ON THE DISCRETIZATION OF SOME NONLINEAR FOKKER-PLANCK-KOLMOGOROV EQUATIONS AND APPLICATIONS

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On the discretization of some nonlinear

fokker-planck-kolmogorov equations and applications

Elisabetta Carlini, Francisco José Silva

To cite this version:

Elisabetta Carlini, Francisco José Silva. On the discretization of some nonlinear fokker-planck- kolmogorov equations and applications. 2017. �hal-01571545�

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FOKKER-PLANCK-KOLMOGOROV EQUATIONS AND APPLICATIONS

ELISABETTA CARLINI AND FRANCISCO J. SILVA

Abstract. In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the non-negativity of the solution, conserves the mass and, as the discretization parameters tend to zero, has limit measure-valued trajectories which are shown to solve the equation. This convergence result is proved by assuming only that the coefficients are continuous and satisfy a suitable linear growth property with respect to the space variable. In particular, under these assumptions, we obtain a new proof of existence of solutions for such equations.

We apply our results to several examples, including Mean Field Games systems and variations of the Hughes model for pedestrian dynamics.

AMS-Subject Classification: 35Q84, 65N12, 65N75.

Keywords: Nonlinear Fokker-Planck-Kolmogorov equations, Numerical Analysis, Semi-Lagrangian schemes, Markov chain approximation, Mean Field Games.

1. Introduction

In this article we consider the nonlinear Fokker-Planck-Kolmogorov (FPK) equation:

∂tm−¹₂ P

1≤i,j≤d

∂_x²_i_,x_j(ai,j(m, x, t)m) + div (b(m, x, t)m) = 0, m(0) = m¯0,

(F P K) where, denoting by P₁(R^d) (respectively P₂(R^d)) the space of probability measures on R^d with first (respectively second) bounded moments, ¯m₀∈ P₂(R^d) and

b:C([0, T];P₁(R^d))×R^d×[0, T]→R^d, ai,j(m, x, t) :=Pr

k=1σik(m, x, t)σjk(m, x, t) ∀i, j= 1, . . . , d,

σi,j :C([0, T];P1(R^d))×R^d×[0, T]→R, ∀i= 1, . . . , d, j = 1, . . . , r.

Equation (F P K) is understood as an equation for measures, in the sense that we seek for a solution m in the space C([0, T];P1(R^d)). Note that the coefficients b and ai,j depend, a priori, on the values m(t)∈ P₁(R^d) in the entire time interval [0, T]. The notion of weak solution to this equation, as well as the assumptions we impose on the coefficientsb andσi,j, will be detailed in Section 2.

Equation (FPK) has been mostly studied in the linear case, i.e. when b(m, x, t) = b(x, t) and σi,j(m, x, t) = σi,j(x, t) for all i = 1, . . . , d and j = 1, . . . , r. This is in part due to the close relation between solutions to (F P K) and solutions to the standard Stochastic Differential Equation (SDE) (1.1) dX(t) =b(X(t), t)dt+σ(X(t), t)dW(t), X(0) =x,

where σ is the matrix d×r matrix whose (i, j) entry isσi,j, W is an r-dimensional Brownian motion and x∈R^d. Indeed, under some assumptions onb and σ_i,j, it is possible to show a correspondence of solutions to (F P K) and the time marginal laws of weak solutions to (1.1) for almost everyx∈R^d with respect to (w.r.t) ¯m0 (see e.g. [46, 31, 11] and the references therein). We refer the reader to [11] for a systematic account of the theory of linear FPK equations and their probabilistic interpretation. When b(m, x, t) = b(m(t), x, t) and σi,j(m, x, t) = σi,j(m(t), x, t) the associated FPK equation is often called

“Sapienza”, Universit`a di Roma, Dipartimento di Matematica Guido Castelnuovo, 00185 Rome, Italy (car- [email protected]).

Institut de recherche XLIM-DMI, UMR-CNRS 7252 Facult´e des sciences et techniques Universit´e de Limoges, 87060 Limoges, France ([email protected]) .

1

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McKean-Vlasov equation and several results exist concerning the well-posedness of the equation and its probabilistic interpretation (see e.g. [33, 51]). In the case of general nonlinear coefficients, the article [12] provides an existence result when σi,j = 0 and in the articles [49, 50] sufficient conditions on the coefficients defining (F P K) are given in order to ensure the existence of solutions in the second order case. The uniqueness of solutions to (F P K) is a difficult matter. The reader is referred to [46, 31] for the analysis in the linear case with rough coefficients, which borrow some ideas from [29, 4] dealing with the analogous problem whenσi,j = 0, and to [47, 48, 13] for the nonlinear case.

