Cyril Imbert
email: imbert@ceremade.dauphine.fr

Sylvia Serfaty
email: serfaty@ann.jussieu.fr

Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations

Keywords: Repeated games, integral curvature flows, parabolic integro-differential equations, viscosity solutions, geometric flows Mathematical Subject Classifications. 35C99, 53C44, 90D10, 49K25

published or not. The documents may come

Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations

Ω 0 ⊂ R N , they prove that the repeated game invented by Joel Spencer (originally called "pusher-chooser" game, now sometimes known as the "Paul-Carol" game) [START_REF] Spencer | Balancing games[END_REF] converges towards the mean curvature motion of ∂Ω 0 . In a second paper [START_REF] Kohn | A deterministic-control-based approach to fully non-linear parabolic and elliptic equations[END_REF], they construct analogous approximations of general fully non-linear parabolic and elliptic equations.

This paper is concerned with extending this approach to several non-local evolutions. In particular we construct a zero-sum repeated game with two players for a geometric motion originally introduced to describe dislocation dynamics [START_REF] Alvarez | Dislocation dynamics: short-time existence and uniqueness of the solution[END_REF]. This motion also appears in [START_REF] Caffarelli | Convergence of Nonlocal Threshold Dynamics Approximations to Front Propagation[END_REF] where threshold dynamics associated with kernels decaying slowly at infinity are considered. It was recently reformulated by the first author [START_REF] Imbert | Level set approach for fractional mean curvature flows[END_REF] in order to deal with singular interacting potentials. Such a motion is referred to as the integral curvature flow; it also appears in [START_REF] Imbert | Phasefield theory for fractional reaction-diffusion equations and applications[END_REF][START_REF] Caffarelli | Non local minimal surfaces[END_REF]. See the introduction of [START_REF] Imbert | Level set approach for fractional mean curvature flows[END_REF] for more details.

In order to construct such a game, we start with the simpler guiding case of the eikonal equation associated with a changing sign velocity, for which we give a game approximation. We are guided by the ideas of Evans and Souganidis [START_REF] Evans | Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations[END_REF]; they proved in particular that the solution of the eikonal equation can be represented by the value function of a differential game. Our first task is thus to give a discrete version of such a representation.

The specificity of the integral curvature flow is that it is non-local in the sense that its normal speed at a boundary point x not only depends on the front close to x (such as the outer normal unit vector or the curvature tensor) but also on the whole curve. Indeed, the integral curvature is a singular integral operator. This is the reason why we also contruct a game to approximate general fully non-linear parabolic equations involving singular integral terms.

The framework of viscosity solutions [START_REF] Crandall | Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre[END_REF][START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] and the level-set approach [START_REF] Osher | Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations[END_REF][START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF][START_REF] Evans | Motion of level sets by mean curvature. I[END_REF] are used in order to define properly the various geometric motions. We recall that the levelset approach consists in representing the initial interface as the 0-level set of a (Lipschitz) continuous function u 0 , looking for the evolving interface under the same form, proving that the function u(t, x) solves a partial differential equation and finally proving that the 0-level set of the function u(t, •) only depends on the 0-level set of u 0 . The proofs of convergence follow the method of Barles-Souganidis [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF] i.e. use the stability, monotonicity and consistency of the schemes provided by our games.

There are several motivations for constructing such games. First, it shows that viscosity solutions of an even wider class of equations have a deterministic control representation; while previously this was known to be true only for first order Hamilton-Jacobi equations, and then since [START_REF] Kohn | A deterministic-control-based approach to motion by curvature[END_REF][START_REF] Kohn | A deterministic-control-based approach to fully non-linear parabolic and elliptic equations[END_REF] for general local second order PDE's. Seen differently, it shows that a wide class of non-local evolutions have a minimax formulation. Then, these games can serve to build robust numerical schemes to approximate the solutions to the equations. Finally, although this has not been achieved very much, they could in principle serve to obtain new qualitative information on the solutions to the PDE's.

The game we present for integral curvature flow, even though this is a geometric evolution, is much more complicated that the Paul-Carol game studied in [START_REF] Kohn | A deterministic-control-based approach to motion by curvature[END_REF]. It would be very nice to find a game whose rules are simpler and which would be a natural generalization of the Paul-Carol game. However, we do not know at this stage whether this is possible.

The paper is organized as follows. In Section 2, we present the various equations that we study, state the definitions, present the games and give the main convergence results: first for parabolic integro-differential equations (in short PIDE), second for eikonal equations, and third for integral curvature flows. In Section 3, we return to these theorems in order and give their proofs.

Notation. The unit ball of R N is denoted by B. A ball of radius r centered at x is denoted by B r (x). The function 1 A (z) is defined as follows: 1 A (z) = 1 if z ∈ A and 0 if not. The unit sphere of R N is denoted by S N -1 . The set of symmetric real N × N matrices is denoted by S N .

Given two real numbers a, b, a∧b denotes min(a, b) and a∨b denotes max(a, b). Moreover, a + denotes max(0, a) and a -= max(0, -a).

The time derivative, space gradient and Hessian matrix of a function φ are respectively denoted by ∂ t φ, Dφ and D 2 φ.

C 2 b (R N ) denotes the space of C 2 bounded functions such that their first and second derivatives are also bounded.

Acknowledgements. The first author was partially supported by the ANR project MICA from the French Ministry of Research, the second by an EURYI award.

Main Results

This section is devoted to the description of the games we introduce to approximate the various geometric motions or solutions of parabolic PIDE.

Following [START_REF] Kohn | A deterministic-control-based approach to motion by curvature[END_REF][START_REF] Kohn | A deterministic-control-based approach to fully non-linear parabolic and elliptic equations[END_REF], in each game there are two opposing players Paul and Carol (or sometimes Helen and Mark). Paul starts at point x at time t > 0 with zero score. At each step n, the position x n and time t n are updated by using a small parameter ε > 0: (t n , x n ) = (t n (ε), x n (ε)). The game continues until the running time t N is larger than a given final time T . At the end of the game, Paul's final score is u T (x N ) where u T is a given continuous function u T defined on R N , and x N is the final position. Paul's objective is to maximize his final score and Carol's is to obstruct him.

We define the value function u ε of the game starting at x at time t as (2.1) u ε (t, x) = max (final score for Paul starting from (t, x)) .

The main results of this paper assert that the value functions associated with the games described in the next subsections converge to solutions of the corresponding evolution equations. As it is natural for control problems, the framework to use is that of viscosity solutions. We present the games in increasing order of complexity, so we start by presenting the results for parabolic integro-differential equations (PIDE).

General Parabolic Integro-Differential Equations

The parabolic non-linear integro-differential equations at stake in this paper are of the following form

(2.2) -∂ t u + F (t, x, Du, D 2 u, I[x, u]) = 0 in (0, T ] × R N
where T > 0 is a final time, F is a continuous non-linearity satisfying a proper ellipticity condition (see below) and I[x, U] is a singular integral term defined for U : R N → R as follows

(2.3) I[x, U] = [U(x + z) -U(x) -DU(x) • z1 B (z)]ν(dz)
where we recall B is the unit ball, 1 B (z) = 1 if |z| < 1 and 0 if not, and ν is a non-negative singular measure satisfying

(2.4) B |z| 2 ν(dz) < +∞, R N \B ν(dz) < +∞ .
We also assume for simplicity that ν(dz) = ν(-dz) but this is not a restriction. Such measures are referred to as (symmetric) Lévy measures and associated integral operators I[x, U] as Lévy operators. Such equations appear in the context of mathematical finance for models driven by jump processes; see for instance [START_REF] Cont | Financial modelling with jump processes[END_REF]. Because of the games we construct, a terminal condition is associated with such a parabolic PIDE. Given a final time T > 0, the solution u of (2.2) is submitted to the additional condition

(2.5) u(T, x) = u T (x)
where u T : R N → R is the terminal datum. The equation is called parabolic when the following ellipticity condition is fulfilled

(2.6) A ≤ B, l ≤ m ⇒ F (t, x, p, A, l) ≥ F (t, x, p, B, m) ,
where A ≤ B is meant with respect to the order on symmetric matrices. Under this condition, the equation with terminal condition (2.5) is well-posed in (0, T ] × R N .

Viscosity solutions for PIDE

In this section, we recall the definition and framework of viscosity solutions for (2.2) [START_REF] Soner | Optimal control with state-space constraint. II[END_REF][START_REF] Sayah | Équations de Hamilton-Jacobi du premier ordre avec termes intégrodifférentiels. I. Unicité des solutions de viscosité[END_REF]. Since we will work with bounded viscosity solutions, we give a definition in this framework. Remark 2.1. If the mesure ν is supported in B R , then the global maximum/minimum 0 of uφ at (t, x) can be replaced with a strict maximum/minimum 0 in (0, T ) × B R ′ (x) for any R ′ ≥ R. Indeed, changing φ outside B R (x) does not change the value of I[x, φ] in this case.

On the one hand, in order for the value of the repeated game we are going to construct to be finite, we need to make some growth assumption on the nonlinearity F . On the other hand, in order to get the convergence of the value of the repeated game, the comparison principle for (2.2) has to hold. For these reasons we assume that F satisfies the ellipticity condition given above together with the following set of assumptions (see [START_REF] Barles | Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited[END_REF]):

Assumptions (A). • (A0) F is continuous on R × R N × R N × S N × R. • (A1) There exist constants k 1 > 0, k 2 > 0 and C > 0 such that for all (t, x, p, A) ∈ R × R N × R N × R N × S N , we have |F (t, x, p, A, 0)| ≤ C(1 + |p| k 1 + |X| k 2 ) .
• (A2-1) For all R > 0, there exist moduli of continuity ω, ω R such that, for all |x|, |y| ≤ R, |v| ≤ R, l ∈ R and for all X, Y ∈ S N satisfying (2.9)

X 0 0 -Y ≤ 1 ε I -I -I I + r(β) I 0 0 I
for some ε > 0 and r(β) → 0 as β → 0 (in the sense of matrices in S 2N ), then, if s(β) → 0 as β → 0, we have

(2.10) F (t, y, v, ε -1 (x -y), Y, l) -F (t, x, v, ε -1 (x -y) + s(β), X, l) ≤ ω(β) + ω R (|x -y| + ε -1 |x -y| 2 )
or

• (A2-2) For all R > 0, F is uniformly continuous on [-R, R] × R n × B R × D R × R where D R := {X ∈ S N ;
|X| ≤ R} and there exist a modulus of continuity ω R such that, for all x, y ∈ R N , |v| ≤ R, l ∈ R and for all X, Y ∈ S N satisfying (2.9) and ε > 0, we have

(2.11) F (t, y, v, ε -1 (x-y), Y, l)-F (t, x, v, ε -1 (x-y), X, l) ≤ ω R (|x-y|+ε -1 |x-y| 2 ) .
• (A3) F (t, x, u, p, X, l) is Lipschitz continuous in l, uniformly with respect to all the other variables.

Assumptions (A0)-(A1) are all we need to show that the relaxed semi-limits of our value functions are viscosity sub-(resp. super-)solutions to (2.2). Assumptions (A2)-(A3) are meant to ensure that a comparison principle holds for (2.2), i.e. that viscosity sub-solutions are smaller than viscosity super-solutions, which guarantees the final convergence.

