Well-Posedness of a Fractional Mean Field Game System with Non-Local Coupling

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Well-Posedness of a Fractional Mean Field Game System with Non-Local Coupling

Olav Ersland

Applied and Engineering Mathematics

Supervisor: Espen Robstad Jakobsen, IMF

Department of Mathematical Sciences Submission date: July 2017

Norwegian University of Science and Technology

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Abstract

We prove existence and uniqueness of classical solutions for a fractional Mean Field Game system with non-local coupling, where the fractional exponent is greater than1/2. To our knowledge this is not proven before in the literature, and is therefore a new result. In addition, we show regularity in time and space for the fractional Hamilton- Jacobi equation, and use this result to show regularity for the fractional Fokker-Planck equation.

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Sammendrag

Vi beviser eksistens og entydighet av klassiske løsninger for et fraksjonelt Mean Field Game system med ikke-lokal kobling, der den fraksjonelle eksponenten er større enn1/2. Til vår kunnskap er dette ikke vist tidligere i litteraturen, og er dermed et nytt resultat.

Vi viser også regularitet i tid og rom for den fraksjonelle Hamilton-Jacobi-ligningen, og bruker dette resultatet for å vise regularitet for den fraksjonelle Fokker-Planck-ligningen.

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Preface

This thesis is the conclusion of my Master's project in Industrial Mathematics at the Applied Physics and Mathematics study programme at the Norwegian University of Sci- ence and Technology (NTNU). The project was carried out during the spring of 2017.

Techniques I have learned in functional analysis, and in courses on partial dierential equations, have proved to be very useful during the work with this thesis. Much of the results have been obtained using standard techniques from functional analysis.

In the end, I would like to thank my supervisor Espen Robstad Jakobsen, professor at the Department of Mathematical Sciences at NTNU, for very good supervision and for pushing me in my work. Also a thanks to my fellow students for their company and coe breaks, that made writing this thesis a very cheerful endeavour.

Trondheim 2017-07-11 Olav Ersland

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Introduction

1.1 Fractional Mean Field Games with non-local coupling

The object of this thesis is to prove existence and uniqueness of solutions for a fractional Mean Field Game (MFG) system with non-local coupling.

Mean Field Games is a relatively new eld of mathematics, and was introduced al- most simultaneously by Lasry and Lions [12] , and Caines, Huang and Malhamé [7].

The idea of Mean Field Games is to model dierential games with instinguishable (sym- metric) players, where the amount of players tend to innity, and each player becomes accordingly small. The average player wants to optimize some cost function in a noisy environment, where the information available is the distribution of other players and the position of itself.

Until very recently, most of the litterature on MFG have modelled the noisy environment as a standard diusion process, but a recent paper by Cesaroni et al. [4] discusses a stationary MFG system where the noisy environment is modelled by pure jump Lévy processes. They look at the stationary case, that is, where one assumes that a Nash equilibrium has occured: a state where no player would spontaneously change their position, knowing the distribution of the other players.

We look at the case where the players still want to change their positions, based on the information they receive on the density of other players: A time dependent case. This is something that, to the best of our knowledge, is not yet presented in the literature.

The system of PDE's that describe this system is given by







−∂_tu+ (−∆

T^d)^α² u+H(x, u, Du) =F(x, m(t)) in (0, T)×T^d

∂_tm+ (−∆_Td)^α² m−div(mD_pH(x, u, Du)) = 0 in (0, T)×T^d m(0) =m0, u(x, T) =G(x, m(T))

(1.1)

whereα∈(1,2), and(−∆

T^d)^α/2uis the fractional Laplacian on the torus. The functions F and G are both non-local coupling functions. The function H is called the Hamilto-

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nian, and is convex in the last variable.

The rst equation in (1.1) is known as the fractional Hamilton-Jacobi equation, and is solved backwards in time, while the second one is the fractional Fokker-Planck equation, and is solved forwards in time. We seek classical solutions, that is, a pair u, m∈C^1,2 (0, T)×T^d

that solves the system (1.1) simultaneously.

The content of this thesis is as follows:

1.2 Outline of thesis

Chapter 2: Preliminaries

Here we present some theory on the fractional Laplacian, both on R^d and on the torus T^d. We then present some theory on the probability space P T^d

endowed with the Kantorovitch-Rubinstein metric d1. The last part of the Preliminaries consists of pre- senting some xed point theorems, Hölder spaces and compact embedding theorms.

Chapter 3: Fractional MFG systems with nonlocal coupling

In this chapter we prove the existence and uniqueness of classical solutions for the fractional MFG system (1.1), under suitable assumptions on the HamiltonianH, the coupling functionsF, Gand the initial conditions m₀.

Chapter 4: Regularity for the fractional Hamilton-Jacobi equation

We present some regularity results for the fractional Hamilton-Jacobi equation, with a HamiltonianH of a quite general form. We prove regularity in time and space by using Duhamel's formula, combining it with known regularity of the unique viscosity solution for the fractional Hamilton-Jacobi equation.

Chapter 5: The fractional Fokker-Planck equation

By rewriting the fractional Fokker-Planck equation into divergence free form, we can write it on the form of a fractional Hamilton-Jacobi equation. We then show under which conditions this system admits a unique solution with sucient regularity for the MFG-existence proof.

Chapter 6: Estimates of ∂x^βH

We show a way to represent the derivative∂_x^βH(s, x, u(s, x)w(s, x)), and use this rep- resentation to give some estimates that are used in Chapter 4.

Concluding remarks

The main results of this thesis is presented, along with suggestions for further work.

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3 1.2. Outline of thesis Appendix

We give the proof of some Lemma's stated in the report, that are a bit too long for being written in the main report.

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Chapter 2

Preliminaries

2.1 The fractional Laplace operator

Assume that we have a function u :R^d→ R. There are several and equivalent ways of denining the fractional Laplace operator on this function, as shown in [11], but we will limit ourselves to only one of them.