Let us now comment on the numerical approximation of FPK equations. One of the most popular numerical schemes in the linear case is the one introduced by Chang and Cooper in [23]. An interesting feature of this finite difference scheme is that the discrete solution preserves some intrinsic properties of the analytical one such as non-negativity and conserves the mass of the initial condition. Starting from this article, several improvements have been obtained in subsequent works, see for instance [60, 30], where high order finite difference schemes have been proposed also for the nonlinear case. Let us also mention [7] dealing with the application of this scheme in the context of stochastic optimal control problems.

Finally, finite element approximation have also been discussed in [58].

In the ‘70s, Kushner has provided a systematic procedure to discretize the solution of a SDE by a discrete-time, discrete-state space Markov chain. The method the author proposes induces finite difference schemes for the associated Kolmogorov backward and forward equations (see e.g. [39, 40, 41]) and so a finite difference discretization of (F P K) in the linear case. A proof of convergence of the scheme by using probabilistic tools (weak convergence of probability measures) is provided under the assumption that the coefficients of the SDE are bounded and uniformly continuous. More recently, in the context of Mean Field Games (MFGs) systems (see [45, 35]), Achdou and Capuzzo-Dolcetta introduced in [2] a semi-implicit finite difference scheme for a linear FPK equation. The scheme is obtained by computing the adjoint scheme of a monotone and consistent discretization of the corresponding dual equation, i.e. the Kolmogorov backward equation. Finally, in the first order caseσ_i,j= 0, we refer the reader to the recent articles [27, 57] dealing with explicit upwind finite volume schemes for the linear equation and to [42] for a similar scheme in the nonlinear and nonlocal case. Let us underline that all the schemes mentioned above share some of the good features of the Chang-Cooper scheme. Indeed, the approximated solutions are non-negative and conserve the initial mass. On the other hand, the main drawback of finite difference and finite element schemes is that, when implemented in their explicit form, they have to satisfy a CFL condition, which implies a strong restriction on the size of the time steps.

A different class of methods in the linear case is the so-called path integration method, introduced in [54]. These are explicit schemes where the marginal laws of the solution of (1.1) are approximated via an Euler-Maruyama discretization of (1.1) using Gaussian one step transition kernels. Recently, in [24], a convergence result for the discrete-time marginal laws in theL¹ strong topology is proved in the framework of a linear and uniformly elliptic FPK equation with unbounded coefficients.

Inspired by the papers [21, 22], dealing with the approximation of Mean Field Games (MFGs), our aim in this article is to provide a discretization of the general (F P K) and to establish some convergence results.

In the linear case, the scheme we propose can be seen as a particular discrete-time, discrete-state space Markov chain approximation of (1.1) and can be obtained as the dual scheme to the Semi-Lagrangian (SL) scheme proposed in [15] for the associated linear Kolmogorov backward equation. In this sense, our discretization is related to the one proposed by Kushner in [39], but using a different Markov chain approximation that allows us to avoid the CFL condition and hence consider large time steps. For this reason, we find that Semi-Lagrangian scheme is a good appellation for our discretization. More importantly, our scheme naturally adapts to the general (F P K) equation, preserves also the positivity, conserves the total mass and allows us to obtain convergence results under rather general assumptions on b andσi,j. Namely, continuity ofb andσi,j and sublinear growth w.r.t. the space variablex, uniformly w.r.t. mandt, are sufficient conditions to prove that if the time stephand space stepρtend to zero and satisfy that ρ²/h →0, then every limit point of the approximated solutions (there exists at least one) solves (F P K). Naturally, if the (F P K) equation admits a unique solution, then we get the convergence of the whole sequence of approximated solutions. As a by-product of this result, we obtain a new proof of existence of solutions to (F P K).

Note also that the initial condition ¯m₀is rather general, we can consider for instance singular measures (e.g. Dirac masses) as initial distributions. Moreover, as we will see in Section 4, we can also construct

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our scheme by using suitable approximations of the coefficients b and σi,j and the convergence result remains valid. Two nonlinear examples in Section 5 show that this is an important issue. Indeed, in these examples we do not have access to an explicit expression of the coefficients and they have to be approximated by suitable and computable coefficients.

Let us point out that a different SL scheme for the (F P K) equation has been proposed in [38] in the linear case. In this article, the advection part and the diffusion reaction term are approximated separately by using two fractional steps. Furthermore, in order to obtain a conservative scheme, the Semi-Lagrangian method applied to the advection part needs to be adjusted. Since our scheme is derived directly from the probabilistic interpretation of (F P K), it has the advantage that the advection and diffusion terms can be treated together and the conservation of the mass is automatically verified (see also the paper [14], where a conservative SL scheme for a parabolic equation in divergence form is studied).