The game for PIDE

We are given positive parameters ε, R > 0. A truncated integral operator I R [x, Φ] is defined by replacing in (2.3) ν(dz) with 1 B R (z)ν(dz). We also consider a positive real number α ∈ (0, (max(1, k 1 , k 2 )) -1 ) where the constants k 1 , k 2 appear in Assumption (A1). In this setting, for the sake of consistency with [START_REF] Kohn | A deterministic-control-based approach to fully non-linear parabolic and elliptic equations[END_REF] where a financial interpretation was given, the players should be Helen (standing for hedger) and Mark (standing for market), with Helen trying to maximize her final score under the opposition of Mark.

Game 1 (Parabolic PIDE). At time t ∈ (0, T ), the game starts at x and Helen has a zero score. Her objective is to get the highest final score.

Helen chooses a function

Φ ∈ C 2 b (R N ) such that Φ ∞ ≤ ε -α , |DΦ(x)| ≤ ε -α and |D 2 Φ(x)| ≤ ε -α .
2. Mark chooses the new position y ∈ B R (x).

Helen's score is increased by

Φ(x) -Φ(y) -εF (t, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) .
Time is reset to t + ε. Then we repeat the previous steps until time is larger than T . At that time, Helen collects the bonus u T (x), where x is the current position of the game.

Theorem and comments

It is possible to construct a repeated game that approximates a PIDE where F also depends on u itself, but its formulation is a bit more complicated. This is important from the point of view of applications but since, with the previous game at hand, ideas from [START_REF] Kohn | A deterministic-control-based approach to fully non-linear parabolic and elliptic equations[END_REF] can be applied readily, we prefer to present it in this simpler framework.

The dynamic programming principle is, in this case, (2.12)

u ε (t, x) = sup Φ∈C 2 (R N ) Φ ∞ ,|DΦ(x)|,|D 2 Φ(x)|≤ε -α inf y∈B R (x) u ε (t + ε, y) + Φ(x) -Φ(y) -εF (t, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) .
We will give below an easy formal argument that allows to predict the following convergence result.

Theorem 1. Assume that F is elliptic and satisfies (A0) and (A1). Assume also that u T ∈ W 2,∞ (R N ). Then the upper (resp. lower) relaxed semi-limit u (resp. u) of (u ε ) ε>0 is a sub-solution (resp. super-solution) of (2.2) and

u(T, x) ≤ u T (x) ≤ u(T, x) .
In particular, if F also satisfies (A2), (A3), then u ε converges locally uniformly in R × R N towards the viscosity solution u of (2.2), (2.5) as ε → 0 and R → +∞ successively. Remark 2.2. As we mentioned, the second statement follows from the fact that (A2)-(A3) together with (A0) imply that the comparison principle for (2.2) holds true in the class of bounded functions.

Remark 2.3. We are in fact going to prove that under the same assumptions, u ε converges locally uniformly in R × R N as ε → 0 towards the viscosity solution of (2.2), (2.5) where I is replaced with the truncated integral operator. Theorem 1 is then a direct consequence of this fact by using stability results such as the ones proved in [START_REF] Barles | Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited[END_REF].

Remark 2.4. We assume that u T lies in W 2,∞ (R N ) for simplicity but one can consider terminal data that are much less regular, for instance bounded and uniformly continuous. However, this implies further technicalities that we prefer to avoid here.

Formal argument for Theorem 1. We assume that u ε is smooth. It is enough to understand why the following equality holds true

(2.13) u ε (t, x) = u ε (t + ε, x) -εF (t, x, Du ε (t + ε, x), D 2 u ε (t + ε, x), I[x, u ε (t + ε, •)]) + o(ε).
Indeed, after rearranging terms, dividing by ε and passing to the limit, we get

-∂ t u(t, x) + F (t, x, Du(t, x), D 2 u(t, x), I[x, u(t, •)]) = 0 .
It is easy to see that if Helen chooses Φ = u ε (t + ε, •), Mark cannot change the score by acting on y. Indeed, the dynamic programming principle implies that u ε (t, x) is larger than the right-hand side of (2.13).

It turns out that it is optimal for Helen to choose Φ = u ε (t + ε, •). In other words, the converse inequality holds true (and thus (2.13) holds true too). To see this, the dynamic programming principle tells us that it is enough to prove that, for Φ ∈ C 2 (R N ) fixed (with proper bounds), we have inf y∈B(x,R)

{u ε (t + ε, y) + Φ(x) -Φ(y) -εF (t, x, DΦ(x), D 2 Φ(x), I[x, Φ])} ≤ u ε (t + ε, x) -εF (t, x, Du ε (t + ε, x), D 2 u ε (t + ε, x), I[x, u ε (t + ε, •)]) + o(ε).
The following crucial lemma permits to conclude. We recall that we assume that the singular measure is supported in B(0, R) for some R > 0.

Lemma 2.1 (Crucial lemma for PIDE). Let F be continuous and Φ, ψ ∈ C 2 (R N ) be two bounded functions. Let K be a compact subset of R N and let x ∈ K. For all ε > 0, there exists y = y ε ∈ B R (x) such that

(2.14) ψ(y) + Φ(x) -Φ(y) -εF (t, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) ≤ ψ(x) -εF (t, x, Dψ(x), D 2 ψ(x), I R [x, ψ]) + o(ε)
where the o(ε) depends on F, ψ, Φ and K but not on t, x, y.

The rigourous proof of this lemma is postponed until Subsection 3.1. However, we can motivate this result by giving a (formal) sketch of its proof. Assume that the conclusion of the lemma is false. Then there exists η > 0 and we have for all y ∈ K ψ(y)ψ(x) > Φ(y) -Φ(x) + ε(F (. . . ) -F (. . . )) + ηε.

In particular, ψ(y)ψ(x) > Φ(y) -Φ(x) + O(ε). This implies (at least formally)

Dψ(x) = DΦ(x) + o(1) D 2 ψ(x) ≤ D 2 Φ(x) + o(1) I[x, ψ] ≤ I[x, Φ] + o(1).
Then the ellipticity of F implies that F (. . . ) -F (. . . ) ≥ o(1) and we get the following contradiction: 0 ≥ ηo(1) + ηε.

One can observe that this very simple game is a natural generalization of the game constructed in [START_REF] Kohn | A deterministic-control-based approach to fully non-linear parabolic and elliptic equations[END_REF] for fully non-linear parabolic equations. Indeed, if F does not depend on I[Φ], then all is needed is proxies for DΦ(x), D 2 Φ(x). So instead of choosing a whole function Φ, Helen only needs to choose a vector p (proxy for DΦ(x)) and a symmetric matrix Γ (proxy for D 2 Φ(x)), and replace Φ(y) -Φ(x) in the score updating by its quadratic approximation

p • (y -x) + 1 2 Γ(y -x), (y -x) .
One then recovers the game of [START_REF] Kohn | A deterministic-control-based approach to fully non-linear parabolic and elliptic equations[END_REF] (except there y is constrained to B ε 1-α (x)). Of course it is natural that for a non-local equation, local information at x does not suffice and information in the whole space needs to be collected at each step.

Level-set approach to geometric motions

Before stating our results for the geometric flows (eikonal equations and integral curvature flow), we recall the level set framework for such geometric evolutions. The level-set approach [START_REF] Osher | Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations[END_REF][START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF][START_REF] Evans | Motion of level sets by mean curvature. I[END_REF] consists in defining properly motions of interfaces associated with geometric laws. More precisely, given an initial interface Γ 0 , i.e. the boundary of a bounded open set Ω 0 , their time evolutions {Γ t } t>0 and {Ω t } t>0 are defined by prescribing the velocity V of Ω t at x ∈ Γ t along its normal direction n(x) as a function of time t, position x, normal direction n(x), curvature tensor Dn(x), or even the whole set Ω t at time t. The geometrical law thus writes

(2.15) V = G(t, x, n(x), Dn(x), Ω t ) .
The level-set approach consists in describing Γ 0 and {Γ t } t>0 as zero-level sets of continous functions u 0 (such as the signed distance function to Γ 0 ) and u(t, •) respectively

Γ 0 = {x ∈ R N : u 0 (x) = 0} and Ω 0 = {x ∈ R N : u 0 (x) > 0} Γ t = {x ∈ R N : u(t, x) = 0}
and

Ω t = {x ∈ R N : u(t, x) > 0} .
The geometric law (2.15) translates into a fully non-linear parabolic equation for u:

(2.16)

∂ t u = G(t, x, Du, (I -Du ⊗ Du)D 2 u, Ω t )|Du| := -F (t, x, Du, D 2 u, Ω t ) (where p = |p| -1 p for p ∈ R N , p = 0) supplemented with the initial condition u(0, x) = u 0 (x).
If proper assumptions are made on the nonlinearity F , the level-set approach is consistent in the sense that, for two different initial conditions u 0 and v 0 with the same 0-level set, the associated (viscosity) solutions u and v have the same zero-level sets at all times as well.

The interested reader is referred to [START_REF] Osher | Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations[END_REF][START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF][START_REF] Evans | Motion of level sets by mean curvature. I[END_REF] for fundamental results, [START_REF] Barles | Front propagation and phase field theory[END_REF] for extensions and [START_REF] Souganidis | Front propagation: theory and applications[END_REF] for a survey paper.

In the present paper, we deal with terminal conditions instead of initial conditions. This is the reason why, for a given terminal time T > 0, we consider the equation -∂ t u + F = 0 supplemented with the terminal condition (2.5). We will consider two special cases of (2.16)

• the eikonal equation (2.17) -∂ t u -v(x)|Du| = 0
• and the integral curvature equation

(2.18) -∂ t u -κ[x, u]|Du| = 0
where κ[x, u] is the integral curvature of u at x (see below for a definition).

Eikonal equation

The first geometric law (2.15) we are interested in is the simple case where V = v(x) and v : R N → R is a Lipschitz continuous function and we do not assume that it has a constant sign. In this case, the geometric equation (2.16) reduces to the standard eikonal equation (2.17).

The solution of an eikonal equation can be represented as the value function of a deterministic control problem when v has a constant sign [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF]. If v changes sign, it can be represented as the value function of a deterministic differential game problem, i.e., loosely speaking, a control problem with two opposing players [START_REF] Evans | Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations[END_REF].

We recall the definition of a viscosity solution to the eikonal equation (2.17).

Definition 2 (Viscosity solution for (2.17)). Given a function u : (0, T ) × R N → R, we say that 1. It is a viscosity sub-solution of (2.17) if it is upper semi-continuous and if for every test-function φ ∈ C 2 such that uφ admits a local maximum at (t, x) ∈ (0, T ) × R N , we have

(2.19) -∂ t φ(t, x) -v(x)|∇φ|(t, x) ≤ 0.

It is a viscosity super-solution of (2.17) if it is lower semi-continuous and if for every test-function

φ ∈ C 2 such that u -φ admits a local minimum at (t, x) ∈ (0, T ) × R N , we have (2.20) -∂ t φ(t, x) -v(x)|∇φ|(t, x) ≥ 0.

It is a viscosity solution of (2.17) if it is both a sub and super-solution.

Here, we construct a semi-discrete approximation by discretizing time. We use two opposing players Paul and Carol: Paul can take advantage of v ≥ 0 to move, while when v ≤ 0 it is Carol who takes advantage to move (in the opposite direction). We recall that (•) + denotes the positive part and (•) -the negative part of a quantity.

The game for the eikonal equation

In order to describe the game, we introduce the following cut-off function: for ε > 0 and r > 0, we define

(2.21) C ε (r) = (r ∨ ε 3 2 ) ∧ ε 1 2 =    ε 3 2 if 0 < r < ε 3 2 , r if ε 3 2 < r < ε 1 2 , ε 1 2 if r > ε 1 2 .
This function is non-decreasing and for every r we have ε

3 2 ≤ C ε (r) ≤ ε 1 2 .
Game 2 (Eikonal equation). At time t ∈ (0, T ), Paul starts at x with zero score. His objective is to get the highest final score.