One can dene it is as a singular integral. Let α ∈ (0,2). Then the fractional Laplacian (−∆)^α/2u can be written as (for an arbitrary r >0)

(−∆)^α/2u(x) =c(d, α) Z

Br

u(x+z)−u(x)− ∇u(x)·z

|z|^d+α dz +

Z

R^d\Br

u(x+z)−u(x)

|z|^d+α dz (2.1) !

where c(d, α) is a constant. For the case α ∈ (1,2), the expression (2.1) one can simplifed to (Theorem 1. in [6])

(−∆)^α/2u(x) =c(d, α) Z

R^d

|z|^d+α dz

(2.2)

Note that these integrals are singular nearz= 0, so that they are understood in the sense of Cauchy principal value.

2.1.1 Fractional Laplacian on the torus

Having given a denition of the fractional Laplacian on the whole space R^d, we want to look into how it is dened on the torus,T^d. This is natural, since we will later look at a Mean Field Game system dened on the torus.

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The d-dimensional torus can be dened as the quotient space T^d:=R^d/Z^d

(2.3)

or equivalently, as the product space ofdcircles T^d:=S¹× · · · ×S¹

| {z }

d times

(2.4)

A thing worth knowing about the torus, is that Lemma 2.1. The torusT^d is compact.

This follows easily from S¹ being compact due to Heine-Borel, and then from that product spaces of compact spaces are also compact.

If a function f :T^d→ R, then the function f has a periodic extension toR^d, which we will just callf. For this function we have that for all x∈R^d andz∈Z^d

f(x+z) =f(x)

Using the periodic extensions for functions dened on the torus, we can dene the fractional Laplacian for functions on the torus. This is because the earlier denitions (2.1) and (2.2) still works for functions u : T^d → R, when we look at their periodic extensions toR^d. So, the denition is the same, with the only dierence that nowx∈T^d

(−∆T^d)^α/2u(x) =c(d, α) Z

Br

|z|^d+α dz +

Z

R^d\Br

u(x+z)−u(x)

|z|^d+α dz

!

, x∈T^d

For the torus, we also present another way to work with the fractional Laplacian, and that is through the use of Fourier series.

The Fourier series of the function u:T^d→R is given by (see [13]) u(x) = X

n∈Z^d

cn(u)e^in·x , x∈T^d

wheren·x=n₁x₁+· · ·+n_dx_d, and the Fourier coecients are dened as c_n(u) = 1

(2π)^d Z

T^d

u(x)e^−in·xdx

Then the Fourier series for the fractional Laplace on the torus is given by (for α ∈ (0,2))

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7 2.1. The fractional Laplace operator

(−∆T^d)^α/2u(x) = X

n∈Z^d

c(α, d)|n|^αcn(u)e^in·x , x∈T^d (2.5)

We will now state some properties of the fractional Laplacian on the torus, and it begins with the following interpolation Lemma

Theorem 2.1. (Hölder estimates, Theorem 2.6 in [14]). Assume that α ∈ (0,2) and σ ∈(0,1].

Letv ∈C^1,σ T^d

andα≥σ, withσ−α+ 1>0. Then (−∆

T^d)^α² v∈C^0,σ−α+1 T^d and

k(−∆_Td)^α² vk_C0,σ−α+1(T^d)≤Ckvk_C1,σ(T^d) (2.6)

A consequence of Theorem 2.1 is that, if we have a functionv∈C² T^d

, we get the interpolation

k(−∆

T^d)^α² vk_L∞(T^d) ≤Ckvk_C2(T^d) (2.7)

which is an estimate that we use a lot.

The next thing we want to say about the fractional Laplacian on the torus, is something about the identity h(−∆)^α/2u, vi_L2(T^d) =hu,(−∆)^α/2vi_L2(T^d), foru, v∈C² T^d

. This is a result we need in the uniqueness proof for classical solutions of the MFG system.

Lemma 2.2. Assume that f, g∈C^∞ T^d

. Then the following identity holds:

Z

T^d

(−∆T^d)^α/2f(x)g(x) = Z

T^d

f(x) (−∆

T^d)^α/2g(x)dx

Proof. Since f, g ∈ C^∞ T^d

, one can show that the corresponding Fourier series, and the Fourier series of (−∆

T^d)^α/2f and (−∆

T^d)^α/2g converges absolutely (see [13]).

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Therefore, one can interchange integration and summation to obtain the result Z

T^d

(−∆T^d)^α² f(x)g(x)dx= Z

T^d



 X

n,m∈Z^d

c(α, d)|n|^αcn(f)e^inxcm(g)e^imx



dx

= X

n,m∈Z^d

c(α, d)|n|^αcn(f)cm(g) Z

T^d

e^i(n+m)·xdx

(*)= X

n+m=0

c_n(f)c(α, d)|m|^αc_m(g) Z

T^d

e^i(n+m)·xdx

(*)= X

n,m∈Z^d

cn(f)c(α, d)|m|^αcm(g) Z

T^d

e^i(n+m)·xdx

= Z

T^N

f(x) (−∆

T^d)^α² g(x)dx.

The equality marked with (*) comes from the fact that Z

T^d

e^i(n+m)·xdx= 0, forn+m6= 0

We can generalize the result from Lemma 2.2, to functionsu, v∈C² T^d

by using a density argument.

Lemma 2.3. Let f, g∈C² T^d

. Then the following identity holds, forα∈(1,2). Z

T^d

(−∆_Td)^α/2f(x)g(x) = Z

T^d

f(x) (−∆_Td)^α/2g(x)dx

Proof. The proof is given in the Appendix.

2.2 Measures and distance

In this section we want to say something about the space of Borel probability measures on the torus, and give a denition of a metric d₁ dened on this space. We will just list the results we need.

Denition 2.1. Let X be a separable metric space. We denote P(X) to be:

P(X) :=the family of all Borel probability measures on X.

Theorem 2.2. (Prokhorov, from Ambrosio thm 5.1.3) If a set K⊂P(X) is tight, i.e.

(2.8) ∀ >0 ∃K compact in X such that µ(X\K)≤ ∀µ∈K,

thenK is relatively compact inP(X). Conversely, if there exists an equivalent complete metric for X, i.e. X is a so called Polish space, then every relatively compact subset of P(X) is tight.

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9 2.3. Analysis Comment: What is meant, is thatK is relatively compact with respect to the narrow topology on P(X).