We study in this work several applications of the scheme. We first consider two linear equations. The first one deals with a FPK equation where the underlying dynamics models a damped noisy harmonic oscillator, while the second FPK equation is of first order and describes the distribution of a prey-predator system modeled by a Lotka-Volterra system including effects of seasonality. Even if these two examples are simple, we have chosen them because of the following features. In the first model the exact solution admits an explicit expression, which allows us to quantify exactly the error of the approximation. In the second model, we consider a large time horizon in order to capture the asymptotic behavior of the system, which allows us to show the benefits of being able to chose large time steps. Next, we consider two nonlinear models. In the first one, we apply our scheme to a particular non-degenerated FPK arising in MFGs. The resulting approximation is similar to the one proposed in [21, 22], the main difference being that the non-degeneracy of the system allows us to prove the convergence of the approximation in general dimensions. In the second model, we propose a variation of the Hughes model for pedestrian dynamics (see [36]), where, differently from MFGs, agents do not forecast the evolution of the crowd in order to choose their optimal trajectories. We prove an existence result for the associated FPK, as well as the convergence of the proposed discretization.

The article is organized as follows. In Section 2 we introduce the main notations and recall some fun- damental results about the spaceC([0, T];P1(R^d)), which are the keys to establish the convergence result.

Section 3 presents the scheme, first in the linear case, for pedagogical reasons, and then in the general nonlinear case. In Section 4 we prove our main results, concerning the convergence of the discretization.

Finally, in Section 5 we consider the application of the scheme to the models described in the previous paragraph.

Acknowledgements: The first author acknowledges financial support by the Indam GNCS project

“Metodi numerici per equazioni iperboliche e cinetiche e applicazioni”. The second author is partially supported by the ANR project MFG ANR-16-CE40-0015-01 and the PEPS-INSMI Jeunes project “Some open problems in Mean Field Games” for the years 2016 and 2017.

Both authors acknowledge financial support by the PGMO project VarPDEMFG.

2. Preliminaries

We denote by P(R^d) the space of probability measures on R^d. Given a Borel measurable function Ψ :R^d→R^d

0 andµ∈ P(R^d), we denote by Ψ]µ∈ P(R^d

0) the probability measure defined as Ψ]µ(A) :=

µ(Ψ⁻¹(A)) for all A ∈ B(R^d

0). Given p ∈ [1,∞[, the set Pp(R^d) denotes the subset of P(R^d) with boundedpmoments, i.e.

Pp(R^d) :=

µ∈ P(R^d) ; Z

R^d

|x|^pdµ(x)<∞

. Define

dp(µ1, µ2) := inf (Z

R^d×R^d

|x−y|^pdγ(x, y) ¹_p

; γ∈ P(R^d×R^d), π1]γ=µ1, π2]γ=µ2

) ,

whereπ_i:R^d×R^d →R^d (i= 1,2) is defined asπ_i(x₁, x₂) =x_i. It is well known thatd_p is a distance in Pp(R^d) (see e.g. [59, Theorem 7.3]) and that Pp(R^d), dp

is a separable complete metric space (see e.g.

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[6, Proposition 7.1.5]). Moreover, (2.1) d_p(µ₁, µ₂)^p≤inf

Z

R^d

|x−T(x)|^pdµ₁(x) ; T :R^d→R^dis Borel measurable and T ]µ₁=µ₂

, with equality ifµ1 has no atoms (see [3, Theorem 2.1]). Finally, let us mention an important result that says thatd1 corresponds to the Kantorovic-Rubinstein metric, i.e.

d1(µ1, µ2) = sup Z

R^d

f(x)d(µ1−µ2)(x) ; f ∈Lip₁(R^d)

,

where Lip₁(R^d) denotes the set of Lipschitz functions defined inR^dwith Lipschitz constant less or equal than 1 (see e.g. [59]).

Now, letC ⊆C([0, T];P1(R^d)) and suppose that there exists a modulus of continuity ¯ω: [0,+∞[→R, i.e. ¯ω≥0, ¯ω is continuous and ¯ω(0) = 0, such that

(2.2) sup

µ∈C

d₁(µ(t₁), µ(t₂))≤ω(|t¯ ₁−t₂|) ∀t₁, t₂∈[0, T].

Assume in addition that there existsC >0 such that

(2.3) sup

µ∈C

sup

t∈[0,T]

Z

R^d

|x|²dµ(t)(x)≤C.

Since the set

µ∈ P1(R^d) ; R

R^d|x|²dµ(x)≤C is compact inP1(R^d) (see [6, Proposition 7.1.5]), (2.3), (2.2) and the Arzel´a-Ascoli theorem yield the following result.