1. Either B ε (x) ∩ {v > 0} = ∅, then Paul chooses a point x P ∈ B ε (x) ∩ {v > 0} and time gets reset to t P = t + C ε [ε(v + (x P )) -1 ]. Or B ε (x) ∩ {v > 0} = ∅,
then Paul stays at x P = x and time gets reset to

t P = t + ε 2 . 2. Either B ε (x P ) ∩ {v < 0} = ∅, then Carol chooses a point x C ∈ B ε (x P ) ∩ {v < 0} and time gets reset to t C = t P + C ε [ε(v -(x C )) -1 ].
Or B ε (x P ) ∩ {v < 0} = ∅, then Paul stays at x C = x P and time gets reset to t C = t P + ε 2 .

The players repeat the two previous steps until

t C ≥ T . Paul's final score is u T (x C
) where x C s the final position of the game.

Result and remarks

The previous game can be translated as follows: let for short E + and E -denote the sets

(2.22) E ± (x) = B ε (x) ∩ {±v > 0} if B ε (x) ∩ {±v > 0} = ∅ {x} if not. With this notation (2.23) u ε (t, x) = sup x P ∈E + (x) inf x C ∈E -(x P ) {u ε (t C , x C )} , where (2.24)
       t P = t + C ε [ε(v + (x P )) -1 ] if B ε (x) ∩ {v > 0} = ∅ ε 2 if not t C = t P + C ε [ε(v -(x C )) -1 ] if B ε (x P ) ∩ {v < 0} = ∅ ε 2 if not and u ε (t, x) = u T (x) if t ≥ T .
We will refer to (2.23) as the dynamic programming principle for Game 2 even if only one time step is considered. We next claim that the following convergence result holds true; again the limiting equation can be predicted by a formal argument from (2.23) (see below).

Theorem 2. Assume that v is Lipschitz continuous and u T is bounded and Lipschitz continuous. Then the function u ε converges locally uniformly as ε → 0 towards the unique viscosity solution of (2.17), (2.5).

Remark 2.5. Let us mention that the parameters α = 1 2 and β = 3 2 in the definition of C ε (2.21) really only need to satisfy 1 < α < β < 2.

We next give the formal argument which permits to predict the convergence result for the eikonal equation.

Formal argument for Theorem 2. We first rewrite the dynamic programming principle as follows 0 = sup

x P ∈E + (x) u ε (t P , x P ) -u ε (t, x) + inf x C ∈E -(x P ) {u ε (t C , x C ) -u ε (t P , x P )} .
We only treat the case v(x) > 0 because the argument is completely analogous in the case v(x) < 0. Hence, for ε small enough, B ε (x) ⊂ {v > 0}, B ε (x P ) ∩ {v < 0} = ∅ and (t C , x C ) = (t P + ε 2 , x P ). The previous equality then yields (approximating C ε (r) by r)

0 = sup x P ∈Bε(x) u ε (t P , x P ) -u ε (t, x) + O(ε 2 ) = sup x P ∈Bε(x) ∂ t u ε (t, x)(t P -t) + Du ε (t, x)(x P -x) + o(ε) = ε v(x P ) (∂ t u ε (t, x) + v(x P )|Du ε (t, x)|) + o(ε) = ε v(x P ) (∂ t u ε (t, x) + v(x)|Du ε (t, x)|) + o(ε) .
Hence, dividing by ε/v(x P ) and letting ε → 0, we obtain formally

∂ t u(t, x) + v(x)|Du|(t, x) = 0 .

Integral curvature flow 2.4.1 Definitions

Even if the authors do not use this word, the notion of integral curvature is considered in papers such as [START_REF] Da Lio | Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics[END_REF][START_REF] Forcadel | Homogenization of some particle systems with two-body interactions and of the dislocation dynamics[END_REF][START_REF] Caffarelli | Convergence of Nonlocal Threshold Dynamics Approximations to Front Propagation[END_REF][START_REF] Imbert | Level set approach for fractional mean curvature flows[END_REF][START_REF] Caffarelli | Non local minimal surfaces[END_REF]. Here is the definition we will take. Consider a function K : R N → (0, +∞) such that (2.25)

             K is even, supported in B R (0) K ∈ W 1,1 (R N \ B δ (0)) for all δ > 0 B δ (0) K = o 1 δ Q(r,e) K < +∞ for all r > 0, e ∈ S N -1 Q(r,e) K = o 1 r
where Q(r, e) is a paraboloid defined as follows

Q(r, e) = {z ∈ R N : r|z • e| ≤ |z -(z • e)e| 2 }.
Interesting examples of such K's include

K(z) = C(z) 1 + |z| N +α or K(z) = C(z) |z| N +α
for some cut-off function C : R N → R which is even, smooth and supported in B R (0). Remark 2.6. It is not necessary to assume that K has a compact support in order to define the non-local geometric flow. However, we need this assumption in order to construct the game and prove that it approximates the geometric flow. We can then later follow what we did when dealing with PIDE: approximate any integral curvature flow by first approximating K by kernels K R compactly supported in B R (0) and by taking next the limit of the corresponding value functions as ε → 0 and R → ∞ respectively.

Consider U ∈ C 2 such that DU(x) = 0. We define

κ * [x, U] = K * 1 {U ≥U (x)} -K * 1 {U <U (x)} κ * [x, U] = K * 1 {U >U (x)} -K * 1 {U ≤U (x)} .
These functions coincide if for instance DU = 0 on {U = U(x)}. They define the integral curvature of the "hypersurface" {U(z) = U(x)} at the point x. The reader can notice that this "hypersurface" is oriented via the sign of the function U. The classical curvature can be recovered if K(z) = 1-α |z| N+α and α → 1, α < 1; see [START_REF] Imbert | Level set approach for fractional mean curvature flows[END_REF]. Functions κ * and κ * enjoy the following properties (see [START_REF] Imbert | Level set approach for fractional mean curvature flows[END_REF]):

{U ≥ U(x)} ⊂ {V ≥ V (x)} ⇒ κ * [x, U] ≤ κ * [x, V ], (2.28) {U > U(x)} ⊂ {V > V (x)} ⇒ κ * [x, U] ≤ κ * [x, V ].
We next make precise the notion of viscosity solutions for (2.18), (2.5) that will be used in the present paper.

Definition 3 (Viscosity solutions for (2.18)). Given a function u : (0, T ) × R N → R, we say that 1. It is a sub-solution of (2.18) if it is upper semi-continuous and if for every test-function φ ∈ C 2 such that uφ admits a strict maximum in (0, T ) × B R+1 (x) at (t, x), we have

(2.29) -∂ t φ(t, x) -κ * [x, φ(t, •)]|Dφ|(t, x) ≤ 0 if Dφ(t, x) = 0 and -∂ t φ(t, x) ≤ 0 if Dφ(t, x) = 0;
2. It is a super-solution of (2.18) if it is lower semi-continuous and if for every testfunction φ ∈ C 2 such that uφ admits a strict minimum in (0, T ) × B R+1 (x) at (t, x), we have

(2.30) -∂ t φ(t, x) -κ * [x, φ(t, •)]|Dφ|(t, x) ≥ 0 if Dφ(t, x) = 0 and -∂ t φ(t, x) ≥ 0 if Dφ(t, x) = 0;
3. It is a solution of (2.18) if it is both a sub and super-solution.

It is proved in [START_REF] Imbert | Level set approach for fractional mean curvature flows[END_REF] that a comparison principle holds true for such super-and subsolutions.

The game for the integral curvature equation

In the following game, the two players choose successively hypersurfaces and points on these hypersurfaces or close to them. In the present paper, a hypersurface refers to the 0-level set of a smooth function φ. We recall the definition of the cut-off function we considered in the repeated game for the eikonal case.

(2.31)

C ε (r) = (r ∨ ε 3 2 ) ∧ ε 1 2 =    ε 3 2 if 0 < r < ε 3 2 , r if ε 3 2 < r < ε 1 2 , ε 1 2 if r > ε 1 2 .
We also recall that R is the size of the support of κ as in (2.25).

Game 3 (Integral curvature equation). At time t ∈ (0, T ), Paul starts at x with zero score. His objective is to get the highest final score.

Paul chooses a point x +

P ∈ B ε (x) and a hypersurface Γ + passing through x + P defined by

Γ + = {z ∈ R N : φ + (z) = φ + (x + P )} with φ + ∈ C 2 (R N ), oriented through the requirement φ + (x) ≤ φ + (x + P ). • If Dφ + (x + P ) = 0 and κ * [x + P , Γ + ] > 0, Carol chooses the new position point x + C in the half-space delimited by Γ + i.e. in {z ∈ B R (x + P ) : φ + (z) ≥ φ + (x + P )}. Time gets reset to t + = t + C ε (εκ * [x + P , Γ + ] -1 ). • If Dφ + (x + P ) = 0 or κ * [x + P , Γ + ] ≤ 0, then the game stays at x: x + C = x. Time gets reset to t + = t + ε 2 .

From the new position x +

C and time t + determined above, Carol chooses a point

x - C ∈ B ε (x + C ) and a hypersurface Γ -passing through x - C defined by Γ -= {z ∈ R N : φ -(z) = φ -(x - C )} with φ -∈ C 2 (R N ), and oriented through the requirement φ -is such that φ -(x + C ) ≥ φ -(x - C ). • If Dφ -(x - C ) = 0 and κ * [x - C , Γ -] < 0, Paul chooses the new position point x - P in the half-space delimited by Γ -i.e. in {z ∈ B R (x - C ) : φ -(z) ≤ φ -(x - C )}. Time gets reset to t -= t + + C ε (ε|κ * [x - C , Γ]| -1 ). • If Dφ -(x - C ) = 0 or κ * [x - C , Γ -] ≥ 0, then the game stays at x + C (x - C = x + C
) and time gets reset to t -= t + + ε 2 .

3. Then previous steps are repeated as long as t -< T . Paul's final score is u T (x - P ). Remark in particular that in Step 1, the value of the function φ + is successively increased while in Step 2, the value of the function φ -is successively decreased. Precisely,

φ + (x) ≤ φ + (x + P ) ≤ φ + (x + C ) , φ -(x + C ) ≥ φ -(x - C ) ≥ φ -(x - P ) .

Results and remarks

In order to state the dynamic programming principle, we first introduce admissible sets of points and half-spaces for both players. Precisely, we consider

(2.32) C ± (x) = {(y, ϕ) ∈ B ε (x) × C 2 (R N ) : ±ϕ(y) ≥ ±ϕ(x)}, (2.33)
P + (x, y, ϕ) = {z ∈ B R (y) : ϕ(z) ≥ ϕ(y)} if Dϕ(y) = 0 and κ * [y, ϕ] > 0 {x} if not, (2.34)
P -(x, y, ϕ) = {z ∈ B R (y) : ϕ(z) ≤ ϕ(y)} if Dϕ(y) = 0 and κ * [y, ϕ] < 0 {x} if not.
Hence, the dynamic programming principle associated to the game is (2.35)

u ε (t, x) = sup (x + P ,φ + )∈C + (x) inf x + C ∈P + (x,x + P ,φ + ) inf (x - C ,φ -)∈C -(x + C ) sup x - P ∈P -(x + C ,x - C ,φ -) u ε t -, x - P where (2.36)        t + = t + C ε (εκ * [x + P , Γ + ] -1 ) if Dφ + (x + P ) = 0 and κ * [x + P , Γ + ] > 0, ε 2
if not,

t -= t + + C ε (ε|κ * [x - C , Γ -]| -1 ) if Dφ -(x - C ) = 0 and κ * [x - C , Γ -] < 0, ε 2 if not.
The last main result is

Theorem 3. Assume that u T ∈ W 2,∞ (R N ).
Then the sequence u ε converges locally uniformly as ε → 0 towards the unique viscosity solution of (2.16), (2.5).