Denition 2.2. (The set of Borel probability measures on the torus) We dene P T^d to be:

P

T^d

:=the set of Borel probability measures onT^d On this set, we can dene the following (Kantorovitch-Rubinstein) distance:

d1(µ, ν) = sup Z

T^d

φ(x) (µ−ν)|φ:T^d→R 1−Lipschitz continuous

.

which metricizes the weak topology on P T^d . Lemma 2.4. P T^d

, d1

is a compact metric space.

Proof. We refer to Lemma 4.1.7 of [3], and recall that all r-moments of members of P T^d

are nite.

We also state the following property.

Lemma 2.5. The metric d1 can be dened equivalently as:

d₁(µ, ν) = sup Z

T^d

φ(x) (µ−ν)|φ:T^d→R1−Lipschitz continuous, φ(0) = 0

Proof. Recall the denition of d₁. Take any φ∈ 1−Lip. Assume that φ(0) = k ∈ R.

We can then dene φ˜(x) =φ(x)−k. Thenφ˜∈1−Lip, since

|φ˜(x)−φ˜(y)|=|φ(x)−φ(y)| ≤1· |x−y|

For anyµ, ν ∈P T^d

, we get Z

T^d

φ˜(x)d(µ−ν) (x) = Z

T^d

φ(x)d(µ−ν) (x)−k Z

T^d

d(µ−ν) (x)

= Z

T^d

φ(x)d(µ−ν) (x) This shows that the denitions are equivalent.

2.3 Analysis

In this section we will present some results from analysis. We start with the fundamental theorem of Calculus.

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Theorem 2.3. (Fundamental theorem) Assume that f ∈ C¹ R^N

. Then the following holds forx, y∈R^N:

f(x)−f(y) = Z 1

0

d

dtf(tx+ (1−t)y)dt By using the chain rule, one can also write this as:

f(x)−f(y) =

N

X

i=1

(x_i−y_i) Z 1

0

∂

∂xi

f(tx+ (1−t)y)dt

We also need the following short result for some of the calculations later.

Lemma 2.6. Suppose thatf : [0,∞)×R^N →R is Lipschitz and uniformly bounded. In other words, there exists constants L, M >0 such that:

|f(s, x)−f(t, y)| ≤L(|s−t|+|x−y|) kfk_∞≤M

Then there exists a constant C >0 such that f satises:

|f(s, x)−f(t, y)| ≤C

|s−t|¹² +|x−y|

Proof. The proof consists of two cases,|s−t| ≤1 and |s−t|>1. The case |s−t| ≤ 1 holds trivially as then|s−t| ≤ |s−t|¹². For the case|s−t|>1one can compute

|f(s, x)−f(t, y)| ≤M+M ≤2M

|s−t|¹² +|x−y|

One can then choose C= max (L,2M).

The next theorem we present is the Arzela-Ascoli theorem, which is a useful Theorem from functional analysis.

Theorem 2.4. (Arzela-Ascoli) (p. 234 of [10] ) Let K be a compact space, and (E, d) be a metric space. The space of continuous functions C(K, E) from K to E, endowed with the uniform distance, is a metric space.

A subset A ⊂ C(K, E) is relatively compact in C(K, E) if and only if, for each point x∈K:

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11 2.4. Hölder continuity and Hölder spaces

• (EQ) A is equicontinuous at x, that is for all > 0, there exists a neighbourhood V of x such that:

(2.9) ∀f ∈A ∀y∈V : d(f(x), f(y))<

• (RC) The set A(x) ={f(x)|f ∈A} is relatively compact in (E, d).

The following is a Lemma that is useful for proving convergence of sequences.

Lemma 2.7. Let (X, d) a metric space and K ⊂⊂X a compact subset of X.

Further, let(x_n)⊂K be a sequence, such that all convergent subsequences have the same limit point x^∗ ∈K. Then xn→x^∗.

Proof. By contradiction. Assume that there exists a subsequence (xn_k) that doesn't converge towardsx^∗, i.e:

(2.10) ∃ >0 ∀N ∈N∃n≥N : d(xn, x^∗)>

Starting fromN = 1,2,3, . . . these xn denes a subsequence(xn)⊂(xn). SinceK is compact, it follows that (x_n) has a convergent subsequence, with limit, say x˜ ∈ K. However, since all convergent subsequences of (x_n) have the same limit point, it follows thatx˜=x^∗. But this is a contradiction to the construction (2.10).

The following two xed point theorems are really important for us, and play an important role in this thesis.

Theorem 2.5. (Schauder's xed point theorem)

Let X be a Banach space, K ⊂ X a convex, closed and compact subset. Further, let T :K→K be a continuous map. ThenT has a xed point in K.

Theorem 2.6. (Banach's xed point theorem)

Let (X, d) be a complete metric space, and T :X →X a map. If there exists q ∈ [0,1) such that for all x, y∈X:

d(T(x), T(y))≤qd(x, y) ThenT has a unique xed point x∈X.

2.4 Hölder continuity and Hölder spaces

Since concepts like Hölder continuity and Hölder spaces will be used later on, they will be presented here. We will dene the Hölder-norm, Hölder spaces, and look at compact inclusion of Hölder spaces.

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2.4.1 Denitions

Denition 2.3. A function f : Ω ⊂ X → Y between two metric spaces (X, d_X) and (Y, dY) is Hölder-continuous with exponent α ∈ (0,1] if there exists a constant C > 0 such that:

∀x, y∈Ω : d_Y (f(x), f(y))≤C(d_X(x, y))^α

Denition 2.4. (reference: Def. 1.7 p. 46 of[1]) (Hölder space) Let Ω ⊂ Rⁿ, k ∈ N and β ∈ (0,1]. The Hölder space C^k,β(Ω) is the set of all functions f : Ω → R with f ∈C^k(Ω), such that the following norm is nite:

kfk_Ck,β(Ω):= X

|α|≤k

kD^αfk_C(Ω)+ X

|α|=k

[D^αf]_C0,β(Ω).

Here, we denote by

kD^αfk_C(Ω) := sup{|D^αf(x)||x∈Ω}

the supremumsnorm, and [D^αf]_C0,β := sup

|D^αf(x)−D^αf(y)|

|x−y|

x, y∈Ω, x6=y

a semi-norm.

Later on, we often use the convention of writingC^k+β(Ω), instead ofC^k,β, fork∈N andβ ∈(0,1].