Lemma 2.1. Under the above assumptions,C is a relatively compact subset ofC([0, T];P1(R^d)).

For notational convenience, for ψ = b, σi,j we set ψ[µ](x, t) := ψ(µ, x, t). We say that m ∈ C([0, T];P1(R^d)) solves (F P K) if for all t ∈ [0, T] and ϕ ∈ C₀^∞(R^d), the space of C^∞-functions with compact support, we have

(2.4)

R

R^dϕ(x)dm(t)(x) = R

R^dϕ(x)d ¯m0(x) +Rt 0

R

R^d[b[m](x, s)· ∇ϕ(x)] dm(s)(x)ds +Rt

0

R

R^d

h1 2

P

i,ja_i,j[m](x, s)∂²_x

i,x_jϕ(x)i

dm(s)(x)ds.

(H)We will suppose that:

(i) The mapsbandσare continuous.

(ii) There existsC >0 such that

(2.5) |b[µ](x, t)|+|σ[µ](x, t)| ≤C(1 +|x|) ∀µ∈ P1(R^d), t∈[0, T].

The aim of this article is to study convergent numerical schemes for solutions to (F P K) (if they exists).

As it can be guessed from the references [46, 31, 11] in the linear case, i.e. whenbandσ_i,jdo not depend on m, the existence of solutions to (F P K) should be related with the existence of (weak) solutions to the McKean-Vlasov equation

(2.6) dX(t) =b[m](X(t), t)dt+σ[m](X(t), t)dW(t), X(0) =X0.

In (2.6),W is anr-dimensional Brownian motion defined on a probability space (Ω,F,P),mbelongs to C([0, T];P₁(R^d)) and satisfiesm(t) = Law(X(t)) for allt∈[0, T], where we have denoted by Law(Y) the law induced inR^d by a d-valued random variable Y, and X0 is a random variable, independent ofW, and such that Law(X0) =m0.

This observation, relating formally solutions of (F P K) and (2.6), leads naturally to study the laws of discrete approximations of (2.6), for which existence is not difficult to show, and then to study their limit behavior. This strategy will be followed in the next sections.

Remark 2.1. In this article we do not tackle the study of uniqueness of solutions to (F P K). As it can be seen in[46, 31, 11], in the linear case, the study of uniqueness is already quite complicate in the absence of first order information, w.r.t. the space variable, of b andσ. We refer the reader to [47, 48, 13] for some recent and interesting results in the general nonlinear case.

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3. The fully-discrete scheme

In this section we describe the scheme we propose and study its main properties. In order to introduce the main ideas we will start by considering first the (F P K) equation with σ= 0 andb independent of m, i.e. the first order linear FPK equation, also calledcontinuity equation. Then, we will consider the stochastic caseσ 6= 0 but still with coefficientsb andσ independent of m. Finally, the scheme for the general (F P K) will easily follow by freezing the m dependence of b and σ. We motivate the schemes by assuming stronger assumptions onb and σ, which will imply uniqueness of the underlying SDEs, in order to take advantage of the semi-group properties of the solutions and somehow guess a consistent approximation. However, after the scheme is derived, we can dispense with these assumptions and study the limit behavior under assumption(H)only.

So, let us assume first thatσ≡0 and thatbdoes not depend onm, i.e. b[m](x, t) =b(x, t). In addition to (H), assume thatb is Lipschitz w.r.t. x, uniformly int∈[0, T]. For any 0≤s≤t≤T and x∈R^d, we set Φ(x, s, t) =X(t) whereX is the unique solution of

(3.1) X˙(t⁰) =b(X(t⁰), t⁰) fort⁰∈]s, T[, X(s) =x.

We have that Φ defines a measurable function of (x, s, t) (if t ≤swe simply set Φ(x, s, t) =x). Then, m∈C([0, T];P1(R^d)) defined as

(3.2) m(t)(A) := Φ(·,0, t)]m¯₀(A) ∀A∈ B(R^d), t∈[0, T],

is the unique solution of (F P K) (see [5]). We also have that for allt∈[0, T] andh∈[0, T−t]

(3.3) m(t+h)(A) = Φ(·, t, t+h)]m(t)(A) ∀A∈ B(R^d).

Given N ∈Nwe seth=T /N. Let us consider the following explicit time discretization of (3.2), based on a standard explicit Euler approximation of (3.1) and property (3.3)

(3.4) m0= ¯m0, mk+1:= Φk]mk, where Φk(x) :=x+hb(x, tk) ∀k= 0, . . . , N −1.