Remark 2.7. To avoid further technicalties, we assume that the terminal datum is very regular; it can be shown that the result still holds true for much less regular functions u T such as bounded uniformly continuous ones. This extension is left to the reader.

Open problems

As we mentioned above, the games we have constructed have much more complicated rules than the Paul-Carol game for mean curvature flow. It is then a natural open problem to find simpler games and in particular a game for the integral curvature flow associated with the singular measure ν(dz) = (1α)dz/|z| N +α which converges (in some sense) as α → 1 to the original Paul and Carol game. The reason to look for such as game is that it is known [START_REF] Imbert | Level set approach for fractional mean curvature flows[END_REF] that the integral curvature flow converges towards the mean curvature flow as α → 1.

The same question can be raised for the fractional Laplacian operators in the situation of PIDE's: find a game a natural game associated to ∆ α operators, which coincides with a natural game for α = 1.

3 Proofs of convergence results

Proof of Theorem 1

As explained in Remark 2.3, it is enough to prove the convergence as ε → 0 by assuming that ν is supported in B R for some fixed R > 0.

For fixed ε > 0 and x ∈ R N , the value function u ε (t, x) is finite for t close to T thanks to the following lemma. Proposition 3.2 below is needed to prove that u ε (t, x) is finite for all t ∈ (0, T ).

Lemma 3.1 (The functions u ε are well defined). For all Φ ∈ C 2 (R N ) such that We thus get from (A1) and (A3)

(3.1) Φ ∞ ≤ ε -α , |DΦ(x)| ≤ ε -α , |D 2 Φ(x)| ≤ ε -α ,
-εF (t, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) ≤ Cε(1 + ε -αk 1 + ε -αk 2 + ε -α )
and the lemma follows at once.

Let us define as usual the semi-relaxed limits u = lim inf * ε→0 u ε and u = lim sup * ε→0 u ε . Theorem 1 will follow from the following two propositions.

Proposition 3.1. The functions u and u are finite and are respectively a super-solution and a sub-solution of (2.2). Proposition 3.2. Given R > 0, there exists a constant C > 0 such that for all ε > 0, all (t, x) ∈ (0, T ) × B R , we have

|u ε (T -t, x) -u T (x)| ≤ Ct .
In particular, u and u are finite and they satisfy at time t = T and for all x ∈ R N u(T, x) = u(T, x) = u T (x) .

These two propositions together with the comparison principle imply that u = u, i.e. u ε converges locally uniformly towards a continuous function denoted u. This implies that u is a (continuous) viscosity solution of (2.2) satisfying (2.5). This finishes the proof of the theorem.

We now prove the two propositions.

Proof of Proposition 3.1. We use the general method proposed by Barles and Souganidis [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF] in order to prove that u is a super-solution of (2.2). This is the reason why, given a function U : R N → R, we introduce

S ε [U](t, x) = sup Φ∈C 2 (R N ) Φ ∞,|DΦ(x)|,|D 2 Φ(x)|≤ε -α inf y∈B R (x) U(y) + Φ(x) -Φ(y) -εF (t, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) .
The two important properties of S ε are:

(3.2) it commutes with constants:

S ε [U + C] = R ε [U] + C for any constant C ∈ R; (3.3) it is monotone: if U ≤ V then S ε [U] ≤ S ε [V ].
The dynamic programming principle (2.12) is rewritten as follows

(3.4) u ε (t, x) = S ε [u ε (t + ε, •)](t, x).
We now explain how to prove that u is a super-solution of (2.2). The case of u is proven analogously thanks to a "consistency lemma" (see below Lemma 3.2). Following Definition 1 and Remark 2.1, consider a C 2 bounded test function φ and a point (t 0 , x 0 ) with t 0 > 0 such that uφ admits a strict minimum 0 at (t 0 , x 0 ) on V 0 = (0, T ) × B R+1 (x 0 ). By definition of u, there exists (τ ε , y ε ) such that (τ ε , y ε ) → (t 0 , x 0 ) and u ε (τ ε , y ε ) → u(t 0 , x 0 ) as ε → 0, up to a subsequence. Let then (t ε , x ε ) be a point of minimum of u εφ on V 0 . We have

u ε (t ε , x ε ) -φ(t ε , x ε ) ≤ u ε (τ ε , y ε ) -φ(τ ε , y ε ) → u(t 0 , x 0 ) -φ(t 0 , x 0 ) = 0
hence by definition of u and (t 0 , x 0 ) as a strict local minimum, we conclude that we must have (t ε , x ε ) → (t 0 , x 0 ) as ε → 0. In addition, for all (t, x) ∈ V 0 , we have

u ε (t, x) ≥ φ(t, x) + (u ε (t ε , x ε ) -φ(t ε , x ε )) =: φ(t, x) + ξ ε .
In particular, if ε is small enough, this inequality holds true on (0, T ) × B R (x ε ). From the definition of S ε , the dynamic programming principle (3.4), and the properties (3.2)-(3.3), the previous inequality implies

u ε (t ε , x ε ) = S ε [u ε (t ε + ε, •)](t ε , x ε ) ≥ S ε [φ(t ε + ε, •) + ξ ε ](t ε , x ε ) = S ε [φ(t ε + ε, •)](t ε , x ε ) + ξ ε . Since u ε (t ε , x ε ) = φ(t ε , x ε ) + ξ ε , we get φ(t ε , x ε ) ≥ S ε [φ(t ε + ε, •)](t ε , x ε ).
We claim Proposition 3.1 is proved if the following lemma holds true. Lemma 3.2 (Consistency lemma for PIDE). Consider a C 2 bounded test function ψ. Given a compact subset K of (0, T ) × R N , there then exists a function o(ε) such that o(ε)/ε → 0 as ε → 0 and for all (t, x) ∈ K, we have

S ε [ψ](t, x) = ψ(x) -εF (t, x, Dψ(x), D 2 ψ(x), I R [x, ψ]) + o(ε).
Indeed, applying this lemma to ψ = φ(t ε + ε, •), we are led to

φ(t ε , x ε ) ≥ φ(t ε + ε, x ε ) -εF (t ε , x ε , Dφ(t ε + ε, x ε ), D 2 φ(t ε + ε, x ε ), I R [x ε , φ(t ε + ε, •)]) + o(ε) .
Dividing by ε and letting ε → 0 we obtain (using the C 2 character of φ and continuity of F ), -∂ t φ(t 0 , x 0 ) + F (t 0 , x 0 , Dφ(t 0 , x 0 ), D 2 φ(t 0 , x 0 ), I R [x 0 , φ(t 0 , •)]) ≥ 0.

This allows us to conclude that u is a viscosity super-solution of (2.2). The proof that u is a sub-solution is analogous.

We now turn to the proof of the consistency lemma, i.e. Lemma 3.2.

Proof of Lemma 3.2. This lemma easily follows from Lemma 2.1. Indeed, Lemma 2.1 implies that for ψ ∈ C 2 (R N ), there exists o(ε) depending on ψ such that

S ε [ψ](t, x) ≤ ψ(x) -εF (t, x, Dψ(x), D 2 ψ(x), I R [x, ψ]) + o(ε).
On the other hand, by choosing Φ = ψ in the definition of S ε , we immediately get

S ε [ψ](t, x) ≥ ψ(x) -εF (t, x, Dψ(x), D 2 ψ(x), I R [x, ψ]) .
Combining the two previous inequalities yield the desired result.

We next turn to the proof of the crucial Lemma, i.e. Lemma 2.1.

Proof of Lemma 2.1. By contradiction assume this is wrong, hence there exists η > 0 and ε n → 0 such that for all y ∈ B R (x),

(3.5) Φ(x) -ψ(x) -Φ(y) + ψ(y) > -ε n F (t, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) -F (t, x, Dψ(x), D 2 ψ(x), I R [x, ψ]) + ηε n .
In order to simplify notation, ε n is simply denoted by ε.

Let us first take y = x + ε 1 2 w with w = 1. Inserting into (3.5) and using the C 2 character of Φ and ψ gives -ε

1 2 D(Φ -ψ)(x) • w ≥ -Cε
where C depends only on Φ, ψ, F, x and not on ε. Dividing by ε 1 2 , and using the fact that this is true for every w ∈ R N with w = 1, we find that

(3.6) |D(Φ -ψ)(x)| ≤ Cε 1 2
.

Using now y = x + ε 1 3 w and doing a second order Taylor expansion of Φψ, we find

-ε 1 3 D(Φ -ψ)(x) • w - ε 2 3 2 D 2 (Φ -ψ)(x)w, w ≥ O(ε)
and using (3.6) we obtain

D 2 (Φ -ψ)(x)w, w ≤ O(ε 1 6
) .

Since this is true for any w of norm 1, we deduce that

(3.7) |D 2 (Φ -ψ)(x)| ≤ O(ε 1 6
)

where |A| = λ max (A) denotes here the largest eigenvalue of a symmetric matrix. Finally setting y = x + z in (3.5) with z ∈ B R (0) gives

Φ(x + z) -Φ(x) ≤ ψ(x + z) -ψ(x) + Cε
where the constant C again depends on x, Φ, ψ, F but not on z. Integrating this inequality with respect to z's such that γ ≤ |z| ≤ R (with γ > 0 to be chosen later), we obtain

|z|≥γ (Φ(x + z) -Φ(x))ν(dz) ≤ |z|≥γ (ψ(x + z) -ψ(x))ν(dz) + Cεν(|z| ≥ γ) .
By (2.4), we know that γ 2 ν(|z| ≥ γ) ≤ C ν where C ν is a constant that only depends on ν, we conclude that

(3.8) |z|≥γ (Φ(x + z) -Φ(x))ν(dz) ≤ |z|≥γ (ψ(x + z) -ψ(x))ν(dz) + Cεγ -2
where C depends on x, Φ, ψ, F and ν (we do not change the name of the constant). On the other hand, by using the C 2 regularity of Φ and ψ, we obtain

|z|≤γ (Φ(x + z) -Φ(x) -DΦ(x) • z)ν(dz) ≤ |z|≤γ (ψ(x + z) -ψ(x) -Dψ(x) • z)ν(dz) + 1 2 |D 2 Φ(x) -D 2 ψ(x)| + Cγ |z|≤γ |z| 2 ν(dz)
where C only depends on Φ and ψ. Now choosing γ = ε 1 6 and using (3.7), we get

|z|≤γ (Φ(x + z) -Φ(x) -DΦ(x) • z)ν(dz) ≤ |z|≤γ (ψ(x + z) -ψ(x) -Dψ(x) • z)ν(dz) + Cε 1 6
where C depends only on Φ, ψ, x, F, ν. Finally, from (3.7), we have

ε 1/6 ≤|z|≤1 (-DΦ(x) • z + Dψ(x) • z)ν(dz) ≤ Cε 1/6 .
Combining the above estimates with (3.8), we conclude that

(3.9) I R [x, Φ] ≤ I R [x, ψ] + Cε 1 6
where C depends on Φ, ψ, F, x and ν. Combining (3.6), (3.7) and (3.9), the continuity of F and its monotonicity condition (2.6) yield that (3.10)

F (t, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) -F (t, x, Dψ(x), D 2 ψ(x), I R [x, ψ]) ≥ o(1) .
Inserting this into (3.5) and choosing y = x, we find 0 > εo(1) + ηε a contradiction for ε small enough. Hence the lemma is proved.