2.4.2 Compact embedding theorems

Theorem 2.7. LetΩ⊂Rⁿ be a closed and bounded subset (compact by the Heine-Borel theorem), and let 0≤α < β≤1. Then the embedding:

i:C^0,β(Ω)→C^0,α(Ω) u7→u

is continuous.

Further, ifD⊂C^0,β is a uniformly bounded subset, that is

∃M >0 s.t ∀f ∈D: kfk_C0,β(Ω)≤M then the setD is precompact in C^0,α.

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13 2.4. Hölder continuity and Hölder spaces Proof. Continuity:

From the assumptions diam (Ω)<∞. Then, for any u∈C^0,β(Ω): [u]_C0,α(Ω)= sup

x,y∈Ω,x6=y

|u(x)−u(y)|

|x−y|^α

= sup

x,y∈Ω,x6=y

|x−y|^β

|x−y|^α

|u(x)−u(y)|

|x−y|^β

β>α

≤ (diam (Ω))^β−α[u]_C0,β(Ω)

it follows then that:

kuk_0,α;Ω=kuk_0;Ω+ [u]_0,α;Ω≤ kuk_0;Ω+C(Ω, α, β) [u]_0,β;Ω≤ckuk_0,β;Ω withc= max

1,(diam (Ω))^β−α

, and it follows that the inclusion is continuous.

Compactness:

We want to show that the setDis sequentially compact in C^0,α(Ω). Take any sequence (un)⊂D. The sequence is uniformly bounded by the constantM:

ku_nk_∞≤ ku_nk_0,β;Ω≤M It is also equicontinuous, since ∀ >0choose δ= _M 1/β

, so that ku_n(x)−un(y)k∞≤M|x−y|^β <

whenever |x−y|< δ.

Apply the Arzelà-Ascoli theorem: ∃(u_n_k) ⊂ (u_n) a uniformly convergent subsequence.

Denote the limit byu. Then it holds thatku_n_k −uk∞→0. Further, we have:

|(u_n_k−u) (x)−(u_n_k−u) (y)|

|x−y|^α

=

|(u_n_k−u) (x)−(u_n_k −u) (y)|

|x−y|^β

α/β

|(u_n_k −u) (x)−(u_n_k−u) (y)|^1−β/α

≤ |u_n_k−u|^β/α_C_0,β(2ku_n_k−uk∞)^1−β/α ≤(2M)^β/α(2ku_n_k−uk∞)^1−β/α →0 By taking the supremum of x, y∈Ω, x¯ 6=y on the left side, we obtain:

[u_n_k−u]_C0,αΩ¯ →0 In total we have

ku_n_k−uk_0,α;Ω→0 This shows thatD is precompact inC^0,α.

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Theorem 2.8. (reference: see Thm 8.6, p. 338 of [1]) Let Ω ⊂ Rⁿ be a bounded and closed subset. Let k ∈0,1,2,3,· · · and 0 ≤α < β ≤ 1. If D ⊂C^k,β(Ω) is a uniformly bounded subset, then D is precompact in C^k,α(Ω).

Proof. The result follows from repeated use of the Arzelà-Ascoli theorem. We have the following bound onD, for some M >0:

sup

u∈D

kuk_k,β;Ω≤M

It follows directly for any multiindexα= (α1,· · ·, αn) that∀u∈D: kD^αuk_∞≤M

We also have that for|α|< k:

kD^αu(x)−D^αu(y)k_∞≤M|x−y|

and for|α|=k:

kD^αu(x)−D^αu(y)k_∞≤M|x−y|^β

It follows that the family of functions {D^αu:u∈D,|α| ≤k} is equicontinuos.

Let(un)⊂Dbe a sequence. We want to show that there exists a convergent subsequence (u_n_k)⊂(u_n) with limitu, such thatku_n_k −uk_k,α;Ω →0. The argument is an inductive one. Let m = |α| ≤ k. Assume that ∃(u_n_k) ⊂ (u_n) a convergent subsequence, and a limit pointu, such that ∀|α| ≤m:

ku_n_k−uk_m,0;Ω→0 Denote A_m =

αm1,· · · , αm_Nm as the set of multi-indexes of size|α|=m. We start by looking at the multi-indexαm1 = αm₁₁,· · ·, αm1n

. Denote αm1,1 = αm₁₁+ 1,· · · , αm1n

.

For |m| < k, we observe that the sequence D^α^m¹^,1u_n_k is uniformly bounded and equicontinuous. Thus, by Arzelà-Ascoli, there exists a convergent subsequence

u_n_kj

⊂ (u_n_k), and a limit (still denoted u) such that:

kD^α^m¹^,1u_n_k−D^α^m¹^,1uk_∞→0 Also, if|m|=k−1, we observe that

[D^α^m¹^,1unk−D^α^m¹^,1u]_0,α;Ω→0 as shown in theorem 2.7).

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15 2.4. Hölder continuity and Hölder spaces Use this new subsequence

u_n_kj

and repeat the argument for

αm1,2,· · · , αm1,n,· · · , αm_Nm,1,· · · , αm_Nm,n. Thus it holds form+ 1that∃(unk)⊂(un) a convergent subsequence, and a limit pointu, such that ∀|α| ≤m+ 1:

ku_n_k−uk_m+1,0;Ω→0 and if |m|=k−1:

ku_n_k−uk_k,α;Ω→0

For |m| = 0, we have that the sequence (un) is uniformly bounded and equicontinuos, thus there exists a convergent subsequence(un_k)⊂(un), with limit pointu. So that

ku_n_k−uk_∞→0 This concludes the proof.

2.4.3 Parabolic Hölder spaces

We use the standard denition of parabolic Hölder spaces (see Krylov).