The sequencesmk and Φk (k= 0, . . . , N) depend of course onhbut we have omitted this dependence in order to ease the reading. Let us now introduce some standard notations that will be used for the space discretization. Letρ >0 be a givenspace step, and consider a uniform space grid

G_ρ:={x_i=iρ; i∈Z^d}.

Given a regular triangulation Tρ of R^d, with vertices belonging to Gρ, we consider a P1 basis (βi)_i∈_Zd, i.e. for all i∈Z^d, βi is continuous, affine on each element ofTρ and such that βi(xj) = 1 ifi =j and βi(xj) = 0 otherwise. Moreover, the support supp(βi) ofβi is compact andβi satisfies

0≤β_i≤1 ∀i∈Z^d, and X

i∈Z^d

β_i(x) = 1 ∀x∈R^d. We look for a discretization of (3.4) taking the form

(3.5) m_k= X

i∈Z^d

m_i,kδ_x_i ∀k= 0, . . . , N−1.

For alli∈Z^d, let us define

Ei:=n

x∈R^d; |x−xi|_∞≤ ρ 2 o

.

In Section 4 we will let ρ↓ 0, thus, without loss of generality, we can assume that ¯m0(∂Ei) = 0 for all i∈Z^d. We define the weightsm_i,k of the Dirac masses in (3.5) inductively as

(3.6) mi,0= ¯m0(Ei), mi,k+1= X

j∈Z^d

βi(Φj,k)mj,k ∀k= 0, . . . , N −1, i∈Z^d, where

(3.7) Φi,k:= Φk(xi) =xi+hb(xi, tk) ∀i∈Z^d.

The sequences of weights in (3.6) depends on (ρ, h), but, for notational convenience, we have omitted this dependence.

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Remark 3.1. (i)In order to understand the intuitive meaning of (3.6), take d= 1,ρ= 1 andβi(x) :=

max{1− |x−xi|,0} for all i ∈ Z, x∈ R. Then, the mass mi,k+1, at xi at time tk+1, is obtained by first considering the setAi,k ofj’s such thatΦj,k∈supp(βi)and then adding the massesmj,k(j∈ Ai,k) weighted by1− |Φj,k−xi|. For instance, ifΦj,k=xi+ 1/2 then half of the massmj,k will go toxi and the other half will go to xi+1.

(ii)In this deterministic setting ifd= 1it is easy to check that (3.6)coincides with the scheme proposed in[55].

Now, if σ[m](x, t) = σ(x, t) is not identically zero we can consider the same type of scheme, taking into account that the characteristics curves are stochastic. Indeed, consider a filtered probability space (Ω,F,F,P), anr-dimensional Brownian motionW defined in this probability space and adapted to the filtration F:={Ft}_t∈[0,T]. Define Φ : Ω×R^d×[0, T]×[0, T]→R^d as Φ(ω, x, s, t) =xift≤s and, for s < t, Φ(ω, x, s, t) =X(t, ω), whereX solves

(3.8) dX(t⁰) =b(X(t⁰), t⁰)dt⁰+σ(X(t⁰), t⁰)dW(t⁰) for t⁰∈]s, T[, X(s) =x.

Then, assuming thatb andσ are Lipschitz with respect tox, uniformly int∈[0, T], we have that (see e.g. [31])

(3.9) m(t)(A) :=

Z

Ω

Φ(ω,0, t)]m¯0(A)dP(ω) =E(Φ(·,0, t)]m¯0(A)) ∀A∈ B(R^d), t∈[0, T].

Analogously to (3.3), we have that

(3.10) m(t+h)(A) =E(Φ(·, t, t+h)]m(t)(A)) ∀A∈ B(R^d).

Therefore, if we discretize the Brownian motionW by anr-dimensional random walk withN time steps, the stochastic characteristic

X(t+h) =X(t) + Z t+h

t

b(X(t⁰), t⁰)dt⁰+ Z t+h

t

σ(X(t⁰), t⁰)dW(t⁰), can be approximated with an explicit Euler scheme by

(3.11) X(t+h) =X(t) +hb(X(t), t) +

√

rhσ(X(t), t)Z,

whereZ is anr-valued random variable, independent ofX(t), satisfying that for all`= 1, . . . , r, (3.12) P({Z^`= 1) =P(Z^`=−1) = 1

2d and P



 [

1≤`1<`₂≤r

{Z^`¹ 6= 0} ∩ {Z^`² 6= 0}



= 0.

Relations (3.11)-(3.12) motivate the following extensions of Φi,k, defined in (3.4), (3.13) Φ^`,+_i,k := xi+hb(xi, tk) +√

rhσ`(xi, tk) ∀i∈Z^d, k= 0, . . . , N−1, `= 1, . . . , r, Φ^`,−_i,k := x_i+hb(x_i, t_k)−√

rhσ_`(x_i, t_k) ∀i∈Z^d, k= 0, . . . , N−1, `= 1, . . . , r.