We conclude the proof of the convergence theorem (Theorem 1) by proving that the terminal condition is satisfied (Proposition 3.2).

Proof of Proposition 3.2. It is enough to prove that for some constant C > 0 and for all k ∈ N, we have

(3.11) ∀(t, x) ∈ (0, T ) × R N , |u ε (T -kε, x) -u T (x)| ≤ Ckε .
We argue by induction. The relation (3.11) is clear for k = 0. We assume it is true for k and we prove it for k + 1.

We first consider a family u η T of bounded C 2 functions such that u η Tu T W 2,∞ ≤ η. From the one-step dynamic programming principle (2.12) and the choice Φ = u η T , we easily deduce that

u ε (T -(k + 1)ε, x) ≥ inf y∈B R (x) {u ε (T -kε, y) + u η T (x) -u η T (y) -εF (T -(k + 1)ε, x, Du η T (x), D 2 u η T (x), I R [x, u η T ])} ≥ inf y∈B R (x) {u ε (T -kε, y) -u T (y)} + u T (x) -C 1 ε -2η ≥ -Ckε + u T (x) -C 1 ε -2η
where we used (3.11) and we chose

C 1 = max{F (t, x, p, A, l) : t ∈ (0, T ), x ∈ B R , |p| + |A| ≤ 2 u T W 2,∞ , |l| ≤ C ν u T W 2,∞ } + 1 .
Changing C if necessary in (3.11) we can assume that C ≥ C 1 . Since η is arbitrary, we easily get an estimate from below:

u ε (T (k + 1)ε, x) -u T (x) ≥ -C(k + 1)ε.
Using once again the one-step dynamic programming principle (2.12) and (3.11), we next get

(3.12) u ε (T -(k + 1)ε, x) ≤ sup Φ∈C 2 (R N ) Φ ∞,|DΦ(x)|,|D 2 Φ(x)|≤ε -α inf y∈B R (x) u η T (y) + Φ(x) -Φ(y) -εF (T -(k + 1)ε, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) + Ckε + η .
In order to get the upper bound in (3.11), we use the consistency Lemma 3.2. Applying it to Φ, ψ = u η T , t = T -(k + 1)ε, we get from (3.12)

u ε (T -(k + 1)ε, x) ≤ u T (x) -ε(F (T -(k + 1)ε, x, Du η T , D 2 u η T (x), I R [x, u η T ]) + o(1)) +Ckε + 2η ≤ u T (x) + C 1 ε + Ckε + 2η ≤ u T (x) + C(k + 1)ε + 2η
and since η is arbitrary, the proof of the proposition is now complete.

Proof of Theorem 2

We first remark that for all ε > 0 and all (t, x) ∈ (0, T ] × R N , inf u T ≤ u ε (t, x) ≤ sup u T . We thus can consider the upper and lower relaxed limits u and u (they are finite) and we will prove below the following result.

Proposition 3.3. The functions u and u are respectively a super-solution and a sub-solution of (2.17).

In order to conclude that u ε converges towards the unique solution of (2.17), (2.5), it is then enough to prove that u(T, x) ≤ u T (x) ≤ u(T, x). This is an easy consequence of the following proposition whose proof is postponed too. Lemma 3.3. Given δ, R > 0 there exists C > 0 such that for all t ∈ (Tδ, T ] and all x ∈ B R (0)

(3.13) |u ε (t, x) -u T (x)| ≤ C(T -t + ε 1 2 ) u T Lip .
The comparison principle for (2.17) then permits to conclude.

It remains to prove Proposition 3.3 and Lemma 3.3. In order to do so, we proceed as in the proof of Proposition 3.1 by introducing an operator

S ε [φ]. More precisely, if φ : (0, T ] × R N → R is a bounded function, we let (3.14) S ε [φ](t, x) = sup x P ∈E + (x) inf x C ∈E -(x P ) φ(t C , x C )
where t C is defined by (2.24) and E ± is defined by (2.22). It is convenient to write

t P = t + T P (x, x P ) t C = t P + T C (x P , x C )
where T P and T C are defined as follows

T P (x, x P ) = C ε [ε(v + (x P )) -1 ] if B ε (x) ∩ {v > 0} = ∅ ε 2 if not , (3.15) T C (x P , x C ) = C ε [ε(v -(x C )) -1 ] if B ε (x P ) ∩ {v < 0} = ∅ ε 2 if not .
We also introduce

R ε [φ](t, x) = sup x P ∈E + (x) φ(t P , x P ) R ε [φ](t, x) = inf x C ∈E -(x P ) φ(t C , x C ).
The two important properties of R ε are:

it commutes with constants:

R ε [φ + c] = R ε [φ] + c for any constant c ∈ R; (3.16) it is monotone: if φ ≤ ψ then R ε [φ] ≤ R ε [ψ]. (3.17) We now rewrite S ε as S ε [φ] = R ε [R ε [φ]] .
We remark that R ε and S ε also commute with constants and are monotone. One can also observe that R ε and R ε have opposite values if v is changed into -v and φ into -φ.

Recall that the dynamic programming principle in this context is (2.23) i.e. with the new terminology

(3.18) u ε (t, x) = S ε [u ε ](t, x) = R ε [R ε [u ε ]](t, x).
The core of the proof of of Proposition 3.3 lies in the following "consistency lemma".

Lemma 3.4 (Consistency lemma for the eikonal equation). Consider a C 1 bounded smooth function φ : (0, T ] × R. Given r > 0 small enough and (t 0 , x 0 ) ∈ (0, Tr) × R N , there exists a function o(1) depending only on (ε, r), φ, (t 0 , x 0 ) and the speed function v such that o(1) → 0 as (ε, r) → 0, and the following holds: for all (t, x) ∈ B r (t 0 , x 0 ), there exists

x P , y P , x C , y C ∈ B ε (x), such that R ε [φ](t, x) -φ(t, x) ≤ T P (x, x P ) ∂ t φ(t, x) + v + (x)|Dφ(t, x)| + o(1) , (3.19) R ε [φ](t, x) -φ(t, x) ≥ T P (x, y) ∂ t φ(t, x) + v + (x)|Dφ(t, x)| + o(1) , (3.20) and R ε [φ](t, x) -φ(t, x) ≤ T C (x, y C ) ∂ t φ(t, x) + v -(x)|Dφ(t, x)| + o(1) , (3.21) R ε [φ](t, x) -φ(t, x) ≥ T C (x, x C ) ∂ t φ(t, x) + v -(x)|Dφ(t, x)| + o(1) . (3.22)
We can deduce from this lemma the following one Lemma 3.5 (Consistency lemma for the eikonal equation -second version). We consider a function φ : (0, T ] × R that is bounded and C 1 and (t 0 , x 0 ) ∈ (0, T ) × R N . There exist a function o(1) such that o(1) → 0 as (ε, r) → 0 and positive numbers m ε , M ε such that for all (t, x) ∈ B r (t 0 , x 0 ) and all (ε, r) small enough

(3.23) m ε ∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| + o(1) ≤ S ε [φ](t, x) -φ(t, x) ≤ M ε ∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| + o(1) .
The proofs of the two previous lemmas are postponed. We first explain how to derive Proposition 3.3 from Lemma 3.5.

Proof of Proposition 3.3. We are going show that u is a super-solution. Following Definition 2, let φ be a C 1 function such that uφ admits a minimum 0 at (t 0 , x 0 ) on V 0 = B δ (t 0 , x 0 ). Without loss of generality, we can assume that this minimum is strict, see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. Arguing as in the proof of Proposition 3.1, we deduce that u εφ admits a minimum at (t ε , x ε ) on V 0 with (t ε , x ε ) → (t 0 , x 0 ) as ε → 0; and for all (t, x) ∈ V 0

u ε (t, x) ≥ φ(t, x) + (u ε (t ε , x ε ) -φ(t ε , x ε )) := φ(t, x) + ξ ε
From the properties (3.16), (3.17) of S ε and the dynamic programming principle (3.18), we have

u ε (t ε , x ε ) ≥ S ε [φ + ξ ε ](t ε , x ε ) = S ε [φ](t ε , x ε ) + ξ ε . Since u ε (t ε , x ε ) = φ(t ε , x ε ) + ξ ε it follows that φ(t ε , x ε ) ≥ S ε [φ](t ε , x ε ).
Using Lemma 3.5 applied at (t ε , x ε ) we deduce the existence of m ε such that

m ε (∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| + o(1)) ≤ 0.
Dividing by m ε > 0 then letting ε → 0 and r → 0, we obtain

∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| ≤ 0.
We thus conclude that u is a super-solution. The proof that u is a sub-solution is entirely parallel and Proposition 3.3 is proved.

We now turn to the core of the argument, i.e.

Proof of Lemma 3.4. First we note that it suffices to prove (

follow by changing φ into -φ and v into -v. Consider (t, x) ∈ B r (t 0 , x 0 ) and x P an ε 3 -optimal position starting from x at time t i.e. such that x P ∈ E + (x) and

R ε [φ](t, x) = φ(t P , y) + O(ε 3 ) . First, remark that x P ∈ B ε (x) ⊂ B r+ε (x 0 ). In particular, |v(x P )| ≤ |v(x 0 )| + L v (r + ε) where L v is the Lipschitz constant of the function v. Hence, |v(y)| ≤ 1 ε 1 2
for ε small enough (only depending on L v , r and x 0 ). In particular in view of the definitions (2.21) and (3.15), T P (x, x P ) = ε 2 , T P (x, x P ) = εv + (x P ) -1 or T P (x, x P ) = ε 1 2 . We now distinguish these three cases.

Case T P (x, x P ) = ε 2 . This can only happen if x P = x, v + (x) = 0 and t P = t + ε 2 , so we may write, by Taylor expansion of φ,

R ε [φ](t, x) -φ(t, x) = φ(t P , x P ) -φ(t, x) + O(ε 3 ) = φ(t + ε 2 , x) -φ(t, x) + O(ε 3 ) = ε 2 (∂ t φ(t, x) + v + (x)|Dφ(t, x)| + o ε (1))
and we obtain the desired result (3.19)- (3.20).

Case T P (x, x P ) = ε 1 2 . This case happens only if 0 < v(x P ) ≤ ε 1 2 . This implies in particular that |v(x)| ≤ ε 1 2 + L v ε. Then we simply write R ε [φ](t, x) -φ(t, x) = φ(t + ε 1 2 , x P ) -φ(t, x) + O(ε) = ε 1 2 (∂ t φ(t, x) + o ε (1)) = ε 1 2 (∂ t φ(t, x) + v + (x)|Dφ(t, x)| + o ε (1))
and we obtain the desired result in this case too.