Denition

For points z₁ = (x₁, t₁), z₂ = (x₂, t₂) in R^d+1, dene the parabolic distance between them as:

ρ(z1, z2) =|x₁−x2|+|t₁−t2|^1/2 Let0< α≤1and Q⊂R^d+1. Then we denote

[u]_α,α/2;Q= sup

z1,z2∈Q,z16=z2

|u(z₁)−u(z₂)|

ρ^α(z1, z2) , kuk_α,α/2;Q=kuk_0;Q+ [u]_α,α/2;Q

Denition 2.5. Let Q⊂R^d+1 andα∈(0,1]. The parabolic Hölder spaceC^α,α/2 is the set of functions u:Q→R such that

kuk_α,α/2;Q<∞

Denition 2.6. Let Q⊂R^d+1 and α∈(0,1]. The parabolic Hölder spaceC^2+α,1+α/2 is the set of functions u:Q→R such that

[u]2+α,1+α/2;Q:= [ut]_α,α/2;Q+

d

X

i,j=1

uxixj

α,α/2;Q <∞ and

kuk2+α,1+α/2;Q:=|u|_0;Q+

d

X

i=1

|u_x_i|_0;Q+|u_t|_0;Q+

d

X

i,j=1

|u_x_i_x_j|_0;Q+ [u]2+α,1+α/2;Q<∞

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Compactness

We just state the results here without proofs, since the proof method would be the same as in theorem 2.7 and theorem 2.8.

Theorem 2.9. Let Q ⊂ R^d+1 and 0 ≤ α < β ≤ 1. Let D ⊂ C^2+β,1+β/2(Q) be a uniformly bounded subset. Then the set D is precompact in C^2+α,1+α/2(Q).

Proof. The same technique of proof as in Theorem 2.7 and Theorem 2.8.

We also have the following Lemma that can be proved in a similar way:

Lemma 2.8. Assume thatU is a set of functionsu: [0, T]×T^d→Rsuch that u, Du, D²u, D³u, ∂_tu, ∂_tDu, ∂_tD²u∈C_b

(0, T)×T^d and

∂_tu∈C

1 2,1 b

(0, T)×T^d

Then the setU is compact in C^1,2 (0, T)×T^d .

Proof. The same technique of proof as in Theorem 2.7 and Theorem 2.8.

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Chapter 3

Fractional MFG systems with nonlocal coupling

In this chapter, we want to study a fractional Mean Field Game system with non-local couplings F and G. The system we study is of a quite general form, and to the best of our knowledge, no cases in the literature have proven the existence and uniqueness of solutions for this kind of Mean Field Game system. A newly submitted article by Cesaroni et al. [4], shows existence and uniqueness for the stationary case, where one assumes that the MFG system has reached an equilibrium state (Nash equilibrium) as T → ∞. We look at the time-dependent case, where we don't assume the the players have settled to a steady equilibrium.

3.1 The fractional Mean Field Game system

Our aim is to study the following system of equations, which we call the fractional Mean Field Game system with non-local coupling. The system is on the following form:







−∂_tu+ (−∆

T^d)^α² u+H(x, u, Du) =F(x, m(t)) in (0, T)×T^d

∂_tm+ (−∆

T^d)^α² m−div(mD_pH(x, u, Du)) = 0 in (0, T)×T^d m(0) =m0, u(x, T) =G(x, m(T))

(3.1)

where α ∈ (1,2), and the operator (−∆

T^d)^α² is the fractional Laplace operator on the torus. The functions F and Gare both non-local coupling.

We want to show that under certain assumptions onH, F, G and m0, there exists at least one classical solution for the system (3.1). In other words, we look for a pair (u, m)∈C^1,2 (0, T)×T^d

that satises (3.1). Let us state our assumptions.

17

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Assumptions

We make the following assumtions on the system:

1. (Bounds onF and G) F and Gare continuous inT^d×P T^d . 2. (Lipschitz continuity ofF and G) there exists aC₀ >0 s.t.

|F(x₁, m₁)−F(x₂, m₂)| ≤C₀[|x₁−x₂|+d₁(m₁, m₂)]

∀(x1, m1),(x2, m2)∈T^d×P T^d , and

|G(x₁, m₁)−G(x₂, m₂)| ≤C₀[|x₁−x₂|+d₁(m₁, m₂)]

∀(x1, m1),(x2, m2)∈T^d×P T^d .

3. (Uniform regularity of F and G) There exist constants CF, CG > 0, such that sup_m∈P(T^d)kF(·, m)k_C7

b(T^d) ≤C_F andsup_m∈P(T^d)kG(·, m)k_W7,∞(T^d)≤C_G. 4. The Hamiltonian H:T^d×R×R^d→Rsatises for x∈T^d, u∈[−R, R], p∈B_R:

|D^αH(x, u, p)| ≤C_R

with|α| ≤7 andC_R>0 a positive constant dependent on R. 5. The Hamiltonian H satises for x, y∈T^d, u∈[−R, R], p∈R^d:

|H(x, u, p)−H(y, u, p)| ≤C_R(|p|+ 1)|x−y|

Note that this assumption is automatically satised for Hamiltonians on the form H(u, Du).

6. There exists γ ∈Rsuch that for all x∈T^d, u, v∈R, u < v, p∈R^d, H(x, v, p)−H(x, u, p)≥γ(v−u)

This assumption is automatically satised for Hamiltonians on the formH(x, Du), by choosingγ = 0.

7. The probability measurem₀ is absolutely continuous with respect to the Lebesgue measure (meaning A ⊂ T^d measurable: m0(A) = 0 =⇒ λ(A) = 0), has a W^5,∞ T^d

-continuous density function (still denotedm0).

Theorem 3.1. (Existence of classical solution) Under the assumptions 1.-7., there exists at least one classical solution(u, m) to (3.1).

By using the same type of approach like Cardaliaguet in the proof of Thm 3.1.1 in [3], we will show prove the existence of classical solutions. This result depends upon estimates we have made on the fractional Hamilton-Jacobi equation in later chapters.

First, we will start with a remark on weak solutions of the fractional Fokker-Planck equation.

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19 3.2. On the fractional Fokker-Planck equation

3.2 On the fractional Fokker-Planck equation

The fractional Fokker-Planck equation can be written on the following form:

∂tm+ν(−∆_Td)^α² m−div(mb) = 0 in (0, T)×T^d m(0) =m0

(3.2)

whereν >0is a constant, andb: [0, T]×R^d→R^d is a vector eld that is continuous in time and Hölder continuous in space. In the following Lemma, we dene what we say is a weak solution of the Fokker-Planck equation (3.2).