Inspired by (3.6), relation (3.10) induces the followingexplicitscheme

(3.14)

mi,0 := m¯0(Ei) ∀i∈Z^d, mi,k+1 := _2r¹

r

P

`=1

P

j∈Z^d

h

βi(Φ^`,+_j,k) +βi(Φ^`,−_j,k)i

mj,k ∀i∈Z^d, k= 0, . . . , N−1.

Remark 3.2. Note that the previous scheme is conservative. Indeed, for allk= 0, . . . , N−1, X

i∈Z^d

mi,k+1= X

j∈Z^d

mj,k

2r

r

X

`=1

X

i∈Z^d

h

βi(Φ^`,+_j,k) +βi(Φ^`,−_j,k)i

=X

i∈Z^d

mi,k,

and soP

i∈Z^dmi,k+1=P

i∈Z^dmi,0= 1.

Markov chain interpretation: Note that (3.14) can be interpreted in terms of a discrete-time and countably-state space Markov chain. Indeed, given the initial lawm_·,0onGρ, consider the non-homogeneous

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Markov chain{Xk; k= 0, . . . , N}with values inGρdefined by the previous initial law and the transition probabilities

p^(k)_ji :=P Xk+1 =xi

Xk =xj := 1

2r

r

X

`=1

h

βi(Φ^`,+_j,k) +βi(Φ^`,−_j,k)i

∀i, j∈Z^d, k= 0, . . . , N−1.

Then, (3.14) gives the distribution ofXk for allk= 0, . . . , N. Remark 3.3. (i)Note that ifσ≡0, we recover the scheme (3.6).

(ii) As we will see in Section 4, the Markov chain (Xk)^N_k=0 is a consistent approximation, in the sense of Kushner (see [40]), of the diffusion in (3.8) with s = 0 and with Law(X0) = ¯m0. It is easily seen that, as a function ofm¯0, scheme (3.14) can be formally understood as the dual scheme associated to the Semi-Lagrangian scheme(see[52]) for the Kolmogorov backward equation

∂_tu−¹₂ P

1≤i,j≤d

a_i,j∂_x²

i,x_ju+b· ∇u = 0, u(·, T) = g(·),

as a function ofg∈Cb(R^d) (whereCb(R^d)is the space of bounded continuous functions in R^d).

(iii)In[19, Section 3.1], it is shown that scheme (3.14)can also be constructed from the weak formulation of (F P K) (when bandσ are independent of m).

In the general non-linear case, as we have explained at the end of Section 2, formally,msolves (F P K) iff for allt ∈[0, T], we have thatm(t) = Law(X(t)), whereX solves (2.6) (assuming that (2.6) admits a solution in a weak sense). On the other hand, even in the particular case of regular coefficients and local in time dependence onm, i.e. b[m](x, t) =b(m(t), x, t) andσ[m](x, t) =σ(m(t), x, t), withbandσ regular w.r.t. x, we have thatX is not a Markov process. Nevertheless, loosely speaking again,X solves (2.6) iff Law(X(·))∈C([0, T];P1(R^d)) is a fixed point of the application

(3.15) µ∈C([0, T];P1(R^d))7→ F(µ) := Law(X[µ](·))∈C([0, T];P1(R^d)), whereX[µ](·) solves

(3.16) dX(t) =b[µ](X(t), t)dt+σ[µ](X(t), t)dW(t) fort∈]0, T[, X(0) =X₀.

Since for every fixedµ, X[µ] defines a Markov diffusion, we can apply (3.14) to approximate its law.

Even if the previous discussion is purely formal, it provides the idea to construct a natural discretization of (F P K) by considering a discrete version of the fixed-point problem (3.15), which will be constructed using (3.14). However, sinceb[·](x, t) and σ[·](x, t) act onC([0, T];P₁(R^d)), given ρandhwe first need to extend elements on

S^ρ,h:=







(µi,k)_i∈_Zd, k=0,...,N; µi,k≥0, ∀i∈Z^d, X

i∈Z^d

µi,k = 1, X

i∈Z^d

|xi|µi,k <∞, for allk= 0, . . . , N





 ,

to elements inC([0, T];P₁(R^d)). This can be naturally done by using time interpolation. Givenµ∈ S^ρ,h, we still denote byµthe element ofC([0, T];P₁(R^d)) defined by

