{N},$

which has the constant solution $u\equiv-\bar{H}(0,0)$.

It follows from the preceding remarks that the associated cell problem is

$\sup_{\theta\in\Theta}\\{-L^{\theta}_{y}w(y)-l^{\theta}(y)\\}=\bar{H}(0,0)\quad\textrm{in
}\mathbb{T}^{N},$ (32)

and has a unique $\mathbb{T}^{N}$-periodic solution $w=w(y)$, by the results
of Lemma 4.1.

Define
$\bar{v}(x)=u(x)+\epsilon^{\sigma}w(\nicefrac{{x}}{{\epsilon}})+\epsilon^{\sigma}\|w\|_{\infty}=-\bar{H}(0,0)+\epsilon^{\sigma}w(\nicefrac{{x}}{{\epsilon}})+\epsilon^{\sigma}\|w\|_{\infty}$.
Substituting in (1) and using (32), we have

$\displaystyle\bar{v}(x)+H(x,\nicefrac{{x}}{{\epsilon}},\bar{v})={}$
$\displaystyle-\bar{H}(0,0)+w(\nicefrac{{x}}{{\epsilon}})+\epsilon^{\sigma}\|w\|_{\infty}+H(x,\nicefrac{{x}}{{\epsilon}},\epsilon^{\sigma}w(\nicefrac{{\cdot}}{{\epsilon}}))$
$\displaystyle=$
$\displaystyle{}-\bar{H}(0,0)+w(\nicefrac{{x}}{{\epsilon}})+\epsilon^{\sigma}\|w\|_{\infty}+H(\nicefrac{{x}}{{\epsilon}},\nicefrac{{x}}{{\epsilon}},w)\geq
0.$

Thus by comparison, $u^{\epsilon}\leq\bar{v}$, i.e.,
$u^{\epsilon}-u\leq\epsilon^{\sigma}w(\nicefrac{{x}}{{\epsilon}})+\epsilon^{\sigma}\|w\|_{\infty}\leq
2\epsilon^{\sigma}\|w\|_{\infty}.$ The lower bound is similarly obtained.

(ii) Arguing as in the first part of the proof we have that $u^{\epsilon}$ and
$u\equiv-H(0,0,0)$ are respectively solutions of

$u^{\epsilon}+H(x,\nicefrac{{x}}{{\epsilon}},Du^{\epsilon},u^{\epsilon})=0,\quad\textrm{in
}\mathbb{R}^{N},$ (33)

where
$H(x,y,p,\varphi)=\sup_{\theta\in\Theta}\\{-L^{\theta}_{y}\varphi(x)-f^{\theta}(y)\cdot
p-l^{\theta}(y)\\},$ and

$u+H(x,Du,u)=0\quad\textrm{in }\mathbb{R}^{N}.$

The associated cell problem is

$H(x,y,0,w)=\bar{H}(0,0)\quad\textrm{in }\mathbb{T}^{N},$ (34)

and we define
$\bar{v}(x)=-H(0,0,0)+\epsilon^{\sigma}w(\nicefrac{{x}}{{\epsilon}})+C\epsilon^{\sigma-1}\|w\|_{C^{1}}$,
where $C$ is given by (8). Again, evaluating in (33), we have

$\displaystyle\bar{v}(x)+H(x,\nicefrac{{x}}{{\epsilon}},D\bar{v}(x),\bar{v})$
$\displaystyle\quad=-H(0,0,0)+\epsilon^{\sigma}w(\nicefrac{{x}}{{\epsilon}})+C\epsilon^{\sigma-1}\|w\|_{C^{1}}+H(x,\nicefrac{{x}}{{\epsilon}},\epsilon^{\sigma-1}Dw(\nicefrac{{x}}{{\epsilon}}),\epsilon^{\sigma}w(\nicefrac{{\cdot}}{{\epsilon}}))$
$\displaystyle\quad=-H(0,0,0)+\epsilon^{\sigma}w(\nicefrac{{x}}{{\epsilon}})+C\epsilon^{\sigma-1}\|w\|_{C^{1}}+H(x,\nicefrac{{x}}{{\epsilon}},0,\epsilon^{\sigma}w(\nicefrac{{\cdot}}{{\epsilon}}))$
$\displaystyle\qquad-\epsilon^{\sigma-1}\sup_{\theta\in\Theta}\\{f^{\theta}(\nicefrac{{x}}{{\epsilon}})\cdot
Dw(\nicefrac{{x}}{{\epsilon}})\\}$ $\displaystyle\quad\geq 0.$

We conclude once more that $u^{\epsilon}\leq\bar{v}$, hence
$u^{\epsilon}-u\leq 2C\epsilon^{\sigma-1}\|w\|_{C^{1}}$. ∎

Acknowledgements.

A. R.-P. was partially supported by Fondecyt Grant Postdoctorado Nacional 2019
No. 3190858. E. T. was partially supported by Fondecyt No. 1201897.

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