Lemma 3.1. (Weak solutions of (3.2)) A function m ∈L¹ [0, T]×T^d

is said to be a weak solution to (3.2) if m satises the following for any test function φ∈C_c^∞ [0, T]×R^d

Z

T^d

φ(x, T)dm(T) (x)− Z

T^d

φ(0, x)dm0(x)

= Z _T

0

Z

T^d

∂_tφ(t, x)−ν(−∆

T^d)^α² φ(t, x) +hDφ(t, x), b(t, x)i

dm(t) (x)

Two important properties of weak solutions of the Fokker-Planck equation, is that Lemma 3.2. A classical solution m of (3.2) is also a weak solution.

and

Lemma 3.3. If m is a weak solution of (3.2), then it is unique.

We didn't have time to prove these statements, but it should be possible, according to my supervisor. These Lemma's are essential to our analysis of the Fokker-Planck equation, as they allow us to say that a functionmthat solves Fokker-Planck classically, with m0 being a probability density function, then m is also a probability density function.

The method of proving Lemma 3.2 would probably be to insert a classical solutionminto the denition of weak solution. To prove Lemma 3.3 one can probably use Holmgren's uniqueness theorem, or something similar.

Moving on, we will now introduce the stochastic dierential equation (SDE) related to the fractional Fokker-Planck equation.

dXt=b(Xt, t)dt+ν¹^αdLt, t∈[0, T] X₀ =Z₀

(3.3)

where(L_t) is ad-dimensionalα-stable pure jumps Lévy process, with Lévy measure ν(dz) =c_α dz

|z|^d+α

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where

dL_t= Z

|z|<1

zN˜(dt, dz) + Z

|z|≥1

zN(dt, dz)

Here,N describes a poisson process, andN˜ describes a compensated poisson process.

One can prove the following Lemma to be true:

Lemma 3.4. If L(Z₀) =m₀, then m(t) :=L(X_t) is a weak solution of (3.2) Proof. (Idea of proof)

The proof is a consequence of applying Itô's formula. If φ∈C_c^∞ [0, T]×R^d , then (see Applebaum [2], Thm. 4.4.7):

φ(t, X_t) =φ(0, Z₀) +

Z t 0

[∂tφ(s, Xs) +hDφ(s, Xs), b(Xs, s)i]ds +

Z t 0

Z

|z|≥1

[φ(s−, X_s−+z)−φ(s−, X_s−)]N(ds, dx) +

Z t 0

Z

|z|<1

[φ(s−, X_s−+z)−φ(s−, X_s−)] ˜N(ds, dx) +

Z t 0

Z

|z|<1

[φ(s−, X_s−+z)−φ(s−, X_s−)− h∇φ(s−, X_s−), zi]ν(dx)ds where . When we take the expected value on both sides, the following term vanishes

E

"

Z t 0

Z

|z|<1

[φ(s−, X_s−+z)−φ(s−, X_s−)] ˜N(ds, dx)

#

= 0.

Then, from the denition ofm(t), as the Law ofX_t, we should get the following, recalling the denition of the fractional Laplacian (2.1

Z

T^d

φ(t, x)dm(t) (x) = Z

T^d

φ(0, x)dm0(x) +

Z t 0

Z

T^d

∂_tφ(s, x) +hDφ(s, x), b(s, x)i −ν(−∆_Td)^α² φ(s, x)dm(s) (x) This shows thatm is a weak solution to (3.2).

From this stochastic denition ofm(t), we can obtain the following estimate on the mapt7→m(t)∈P T^d

:

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21 3.3. Proof of existence Lemma 3.5. Let m be a weak solution of the fractional Fokker-Planck equation, with α∈(1,2). Then there exists a constant c₀>0 such that m satises:

d1(m(t), m(s))≤c0(1 +kbk∞)|t−s|¹² ∀s, t∈[0, T] (3.4)

Proof. (Idea of proof)

We can use the SDE (3.3), to obtain estimates we want. One can write Xt−Xs=

Z t s

b(τ, Xτ)dτ+ν^1/α Z t

s

dLt

Then, pick aφ∈1−Lip, and compute d₁(m(s), m(t)) = sup

φ∈1−Lip

Z

T^d

φ(x) (m(s)−m(t)) (dx)

= sup

φ∈1−Lip

{E[φ(Xt)−φ(Xs)]}

≤E

X_t−X_s We get that:

E

X_t−X_s

≤E Z t

s

|b(τ, X_τ)|dτ+ν^1/α|L_t−L_s|

Now, for the rst term inside the expectation, it holds that:

E Z t

s

b(τ, Xτ)dτ

≤ kbk∞|s−t|

For the second term, it should at least hold that (according to my supervisor) E

h

ν^1/α|L_t−Ls|i

≤cν^1/α|s−t|¹² wherec >0 is some constant. So, nally we obtain

d1(m(t), m(s))≤c0(kbk_∞+ 1)|s−t|¹² for some constantc₀ >0, which is what we wanted to show.

3.3 Proof of existence

Having made clear our assumptions, and shown the estimates on the Fokker-Planck equation, we will now show that there exists a pair (u, m) that solves the Mean Field Game system (3.1) classically. We rst begin with the idea of the proof.

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3.3.1 Sketch of proof

The idea of the proof is the same as in the proof of Thm3.1.1 in [3], where they proved existence of solutions for the same system of equations, but with a standard Laplacian instead of the fractional one.

We use Schauder's xed point theorem (see Thm. 2.5) to show existence of solutions:

We look at the Banach spaceC⁰ [0, T],P T^d

, and we deneC ⊂C⁰ [0, T],P T^d which turns out to be a closed, convex and compact subset. Then we dene a map, ψ:C → C by using the fractional Mean Field Games equations, and we show that this is a well-dened and continuous map.

We can then apply Schauder's xed point theorem to conclude that the map ψ has at least one xed point, m ∈ C, and then conclude that this xed point is a classical solution of the system (3.1).

3.3.2 Proof

We begin the proof by dening the set C. The set C

We consider the Banach space C⁰ [0, T],P T^d

endowed with the supremum metric d˜(µ, ν) = sup_t∈[0,T_]d1(µ(t), ν(t)), and we dene the following subset

C:=

( µ∈C⁰

[0, T],P

T^d

: sup

s6=t

d1(µ(s), µ(t))

|s−t|¹² ≤C1

) (3.5)

where the constantC1 >0is later to be determined. For this subsetC we will show the following properties.

Lemma 3.6. C is a closed, convex and compact subset ofC⁰ [0, T],P T^d . Proof. We prove each of the statements one by one.