Case T P (x, x P ) = εv(x P ) -1 . Then v(x P ) ≥ ε 1 2 . This implies in particular that v(x) ≥ ε 1 2 -L v ε.
We write in this case, Taylor expanding φ again

R ε [φ](t, x) -φ(t, x) = φ(t P , x P ) -φ(t, x) + O(ε 3 ) = ε v(x P ) (∂ t φ(t, x) + o ε (1)) + (x P -x) • Dφ(t, x) + O(ε 2 ) .
Hence, we are done if Dφ(t, x) = 0. If not, we can write

R ε [φ](t, x) -φ(t, x) ≤ ε v(x P ) (∂ t φ(t, x) + o ε (1)) + ε|Dφ(t, x)| + O(ε 2 ) ≤ ε v(x P ) (∂ t φ(t, x) + v + (x)|Dφ(t, x)| + o ε (1)) .
To get the reversed inequality in this last case, we consider

y P = x + ε Dφ(t,x) |Dφ(t,x)| . Remark that v(y P ) ≥ v(x) -L v ε ≥ ε 1 2 -2L v ε
> 0 for ε small enough (only depending on L v ). Then, since y P ∈ E + (x), we have

R ε [φ](t, x) -φ(t, x) ≥ φ(t + T P (x, y P ), y P ) -φ(t, x) ≥ T P (x, y P )(∂ t φ(t, x) + o ε (1)) + ε|Dφ(t, x)| + O(ε 2 ) ≥ T P (x, y P )(∂ t φ(t, x) + ε T P (x, y P ) |Dφ(t, x)| + o ε (1)) ≥ T P (x, y P )(∂ t φ(t, x) + max(ε 1 2 , v(y P ))|Dφ(t, x)| + o ε (1)). If v(y P ) ≥ ε 1 2
, then we use the Lipschitz continuity of v + in order to get

R ε [φ](t, x) -φ(t, x) ≥ T P (x, y P )(∂ t φ(t, x) + v + (x)|Dφ(t, x)| + o ε (1)). If v(y P ) < ε 1 2 , then ε 1 2 -L v ε ≤ v(x) ≤ ε 1 2 + L v ε and we conclude that v(x) = o(1)
and the result is obtained in this case too.

Proof of Lemma 3.5. Recall that S ε [φ] = R ε [R ε [φ]]. We distinguish cases.
Case v(x 0 ) > 0. In this case, we can write v(x 0 ) ≥ 2δ 0 > 0. For ε small enough and r ≤ 1 2 , we have for all x P ∈ E + (x) ⊂ B ε (x), δ 0 ≤ v(x P ) ≤ v(x 0 ) + 1 and T P (x, x P ) = ε v(x P ) . We thus obtain from Lemma 3.4 for all (t, x) ∈ B r (t 0 , x 0 ), the existence of

x P ∈ B ε (x) such that R ε [φ](t, x) ≤ φ(t, x) + ε v(x P ) (∂ t φ(t, x) + v + (x)|Dφ(t, x)| + o(1)) ≤ φ(t, x) + ε δ 1 (∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| + o(1))
as (ε, r) → 0, where

δ 1 = δ 0 if ∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| > 0, v(x 0 ) + 1 if ∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| ≤ 0.
Since R ε commutes with constants and is monotone, the previous inequality implies the following one

S ε [φ](t, x) -φ(t, x) ≤ R ε [φ](t, x) -φ(t, x) + ε δ 1 (∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| + o(1)). Now from (3.21) we deduce R ε [φ](t, x) ≤ T C (x, y C ) (∂ t φ(t 0 , x 0 ) + o(1))
for some y C ∈ B ε (x) and since v(x 0 ) ≥ 2δ 0 > 0 we have B ε (x) ∩ {v < 0} = ∅ for r small enough and thus

T C (x, y C ) = ε 2 . It follows that S ε [φ](t, x) -φ(t, x) ≤ ε 2 ∂ t φ(t 0 , x 0 ) + o(ε 2 ) + ε δ 1 (∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| + o(1)) ≤ ε δ 1 (∂ t φ(t 0 , x 0 ) + v(x 0 )|Dφ(t 0 , x 0 )| + o(1)) .
which establishes the upper bound part in (3.23). The case v(x 0 ) < 0 is analogous.

Case v(x 0 ) = 0. By (3.19), we may write

R ε [φ](t, x) ≤ φ(t, x) + T P (x, x P )(∂ t φ(t, x) + o(1)) ≤ φ(t, x) + M ε,1 (∂ t φ(t, x) + o(1))
for some positive constant M ε,1 , since T P is bounded above and below by positive constants depending on ε (the last relation is obtained by discussing according to the sign of ∂ t φ(t, x)). From (3.21) we obtain similarly that

R ε [φ](t, x) ≤ φ(t, x) + M ε,2 (∂ t φ(t, x) + o(1))
for some positive M ε,2 . Combining the two relations we obtain

S ε [φ](t, x) -φ(t, x) ≤ (M ε,1 + M ε,2 )(∂ t φ(t, x) + o(1)).
The lower bound is entirely parallel, and the desired result follows in this case. The proof is now complete.

Proof of Lemma 3.3. It is enough to study the sequence of iterated positions and times starting from (t, x) ∈ (Tδ, T ] × B R (0). The dynamic programming principle (2.23) gives optimal positions x P , x C such that u ε (t, x) = u ε (t C , x C ) for the corresponding time t C . Letting t 0 = t and x 0 = x, and iterating this, we may define for k ∈ {1, . . . , K} optimal positions x k (corresponding to the x C ) and corresponding times t k with K such that t K ≥ T and t K-1 < T such that

u ε (t 0 , x 0 ) = u ε (t k , x k ) = u ε (t K , x K ) = u T (x K ). It follows that u ε (t, x) -u T (x) = u T (x K ) -u T (x) .
We conclude from the previous equality that, in order to prove (3.13), it is enough to prove that for any k ∈ {0, K -1}

(3.24) |x k+1 -x k | ≤ C(t k+1 -t k )
with C not depending on ε and (t, x) (but possibly on δ and R).

We notice that the supremum and the infimum defining u ε may not be attained. In this case, we simply choose first an ε 2 -optimal position, then an ε 3 -optimal position and iterating this, we obtain an error which is smaller than ε as soon as ε ≤ 1 2 . In order to prove such a result, we first remark that it suffices to prove that |x P -x| ≤ C(t Pt) and |x Cx P | ≤ C(t Ct P ) .

Hence, when x P = x and x C = x P , this is automatically satisfied. If not, we always have |x P -x| ≤ ε and |x Cx P | ≤ ε. Hence, we only need to check that for such time steps

C ε (ε|v(x P )| -1 ) ≥ C -1 ε and C ε (ε|v(x C )| -1 ) ≥ C -1 ε for C > 0 well chosen. This is equivalent to showing |v(x i )| ≤ C for i = P, C .
By Lipschitz continuity of v, there exists C R such that for all y ∈ B R+1 (0)

|v(y)| ≤ C R .
Recall that x ∈ B R (0). If the finite sequence (x k ) k=0,...,K remains in B R/2 (x) ⊂ B R+1 (0), we are done: we choose C = C R . We now claim that the finite sequence does remain in B R/2 . We argue by contradiction. If not, consider k 0 , the smallest integer k ≤ K such that x k ∈ B R (x) \ B R/2 (x) ⊂ B R+1 (0). Consider also the number k 1 (≤ k 0 ) of steps such that Paul and Carol move (i.e. x P = x and x C = x P ). This implies that the corresponding time increments are at least ε/C R . This also implies that 2k 1 ε ≥ R/2. Indeed,

R 2 ≤ |x k 0 -x| ≤ k 0 -1 i=0 |x i+1 -x i | ≤ k 1 × (2ε) .
Recalling that t ∈ (Tδ, T ], it is now enough to choose

δ < R 4C R to conclude that t k 0 -t ≥ k 1 ε C R ≥ R
4C R > δ and thus t k 0 > T , and get a contradiction.

Proof of Theorem 3

First we denote u = lim inf * u ε and u = lim sup * u ε . These relaxed semi-limits are finite since we always have inf u T ≤ u ε ≤ sup u T and u T is assumed to be uniformly bounded. The theorem follows as above from the following two results

Proposition 3.4. The functions u and u are respectively a super-solution and a sub-solution of (2.18).

Lemma 3.6. Given R, δ, there exists C > 0 such that for all t ∈ (0, δ) and all x ∈ B R (0)

(3.25) |u ε (t, x) -u T (x)| ≤ C(T -t + ε 1 2 ) .
Lemma 3.6 implies that u(T, x) ≥ u T (x) ≥ u(T, x) and the comparison principle for (2.18) (see [START_REF] Imbert | Level set approach for fractional mean curvature flows[END_REF]) permits to conclude.

It remains to prove Proposition 3.4 and Lemma 3.6. We first introduce some notation, analogous to that of Section 3.2. Given x ∈ R N and φ ∈ C 2 , T P and T C are defined by

T P (x, φ) = C ε (εκ * [x, φ] -1 ) if Dφ(x) = 0 and κ * [x, φ] > 0 ε 2 if not (3.26) T C (x, φ) = C ε (ε|κ * [x, φ]| -1 ) if Dφ(x) = 0 and κ * [x, φ] < 0 ε 2 if not .
It is convenient to write

t + = t + T P (x + P , φ + ) t -= t + + T C (x - C , φ -)
where T P and T C are defined by (3.26).

We now introduce for any arbitrary function φ : (0, T ) × R N → R the following operator

S ε [φ](t, x) = sup (x + P ,φ + )∈C + (x) inf x + C ∈P + (x,x + P ,φ + ) inf (x - C ,φ -)∈C -(x + C ) sup x - P ∈P -(x + C ,x - C ,φ -) φ(t -, x - P )
where t -is defined in (2.36). The dynamic programming principle (2.35) can be rewritten as follows

(3.27) u ε (t, x) = S ε [u ε ](t, x).
For the reader's convenience, we recall here the definitions of C ± (x) and P ± (x, y, ϕ):

C ± (x) = {(y, ϕ) ∈ B ε (x) × C 2 (R N ) : ±ϕ(y) ≥ ±ϕ(x)} , P + (x, y, ϕ) = {z ∈ B R (y) : ϕ(z) ≥ ϕ(y)} if Dϕ(y) = 0 and κ [y, ϕ] > 0 {x} if not, P -(x, y, ϕ) = {z ∈ B R (y) : ϕ(z) ≤ ϕ(y)} if Dϕ(y) = 0 and κ * [y, ϕ] < 0 {x} if not.
Let us also define the following operators

R ε [φ](t, x) = sup (y,ϕ)∈C + (x) inf z∈P + (x,y,ϕ) φ(t + T P (y, ϕ), z) , R ε [φ](t, x) = inf (y,ϕ)∈C -(x) sup z∈P -(x,y,ϕ) φ(t + T C (y, ϕ), z) .
The reader can notice that

S ε [φ](t, x) = R ε [R ε [φ]](t, x) , (3.28) R ε [φ](t, x) = -R ε [-φ](t, x)
In order to get the second equality, we need to remark that

(z, ϕ) ∈ C + (y) ⇔ (z, -ϕ) ∈ C -(y) , z ∈ P + (y, ϕ) ⇔ z ∈ P -(y, -ϕ) , T P (z, ϕ) = T C (z, -ϕ) .
Moreover, the operator R ε is monotone and commutes with constants: Lemma 3.7 (Estimate from below for R ε ). Consider a C 2 function φ : (0, T ] × R N → R and (t 0 , x 0 ) ∈ (0, T ) × R N . There exists a function o(1) depending on φ and (ε, r) such that o(1) → 0 as (ε, r) → 0 and such that for all (t, x) ∈ B r (t 0 , x 0 ) there exists (y, ϕ) ∈ C + (x) such that

φ 1 ≤ φ 2 ⇒ R ε [φ 1 ] ≤ R ε [φ 2 ] (3.29) R ε [φ + c] = R ε [φ] + c (3.
• if Dφ(t 0 , x 0 ) = 0, (3.31) R ε [φ](t, x) -φ(t, x) ≥ T P (y, ϕ) ∂ φ(t 0 , x 0 ) + κ * [x 0 , φ(t 0 , •)] + |Dφ(t 0 , x 0 )| + o(1) , (3.32) |Dϕ(y)| ≥ 1 2 |Dφ(t 0 , x 0 )| and
(3.33) κ * [x 0 , φ(t 0 , •)] + o(1) ≤ κ * [y, ϕ] ≤ κ * [y, ϕ] ≤ κ * [x 0 , φ(t 0 , •)] + o(1);
• if Dφ(t 0 , x 0 ) = 0, (3.31) still holds true with the convention