(3.17) µ(t) :=

t−tk

h

X

i∈Z^d

µi,k+1δx_i+

tk+1−t h

X

i∈Z^d

µi,kδx_i ift∈[tk, tk+1[, for allk= 0, . . . , N−1. Using this notation, define

(3.18) µ∈ S^ρ,h7→ F^ρ,h(µ) := (Law(Xk[µ]))_k=0,...,N ∈ S^ρ,h,

where we computeP(X_k[µ] =x_i) :=m_i,k[µ] recursively with (3.14) with Φ^`,+_j,k and Φ^`,−_j,k replaced by Φ^`,+_i,k[µ] :=x_i+hb[µ](x_i, t_k) +√

rhσ_`[µ](x_i, t_k), Φ^`,−_i,k[µ] :=x_i+hb[µ](x_i, t_k)−√

rhσ_`[µ](x_i, t_k), respectively. Forµ ∈ S^ρ,h let us set νk[µ] := F^ρ,h(µ)

k (k = 0, . . . , N). By definition of the scheme, using that ¯m0∈ P2(R^d) and that ¯m0(∂Ei) = 0, we have

Z

R^d

|x|²dν0[µ](x) = X

i∈Z^d

|xi|²m¯0(Ei) = X

i∈Z^d

Z

E_i

|x−(x−xi)|²d ¯m0(x)≤2 Z

R^d

|x|²d ¯m0(x) +dρ²/2<+∞.

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Moreover, arguing exactly as in the proof of Proposition 4.1 in the next section, under (H)(ii) we obtain the existence ofc >0, independent ofµ, such that

(3.19)

Z

R^d

|x|²dνk[µ](x) = X

i∈Z^d

|xi|²νi,k[µ]≤c ∀k= 0, . . . , N, ∀µ∈ S^ρ,h. In particular,F^ρ,h is well-defined. The discretization of (F P K) we propose is

(3.20) findm∈ S^ρ,h such that m=F^ρ,h(m),

or equivalently, findm∈ S^ρ,hsuch that

(3.21)

m_i,0= ¯m₀(E_i) ∀i∈Z^d, mi,k+1= _2r¹

r

P

`=1

P

j∈Z^d

h

βi(Φ^`,+_j,k[m]) +βi(Φ^`,−_j,k[m])i

mj,k ∀i∈Z^d, k= 0, . . . , N −1.

Now, let us prove the existence of solutions of (3.20). In the following proof we identify S^ρ,h with a subset ofP1(R^d)^N⁺¹ by letting for allµ∈ S^ρ,h

(3.22) µk =X

i∈Z^d

µi,kδx_i ∀k= 0, . . . , N.

Proposition 3.1. There exists at least one solution m^ρ,h∈ S^ρ,h of (3.20).

Proof. As before, for µ ∈ S^ρ,h denote by νk[µ] := F^ρ,h(µ)

k (k = 0, . . . , N). Let c > 0 be such that (3.19) holds. Then, defining

S_c^ρ,h:=







µ∈ S^ρ,h; X

i∈Z^d

|x_i|²µ_i,k ≤c, ∀k= 0, . . . , N





 ,

we have thatS_c^ρ,h is convex andF^ρ,h(S_c^ρ,h)⊆ S_c^ρ,h. Moreover, by [6, Proposition 7.1.5 and Proposition 5.1.8], Fatou’s Lemma and the identification (3.22), we have thatS_c^ρ,his a compact subset ofP1(R^d)^N+1. Finally, if µn ∈ S^ρ,h converge toµ ∈ S^ρ,h, seen as elements of P1(R^d)^N⁺¹, then, using the extension (3.17), assumption(H)(i) implies that Φ^`,+_j,k[µ_n] and Φ^`,−_j,k[µ_n] converge to Φ^`,+_j,k[µ] and Φ^`,−_j,k[µ], respectively, which implies the continuity of F^ρ,h. Since the topology of P1(R^d) is the restriction to P1(R^d) of the topology induced by the modified Kantorovic-Rubinstein norm on the linear space of all bounded Borel measures onR^dwith respect to which all the Lipschitz functions are integrable (see the discussion before Proposition 1.1.4 in [10]), the existence of a solution of (3.20) follows from Schauder’s fixed point theorem.

The computation in Remark 3.2 applies in the nonlinear case and so the scheme is conservative.

Remark 3.4. [Explicit and implicit schemes] Note that if for all t∈[0, T],b[m](x, t) = ˆb(m(· ∧t), x, t) andσ[m](x, t) = ˆσ(m(·∧t), x, t)for some functionsˆbandσˆ defined inC([0, T];P1(R^d))×R^d×[0, T], then the scheme (3.21) is explicit in the time steps and the existence of solution, as well as the uniqueness, of the scheme is straightforward. In the general case, the scheme is implicit in the time steps and, as we have seen in the proof of the previous proposition, the existence of solutions is a consequence of Shauder fixed point theorem. The latter situation is the one we face when we consider MFGs, as we will see in Section 5.3. In the implicit cases, the uniqueness of solutions is generally not true and its fulfilment depends on the problem at hand.