Closed

We have to show that each limit point is contained in the set. Let (µn) ⊂ C and µ_n−→^d^˜ µ^∗ ∈C⁰ [0, T],P T^d

. Then we nd by the triangle inequality:

d1(µ^∗(s), µ^∗(t))≤d1(µ^∗(s), µn(s)) +d1(µn(s), µn(t)) +d1(µn(t), µ^∗(t))

≤C₁|s−t|¹² +d₁(µ^∗(s), µ_n(s)) +d₁(µ^∗(t), µ_n(t))→C₁|s−t|¹² which shows thatµ^∗ ∈ C.

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23 3.3. Proof of existence Convex

Letµ, ν ∈ C andλ∈(0,1). Then:

d1(λµ(s) + (1−λ)ν(s), λµ(t) + (1−λ)ν(t)) = sup

φ∈1−Lip

{ Z

T^d

φ(x) ((λµ(s) + (1−λ)ν(s))−(λµ(t) + (1−λ)ν(t))) (dx)}= sup

φ∈1−Lip

λ

Z

T^d

φ(x) (µ(s)−µ(t)) (dx) + (1−λ) Z

T^d

φ(x) (ν(s)−ν(t)) (dx)

≤λd₁(µ(s), µ(t)) + (1−λ)d₁(ν(s), ν(t))

≤λC1|s−t|¹² + (1−λ)C1|s−t|¹² =C1|s−t|¹². This shows thatλµ+ (1−λ)ν ∈ C, so that convexity holds.

Compact

To show compactness we will use the Arzelá-Ascoli theorem (Thm. 2.4).

We need to show thatC ∈C⁰ [0, T],P T^d

is equicontinuous and relatively compact.

EQ: Given an > 0, we dene δ =

C1

2

. Then we get ∀s, t ∈[0, T] and ∀µ∈ C:

|s−t|< δ =⇒ d₁(µ(s), µ(t))≤C₁|s−t|¹² < C₁ r

C1

2

=, by use of the properties of C.

RC: Let s ∈ [0, T] and dene Ks := {µ(s) :µ∈ C}. We have from denition, that Ks ⊂ P T^d

, and thus it follows from set denitions that the closure Ks is closed in P T^d

. From the compactness ofP T^d

(??), it follows that K_s is compact, and thus from denition of relative compactness, thatK_s is relatively compact.

Thus, by Arzela-Ascoli, we conclude that C is relatively compact inC⁰ [0, T],P T^d Since C also is closed, it follows that it is compact. .

The map ψ

Now, we dene the map ψ:C → C.

Letµ∈ C and denem=ψ(µ)as follows:

Letu be the solution to the fractional Hamilton-Jacobi equation givenµ∈ C −∂_tu+ (−∆

T^d)^α² u+H(x, u, Du) =F(x, µ) in (0, T)×T^d u(x, T) =G(x, µ(T))

(3.6)

We dene the following set:

U :={u:u=u(µ), µ∈ C}

(3.7)

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Further, we denem=ψ(µ)as the solution to the fractional Fokker-Planck equation:

∂tm+ (∆_T^d)^α² m−div(mDpH(x, u, Du(t, x))) = 0 in (0, T)×T^d

m(0,·) =m₀(·) inT^d

(3.8)

We also dene the set:

M:={m:m=m(u), u∈ U } (3.9)

We need to show that the mappingψ has two properties:

1. That the mappingµ→ψ(µ)is well dened.That is, show thatµ∈ C =⇒ ψ(µ)∈ C

2. That the mapping is continuous.

Well dened

Lemma 3.7. The map ψ is well-dened, that is, M ⊂ C. Furthermore the following holds for the setsU andM:

u, Du,· · · , D⁷u, ∂tu, ∂tDu, ∂tD²u∈Cb

(0, T)×T^d

∂_tu∈C

1 2,1 b

(0, T)×T^d

All these quantities are uniformly bounded by a constant U1 > 0, which depends on sup_m∈P(T^d)kG(·, m)k_W7,∞(T^d), α, T, d and the local regularity of F andH.

For M the following estimates holds:

m, Dm,· · · , D⁵m, ∂tm, ∂tDm, ∂tD²m∈C_b

(0, T)×T^d

∂_tm∈C

1 2,1 b

(0, T)×T^d

where all these quantities are uniformly bounded by a constant M1 >0, only dependent on km₀k_W5,∞(T^d), α, T, d, U₁ and the local regularity of H.

Comment

We should give a short comment about why we need as much as C^1,7-regularity in u andC^1,5-regularity inm. The reason comes from our regularity results on the fractional Hamilton-Jacobi equation. We use these results both for the fractional Hamilton-Jacobi equation and the fractional Fokker-Planck equation. To obtain∂tm∈C

1 2,1

b ]0, T[×T^d we needm, Dm,· · ·, D⁵m∈C_b ]0, T[×T^d ,

, due to our computations. Since the Fokker- Planck equation is dependent onu, we need u, Du,· · ·, D⁷u∈Cb ]0, T[×T^d

in order to get enough regularity on m.

This is not ideal of course, but the best we can do for now.

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25 3.3. Proof of existence Proof. Hamilton-Jacobi

Take a µ ∈ C, and look at the fractional Hamilton-Jacobi-equation (3.6). By setting H˜ (t, x, u, p) =H(x, u, p)−F(x, µ(t))in (3.6) we get the expression:

−∂_tu+ (−∆_Td)^α² u+ ˜H(t, x, u, Du) = 0 in (0, T)×T^d u(x, T) =G(x, µ(T))

(3.10)

From the assumptions made on H, F and G, we get that (A0)-(A4) holds. Also the Hölder-assumption (4.2) onHholds, since fors, t∈[0, T], x, y∈T^d, u, v∈[−R, R], p, q ∈ BR

|H˜ (t, x, u, p)−H˜ (s, y, v, q)|

≤ |H(x, u, p)−H(y, v, q)|+|F(x, µ(t))−F(y, µ(s))|

≤LR(|x−y|+|u−v|+|p−q|) +C0(|x−y|+d1(µ(t), µ(s)))

≤LR

|s−t|^1/2+|x−y|+|u−v|+|p−q|

Then, by Theorem 4.2 from the chapter on the fractional HJ-equation, we get there exists a unique uthat solves 3.10, and that it has the following regularity:

u, Du,· · · , D⁷u, ∂tu, ∂tDu, ∂tD²u∈Cb

(0, T)×T^d

∂_tu∈C

1 2,1 b

(0, T)×T^d

where all these quantities are uniformly bounded by a constant U1 >0, which depends on sup_m∈P(T^d)kG(·, m)k_W7,∞(T^d), α, T, d and the local regularity ofF andH.