κ * [x 0 , φ(t 0 , •)] + |Dφ(t 0 , x 0 )| = 0 and T P (y, ϕ) = ε 2 . Lemma 3.8 (Estimate from above for R ε ). Consider a C 2 function φ : (0, T ] × R N → R and (t 0 , x 0 ) ∈ (0, T ) × R N .
There exists a function o(1) such that o(1) → 0 as (ε, r) → 0 and such that for all (t, x) ∈ B r (t 0 , x 0 )

• either

(3.34) R ε [φ](t, x) -φ(t, x) ≤ ε 2 (∂ t φ(t 0 , x 0 ) + o( 1
)) ;

• or Dφ(t 0 , x 0 ) = 0 and

(3.35) R ε [φ](t, x) -φ(t, x) ≤ T P (y, ϕ)(∂ t φ(t 0 , x 0 ) + o(1))
for some (y, ϕ) ∈ C + (x);

• or Dφ(t 0 , x 0 ) = 0 and κ * [x 0 , φ(t 0 , •)] ≥ 0 and

(3.36) R ε [φ](t, x) -φ(t, x) ≤ T P (y, ϕ)(∂ t φ(t 0 , x 0 ) + κ * [x 0 , φ(t 0 , •)]|Dφ(t 0 , x 0 )| + o(1)) for some (y, ϕ) ∈ C + (x) such that T P (y, ϕ) = min ε/κ * [y, ϕ], ε 1 2 with (3.37) 0 < κ * [y, ϕ] ≤ κ * [x 0 , φ(t 0 , •)] + o(1)
.

By using the fact that R ε [φ] = -R ε [-φ] and exchanging the roles of + and -, we then can deduce from the two previous lemmas the two following ones. Lemma 3.9 (Estimate from above for R ε ). Consider a C 2 function φ : (0, T ] × R N → R and (t 0 , x 0 ) ∈ (0, T ) × R N . There exists a function o(1) depending on φ and (ε, r) such that o(1) → 0 as (ε, r) → 0 and such that for all (t, x) ∈ B r (t 0 , x 0 ) there exists (y, ϕ) ∈ C -(x) such that

• if Dφ(t 0 , x 0 ) = 0, (3.38) R ε [φ](t, x)-φ(t, x) ≤ T C (y, ϕ) ∂ t φ(t 0 , x 0 )-κ * [x 0 , φ(t 0 , •)] -|Dφ(t 0 , x 0 )|+o(1) (3.39) |Dϕ(y)| ≥ 1 2 |Dφ(t 0 , x 0 )| and (3.40) κ * [x 0 , φ(t 0 , •)] + o(1) ≤ κ * [y, ϕ] ≤ κ * [y, ϕ] ≤ κ * [x 0 , φ(t 0 , •)] + o(1);
• if Dφ(t 0 , x 0 ) = 0, then (3.38) still holds true with the convention

κ * [x 0 , φ(t 0 , •)] -|Dφ(t 0 , x 0 )| = 0 and T C (y, ϕ) = ε 2 . Lemma 3.10 (Estimate from below for R ε ). Consider a C 2 function φ : (0, T ] × R N → R and (t 0 , x 0 ) ∈ (0, T ) × R N .
There exists a function o(1) such that o(1) → 0 as (ε, r) → 0 and such that for all (t, x) ∈ B r (t 0 , x 0 )

• either

(3.41) R ε [φ](t, x) -φ(t, x) ≥ ε 2 (∂ t φ(t 0 , x 0 ) + o( 1
));

• or Dφ(t 0 , x 0 ) = 0 and

(3.42) R ε [φ](t, x) -φ(t, x) ≥ T C (y, ϕ)(∂ t φ(t 0 , x 0 ) + o(1))
for some (y, ϕ) ∈ C -(x);

• or Dφ(t 0 , x 0 ) = 0 and κ * [x 0 , φ(t 0 , •)] ≤ 0 and

(3.43) R ε [φ](t, x) -φ(t, x) ≥ T C (y, ϕ)(∂ t φ(t 0 , x 0 ) + κ * [x 0 , φ(t 0 , •)]|Dφ(t 0 , x 0 )| + o(1)) for some (y, ϕ) ∈ C -(x) such that T C (y, ϕ) = min ε/|κ * [y, ϕ]|, ε 1 2 with 0 > κ * [y, ϕ] ≥ κ * [x 0 , φ(t 0 , •)] + o(1).
The proofs of Lemmas 3.7 and 3.8 are postponed. We now explain how to derive Proposition 3.4.

Proof of Proposition 3.4. We only prove that u is a sub-solution of (2.18) since a symmetric argument can be used to prove that u is a super-solution. In order to do so, we consider a (t 0 , x 0 ) ∈ (0, T ) × R N and a φ ∈ C 2 such that uφ attains a strict maximum at (t 0 , x 0 ) in (0, T ) × B R+1 (x 0 ). We want to prove that -∂ t φ(t 0 , x 0 )κ * [x 0 , φ(t 0 , •)]|Dφ(t 0 , x 0 )| ≤ 0 if Dφ(t 0 , x 0 ) = 0 and -∂ t φ(t 0 , x 0 ) ≤ 0 if Dφ(t 0 , x 0 ) = 0.

We know that there exists a sequence (t εn , x εn ) such that u εnφ attains a maximum in (0, T ) × B R (x εn ) at (t εn , x εn ). For simplicity, we simply write (t, x) for (t εn , x εn ) and ε for ε n . With the same argument as in the proof of Propositions 3.1 and 3.3, the dynamic programming principle (3.27) and the monotonicity of S ε imply that

(3.44) φ(t, x) ≤ S ε [φ](t, x).
We now estimate S ε [φ](t, x) from above. We distinguish cases.

Case Dφ(t 0 , x 0 ) = 0 and κ * [x 0 , φ(t 0 , •)] > 0. Lemma 3.9 yields an (y, ϕ) ∈ C -(x), with κ * [y, ϕ] > 0 (by (3.40)) for ε small enough, thus T C (y, ϕ) = ε 2 , so that

R ε [φ](t, x) -φ(t, x) ≤ ε 2 (∂ t φ(t 0 , x 0 ) + o(1))
and using properties (3.29)-(3.30), we find

S ε [φ](t, x) = R ε [R ε [φ]](t, x) ≤ R ε [φ](t, x) + ε 2 (∂ t φ(t 0 , x 0 ) + o( 1
≤ S ε [φ](t, x) -φ(t, x) ≤ T P (y, ϕ)(∂ t φ(t 0 , x 0 ) + κ * [x 0 , φ(t 0 , •)]|Dφ(t 0 , x 0 )| + o(1)) + ε 2 (∂ t φ(t 0 , x 0 ) + o(1)) . Since T P (y, ϕ) = min ε/κ * [y, ϕ], ε 1
2 , this can be written as

0 ≤ S ε [φ](t, x) -φ(t, x) ≤ T P (y, ϕ)(∂ t φ(t 0 , x 0 ) + κ * [x 0 , φ(t 0 , •)]|Dφ(t 0 , x 0 )| + o(1))
and dividing by T P (y, ϕ) and letting ε → 0, the desired inequality follows.

Case Dφ(t 0 , x 0 ) = 0 and κ * [x 0 , φ(t 0 , •)] < 0. We apply first Lemma 3.9 and find

R ε [φ](t, x) -φ(t, x) ≤ T C (y, ϕ)(∂ t φ(t 0 , x 0 ) + κ * [x 0 , φ(t 0 , x 0 )]|Dφ(t 0 , x 0 )| + o(1)).
We note that from (3.39) we have Dϕ(y) = 0. Now we cannot have κ * [y, ϕ] ≥ 0, otherwise a contradiction would follow from (3.40) and our assumption κ * [x 0 , φ(t 0 , •)] < 0. We deduce that the case T C (y, ϕ) = ε 2 cannot happen and we must have κ * [y, ϕ] < 0 and ε 2 = o(T C (y, ϕ)). With this piece of information at hand, we can write, as previously

0 ≤ S ε [φ](t, x) -φ(t, x) ≤ R ε [φ](t, x) -φ(t, x) + T C (y, ϕ)(∂ t φ(t 0 , x 0 ) + κ * [x 0 , φ(t 0 , •)]|Dφ(t 0 , x 0 )| + o(1)).
On the other hand, Lemma 3.8 yields (we can only be in the first situation of the lemma)

R ε [φ](t, x) ≤ φ(t, x) + O(ε 2 ) = φ(t, x) + o(T C (y, ϕ)),
and we can write, dividing by T C (y, ϕ),

0 ≤ ∂ t φ(t 0 , x 0 ) + κ * [x 0 , φ(t 0 , •)]|Dφ(t 0 , x 0 )| + o(1)
and the desired inequality is thus obtained in this case too.

Case Dφ(t 0 , x 0 ) = 0 and κ * [x 0 , φ(t 0 , •)] = 0. Once again, we first apply Lemma 3.9 and we obtain with M ε = ε 2 or M ε = T P (y, ϕ). Hence we obtain 0 ≤ (T C (y, ϕ) + M ε )(∂ t φ(t 0 , x 0 ) + o(1))

0 ≤ S ε [φ](t, x) -φ(t, x) ≤ R ε [φ](t,
and we obtain the desired inequality in this case too.

Case Dφ(t 0 , x 0 ) = 0. Then Lemmas 3.9 and 3.8 yield 0 ≤ (ε 2 + T P (y, ϕ))(∂ t φ(t 0 , x 0 ) + o(1))

or 0 ≤ 2ε 2 (∂ t φ(t 0 , x 0 ) + o(1)), and the proof of the proposition is now complete.

We now turn to the proofs of Lemmas 3.7 and 3.8. As the reader shall see, we follow along the lines of proofs of Lemmas 3.4 and 3.5 used in the eikonal case.

i.e. the fact that the infimum can only be achieved in B 2ε (x). To see this, we simply write for z ∈ P + (x, y, ϕ) such that z / ∈ B 2ε (x), φ(t + T P (y, ϕ), z) = φ(t, z) + O(T P (y, ϕ))

≥ φ(t, y) + ε

1 4 + O(ε 1
2 ) ≥ φ(t + T P (y, ϕ), y) + ε and from (3.33) it follows that κ * [x 0 , φ(t 0 , •)] ≤ 0. For z ∈ P + (x, y, ϕ) ∩ B 2ε (x), we have φ(t + T P (y, ϕ), z)φ(t, x) ≥ φ(t + ε 1 2 , z)φ(t, z) + φ(t, z)φ(t, x) + O(ε 3 ) But ϕ(z) ≥ ϕ(y) since z ∈ P + (x, y, ϕ), hence φ(t, z)α ε (z) ≥ φ(t, y) so replacing in the above and using α ε ≥ 0 we are led to φ(t + T P (y, ϕ), z)φ(t, x) ≥ φ(t + ε where the last inequality follows from (3.33) ; and we get (3.31) in all cases.

Assume now that Dφ(t 0 , x 0 ) = 0. Then choose y = x and ϕ(z) = -α ε (z). This is admissible and Dϕ(y) = 0 and T P (y, ϕ) = ε 2 and the conclusion follows easily.

We now turn to the proof of Lemma 3.8.

Proof of Lemma 3.8. We recall that φ(t + T P (y, ϕ), z), φ(t + ε 2 , x) .