We will now look at the Fokker-Planck equation, using the function u we obtained from the Hamilton-Jacobi equation.

Fokker-Planck

We look at the equation ∂_tm+ (−∆

T^d)^α² m−div(mD_pH(x, u, Du(t, x))) = 0 in (0, T)×T^d

m(0,·) =m₀(·) inT^d

(3.11)

We can directly apply Theorem 5.1 from the chapter on the fractional Fokker-Planck equation, to obtain a unique solution mof (3.11) that satises:

m, Dm,· · · , D⁵m, ∂tm, ∂tDm, ∂tD²m∈Cb

(0, T)×T^d

∂_tm∈C

1 2,1 b

(0, T)×T^d

where all these quantities are uniformly bounded by a constant M1>0, only dependent on km₀k_W5,∞(T^d), α, T, d, U₁ and the local regularity ofH.

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Recalling our discussion on the Fokker-Planck equation, we have that m is a weak solution of (3.11) (referring to Lemma 3.2 and Lemma 3.3). Thus, Lemma 3.5 gives us the following estimates onm:

d1(m(t), m(s))≤c0(1 +kD_pH(·, Du)k_∞)|t−s|¹² ∀s, t∈[0, T]

SincekDuk_∞≤U₁, it follows thatkD_pH(·, Du)k_∞≤C₂, whereC₂>0is a constant not dependent onµ. By settingC1 ≥c0(1 +C2) we get the sought after constant in the denition ofC. Further, we obtain thatm=ψ(µ) ∈ C, which shows that the map ψ is well-dened.

Continuity

We now want to check that the mapping is continuous, with respect to the metric d˜ dened onC⁰ [0, T],P T^d

.

For this, letµn∈ C be a given sequence, that converges to a pointµ∈ C. Further, let (u_n, m_n)and (u, m)be the corresponding solutions of the system of equations. We want to show thatm_n∈ C converges to m∈ C, because this in turn implies continuity ofψ. Hamilton-Jacobi

We rst begin by looking at the pairs(µ_n, u_n) and (µ, u). We want to show that un

C^1,2

−−−→u.

From the uniform bounds on functionsu∈ U, it follows thatU ⊂⊂C^1,2 (0, T)×T^d by Lemma 2.8. A consequence of Lemma 2.7 is that, if every convergent subsequence of, a sequence (un) ⊂ U converges, then the whole sequence converges in C^1,2 to the same limit point.

So, we only need to prove the following statement:

Lemma 3.8. Every convergent subsequence(un_k)of(un)(convergent in C^1,2) converges to the same limit point,u=u(µ).

Proof. Let(u_n_k) be a convergent subsequence of(u_n), and (µ_n_k) the corresponding µ-s.

Assume that u_n_k −^C−−^2,1→ u˜ ∈ U. From the assumptions µ_n_k converges to µ. The pair (µ_n_k, u_n_k) satises the fractional Hamilton-Jacobi equation:

−∂_tun_k+ (−∆

T^d)^α² un_k+H(x, un_k, Dun_k) =F(x, µn_k(t)) u_n_k(x, T) =G(x, µ_n_k(T))

(3.12)

Also the limit point(µ,u)˜ satises the fractional Hamilton-Jacobi equation −∂_tu˜−(−∆

T^d)^α² u˜+H(x,uD˜ u) =˜ F(x, µ(t))

˜

u(x, T) =G(x, µ(T)) (3.13)

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27 3.3. Proof of existence To conclude the proof, we need to show that all the terms in equation (3.12) converges pointwise to the terms in equation (3.13).

From the assumptionu_n_k −^C−−^1,2→u˜, we get directly that:

k∂_tunk −∂tuk˜ ₀→0

By using Lemma 2.1 from the Preliminaries, we get k(−∆_Td)^α² unk(t,·)−(−∆_Td)^α² u˜(t,·)k_L∞(T^d)

(2.6)

≤ ku_n_k(t,·)−u˜(t,·)k_C2(T^d)

≤ ku_n_k−uk˜ _C1,2(^(0,T⁾T^d) Further, we have for F that

F(x, µn_k(t))−F(x, µ(t))

≤C0[d1(µn_k(t), µ(t))]

≤C0

hd˜(µnk, µ) i

→0 and we get, by the same method, the same result forG:

G(x, µn_k(T))−G(x, µ(T)) →0

The remaining term to look at is theH(x, Dunk). We know that all u∈ U satises kDuk_∞≤U1as shown in lemma 3.7, so we can use thatHis locally Lipschitz continuous:

|H(x, u_n_k, Du_n_k)−H(x,u, D˜ u)˜ | ≤L_H,U₁(|u_n_k −u|˜ +|Du_n_k −D˜u|)→0

This shows that every term in (3.12) converges pointwise to the corresponding terms in equations (3.13) . The equation (3.13) has a unique solution u˜ ∈ U, referring to theorem 4.2. Thus, all convergent subsequences of (un) have the same limit point. This concludes the proof.

By Lemma 2.8, the set U is compact in C^1,2. Since (u_n)⊂ U, and every convergent subsequence of(un)has the same limit pointu∈ U (Lemma 3.8), we conclude by Lemma 2.7 that u_n−^C−−^1,2→u.

Fokker-Planck

Now, we want to show that mn C^1,2

−−−→ m, based on the result thatun → u∈ C^1,2. The set M ⊂⊂ C^1,2 (0, T)×T^d

, as a consequence of Lemma 2.8. Thus, we will do the same as for the fractional Hamilton-Jacobi equation: We will show that every convergent subsequence (mnk) ⊂(mn) converges to the same limit point, and from here conclude by Lemma 2.7 thatmn

C^1,2

−−−→m. We can prove the following statement:

Well-Posedness of a Fractional Mean Field Game System with Non-Local Coupling