Let ϕ be a fixed C 2 test function.

Case 1. Assume first that the max above is φ(t + ε 2 , x). Then we easily obtain (3.34) as desired.

Case 2. We then turn to the situation where φ(t + T P (y, ϕ), z) > φ(t + ε 2 , x).

We need to prove that (3.35) or (3.36) holds true in this case. So let y ∈ B ε (x) be such that ϕ(y) ≥ ϕ(x), Dϕ(y) = 0 and κ * [y, ϕ] > 0, and such that (3.50) inf z∈P + (x,y,ϕ) φ(t + T P (y, ϕ), z) > φ(t + ε 2 , x)ε 3 .

For any z ∈ P + (x, y, ϕ) = {z ∈ B R (y) : ϕ(z) ≥ ϕ(y)} we may write φ(t + , z)φ(t, x) = φ(t + , z)φ(t + , y) + φ(t + , y)φ(t, x) = φ(t + , z)φ(t + , y) +T P (y, ϕ)(∂ t φ(t 0 , x 0 ) + o ε (1)) + Dφ(t + , y) • (yx) + O(ε 2 ), and using the fact that |y -x| ≤ ε and O(ε 2 ) = o(t +t) since T P (y, ϕ) ≥ ε 3/2 in this case; we obtain (3.51) φ(t + , z)φ(t, x) ≤ φ(t + , z)φ(t + , y) + ε|Dφ(t + , y)| + T P (y, ϕ)(∂ t φ(t 0 , x 0 ) + o ε (1)).

We now evaluate φ(t + , z)-φ(t + , y). In view of (3.48), (3.49) and (3.51), the following lemma permits to conclude.

for n large enough. We obtain a contradiction by taking z = y n .

Either we have T P (y n , ϕ n ) = ε 3/2 or T P (y n , ϕ n ) = ε n /κ * [y n , ϕ n ]. In both situations we have T P (y n , ϕ n ) ≥ ε n /κ * [y n , ϕ n ]. Choosing z = y n in (3.56) yields 0 ≥ T P (y n , ϕ n ) η 2 which is a contradiction.

We next prove that the terminal condition is satisfied at the limit.

Proof of Lemma 3.6. The proof consists in proving the following estimate We remark that (3.57) is a consequence of the following lemma. It remains to prove Lemma 3.13.

Proof of Lemma 3.13. We only prove that for all (t, x) ∈ (T -(k + 1)ε 2 , T ) × R N , we have

u ε (t, x) ≥ u T (x) -C(T -t)
and the reader can check that the proof of the reverse inequality is similar.

It is enough to consider t ∈ (T -(k + 1)ε 2 , Tkε 2 ). We recall that the dynamic programming principle can be written as follows

u ε (t, x) = S ε [u ε ](t, x) = R ε [R ε [u ε ]](t, x).

1 .

 1 Semi-continuity: functions κ * [•, U] and κ * [•, U] are respectively upper and lower semi-continuous κ * [x, U] ≥ lim sup y→x κ * [y, U]; (2.26) κ * [x, U] ≤ lim inf y→x κ * [y, U]; (2.27) 2. Monotonicity property:

  we have -εF (t, x, DΦ(x), D 2 Φ(x), I R [x, Φ]) ≤ Cε γ with γ = 1α max(1, k 1 , k 2 ) ∈ (0, 1) and C depends on F , ν and R. Proof of Lemma 3.1. We consider a bounded C 2 function Φ such that (3.1) holds. From the definition of I R [x, •] (see (2.3)), it is clear that there exists a constant C only depending on R and ν such that |I R [x, Φ]| ≤ Cε -α .

  30) for all c ∈ R. The proof of Proposition 3.4 relies on four consistency lemmas. Before stating them, let us point out that we will write κ[•] + for the positive part of κ[•] and κ[•] -for the negative part (both being nonnegative).

  x)φ(t, x) + T C (y, ϕ)(∂ t φ(t 0 , x 0 ) + o(1)). Lemma 3.8 implies that R ε [φ](t, x)φ(t, x) ≤ M ε (∂ t φ(t 0 , x 0 ) + o(1))

1 2.

 1 z∈P + (x,y,ϕ)∩B 2ε (x) φ(t + T P (y, ϕ), z) and (3.46) follows.Case T P (y, ϕ) = ε By definition of C ε , this happens if κ * [y, ϕ] ≤ ε 1 2

1 2 1 2 1 2 1 2

 1111 , z)φ(t, z) + φ(t, y)φ(t, x) + O(ε 3 ) ≥ ε ∂ t φ(t, x) + ε|Dφ(t, x)| + o ε (ε) ≥ ε (∂ t φ(t, x) + o ε (1)) ≥ ε (∂ t φ(t, x) + κ * [x 0 , φ(t 0 , •)] + |Dφ(t 0 , x 0 )| + o ε (1))and we get (3.31) in this case too.Case T P (y, ϕ) = ε κ * [y,ϕ] + . Observe that from (3.45), κ * (y, ϕ) is bounded above hence T P (y, ϕ) bounded below by cε. As above, we may write, recalling the choice of y = x + ε Dφ(

( 3 .

 3 47) R ε [φ](t, x) = sup (y,ϕ)∈C + (x) inf z∈P + (x,y,ϕ) φ(t + T P (y, ϕ), z)In view of the definition of C + (x) and P + (x, y, ϕ), we can write more precisely(3.48) R ε [φ](t, x) = sup ϕ∈C 2 (R N ) max sup y∈Bε(x):ϕ(y)≥ϕ(x)Dϕ(y) =0, κ * [y,ϕ]>0, inf z∈P + (x,y,ϕ)

( 3 .

 3 49) supy∈Bε(x):ϕ(y)≥ϕ(x) Dϕ(y) =0,κ * [y,ϕ]>0 inf z∈P + (x,y,ϕ)

( 3 .

 3 57) |u ε (t, x)u T (x)| ≤ C(Tt) for t < T and x ∈ R N with C = sup x∈R N max(|κ * [x, u T ]||Du T (x)| + 1, |κ * [x, u T ]||Du T (x)| + 1).

Lemma 3 . 13 .

 313 Consider k ∈ N ∩ (0, ε -2 T ). If, (3.58) ∀(t, x) ∈ (Tkε 2 , T ) × R N , |u ε (t, x)u T (x)| ≤ C(Tt), then (3.59) ∀(t, x) ∈ (T -(k + 1)ε 2 , T ) × R N , |u ε (t, x)u T (x)| ≤ C(Tt).

Thanks to Lemma 3 .

 3 10 and (3.58), we know that there exists (y, ϕ)∈ C + (x) such that R ε [u ε ](t, x) ≥ u T (x) -C(T -(t + T C (y, ϕ))) -T C (y, ϕ)(κ * [x, u T ]|Du T (x)| + 1) ≥ u T (x) -C(Tt).

  )).

	We now use Lemma 3.8. If (3.34) holds true, then (with (3.44))
	0 ≤ S

ε [φ](t, x)φ(t, x) ≤ 2ε 2 (∂ t φ(t 0 , x 0 ) + o(1))

and we conclude that ∂ t φ(t 0 , x 0 ) ≥ 0. The result follows easily in this subcase. The subcase where (3.35) holds works similarly.

If now (3.36) holds true, we get 0

  t,x) |Dφ(t,x)| R ε [φ](t, x)φ(t, x) ≥ inf P (y, ϕ)(∂ t φ(t, x) + o ε (1)) + φ(t, y)φ(t, x) ≥ T P (y, ϕ)(∂ t φ(t, x) + o ε (1)) + ε|Dφ(t, x)| + O(ε 2 ) ≥ T P (y, ϕ) ∂ t φ(t, x) + ε T P (y, ϕ) |Dφ(t, x)| + o ε (1) ≥ T P (y, ϕ) ∂ t φ(t, x) + κ * [y, ϕ] + |Dφ(t, x)| + o ε (1) ≥ T P (y, ϕ) ∂ t φ(t, x) + κ * [x 0 , φ(t 0 , •)] + |Dφ(t 0 , x 0 )| + o ε (1)

z∈P + (x,y,ϕ)∩B 2ε (x) φ(t + T P (y, ϕ), z)φ(t, x) ≥ inf z∈P + (x,y,ϕ)∩B 2ε (x) φ(t + T P (y, ϕ), z)φ(t, z) + φ(t, z)φ(t, x)

≥ T

Proof of Lemma 3.7. We first assume that Dφ(t 0 , x 0 ) = 0. So we can assume that, for ε small enough, |Dφ(t, x)| ≥ θ 0 > 0.

Consider y = x + ε Dφ(t,x) |Dφ(t,x)| and ϕ(z) = φ(t, z)α ε (z) with α ε : R N → [0, +∞) smooth and

We also can write for ε small enough ϕ(y) = φ(t, y) = φ(t, x) + ε|Dφ(t, x)| + O(ε 2 ) ≥ φ(t, x) = ϕ(x) (we used the fact that |Dφ(t, x)| ≥ θ 0 > 0). This means that (y, ϕ) ∈ C + (x) (at least for ε small enough). Remark also that (3.32) holds. Hence

and, using the monotonicity of κ * and its upper semi-continuity (see (2.28) and (2.26)), we conclude that

for ε small enough; the other part of (3. . We treat these cases separately.

Case T P (y, ϕ) = ε 2 . We know that in this case we have κ * [y, ϕ] ≤ 0 and P + (x, y, ϕ) = {x}. Hence,

In the two remaining cases, we have T P (y, ϕ) ≤ ε 1 2 and P + (x, y, ϕ) ∋ y. These two facts imply the following inequality

φ(t + T P (y, ϕ), z).

Lemma 3.11. For any (y, ϕ) ∈ C + (x) with Dϕ(y) = 0, κ * [y, ϕ] > 0, such that (3.50) holds, we have

Moreover, if Dφ(t 0 , x 0 ) = 0, we have

.

There now remains to give the proof of Lemma 3.11. We start by Lemma 3.12. If Dφ(t 0 , x 0 ) = 0, then for any (y, ϕ) as in the above lemma, we have

Proof of Lemma 3.12. We are in the case where (3.50) holds true, hence for all z ∈ {z ∈ B R (y) : ϕ(z) ≥ ϕ(y)}, we have φ(t

Indeed, if z ∈ B R (0) is such that φ(t 0 , x 0 + z) < φ(t 0 , x 0 ), then for ε small enough and (t, x) -(t 0 , x 0 ) small enough, φ(t + T P (y, ϕ), y + z) < φ(t + ε 2 , x)ε 3 and in view of the above this implies ϕ(y + z) < ϕ(y). By properties of the non-local curvature, (3.53) implies (3.54) and the proof of the lemma is complete.

We can now complete the proof of Lemma 3.11 by

Proof of (3.52). Remark first that Lemma 3.12 implies in particular that

since this inequality is trivial when Dφ(t 0 , x 0 ) = 0. We now argue by contradiction. We thus assume that there exists η > 0, ε n → 0, (t n , x n ) → (t 0 , x 0 ) and (y n , ϕ n ) ∈ C + (x n ) such that Dϕ n (y n ) = 0, κ * [y n , ϕ] > 0 and (3.50) holds, and for all z ∈ B R (y n ) ∩ {ϕ n ≥ ϕ n (y n )}, we have

Assume first that there exists a subsequence such that T

We now use that R ε is monotone and commutes with constants (see (3.29) and (3.30)) in order to write

We remark next that (x, u T ) ∈ C + (x) and we write u ε (t, x) ≥ inf If now Du T (x) = 0 or κ * [x, u T ] ≤ 0, then we also have

The proof of Lemma 3.13 is now complete.