4. Convergence analysis

Let us first introduce and recall some classical properties of the linear interpolation operator we consider (see e.g. [25, 56] for further details). LetB(Gρ) the space of bounded functions onGρ and forf ∈B(Gρ) setfi:=f(xi). We consider the following linear interpolation operator

(4.1) I[f](·) := X

i∈Z^d

fiβi(·) for f ∈B(Gρ).

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Given φ∈ Cb(R^d), let us define ˆφ∈ B(Gρ) by ˆφi :=φ(xi) for all i ∈Z^d. Suppose thatφ :R^d → Ris Lipschitz with constantL. Then,

(4.2) I[ ˆφ] is Lipschitz with constant√

dL and sup

x∈R^d

|I[ ˆφ](x)−φ(x)|=c0ρ,

for somec₀>0. On the other hand, ifφ∈ C²(R^d), with bounded second derivatives, then there exists c1>0 such that

(4.3) sup

x∈R^d

|I[ ˆφ](x)−φ(x)|=c₁ρ².

Now, let {Nn}n∈N be a sequence in N such that N_n → ∞ as n → ∞ and set h_n := T /N_n. Given a sequence of space steps ρ_n, such that ρ_n → 0 as n → ∞, we want to study the limit behavior of the extensions to C([0, T];P₁(R^d)), defined in (3.17), of sequences of solutions mⁿ := m^ρⁿ^,hⁿ ∈ S^ρⁿ^,hⁿ of (3.20), withρ=ρ_n andh=h_n (by Proposition 3.1 we now that (3.20) admits at least one solution).

First note that by considering the transport planT(x) =xi ifx∈Ei, and arbitrarily defined in ∂Ei

(because ¯m0(∂Ei) = 0), we have thatT ]m¯0=mⁿ(0). Thus, inequality (2.1) withp= 1 yields (4.4) d1( ¯m0, mⁿ(0))≤

Z

R^d

|x−T(x)|d ¯m0(x) = X

i∈Z^d

Z

Ei

|x−xi|d ¯m0(x)≤√ dρn/2,

which implies thatmⁿ(0)→m¯0 inP1(R^d) asn→ ∞. We will prove in this section that under suitable conditions overρ_nandh_nthe setC:={mⁿ ; n∈N} satisfies (2.2) and (2.3). Therefore, Lemma 2.1 will imply that mⁿ has at least one limit point m∈ C([0, T];P1(R^d)). Finally, we will show thatm solves (F P K).

In the proof of (2.2) and (2.3) we will need some properties of the Markov chainXⁿ, defined by the transition probabilities

p^n,k_ji :=P X_k+1ⁿ =xi

X_kⁿ=xj

:= 1 2r

r

X

`=1

h

βi(Φ^`,+_j,k[mⁿ]) +βi(Φ^`,−_j,k[mⁿ])i

∀i, j∈Z^d,

andk= 0, . . . , N−1. Note that (3.21) implies that the mariginal distributions of this chain are given by mⁿ. Moreover, it is easy to check that (4.2) (resp. (4.3)) implies that if φ:R^d →Ris Lipschitz (reps.

C²with bounded second derivatives), then

(4.5)

E φ(X_k+1ⁿ )

X_kⁿ=xi

=_2r¹ Pr

`=1

h

I[ ˆφ](Φ^`,+_i,k[mⁿ]) +I[ ˆφ](Φ^`,−_i,k[mⁿ])i

= _2r¹ Pr

`=1

h

φ(Φ^`,+_i,k[mⁿ]) +φ(Φ^`,−_i,k[mⁿ])i

+O(ρn), (resp.) E φ(X_k+1ⁿ )

X_kⁿ =xi

=_2r¹ Pr

`=1

h

φ(Φ^`,+_i,k[mⁿ]) +φ(Φ^`,−_i,k[mⁿ])i

+O(ρ²_n).

Proposition 4.1. Suppose that ρ²_n =O(hn). Then, there exists a constantc >0such that

(4.6) sup

n∈N

sup

t∈[0,T]

Z

R^d

|x|²dmⁿ(t)≤c.

Proof. By (3.17), it is enough to show that there existsc >0, independent ofn, such that

(4.7) sup

k=0,...,N_n

Z

R^d

|x|²dmⁿ(t_k)≤c.

For notational convenience we will omit the superscriptn. By definition, Z

R^d

|x|²dm(tk+1)(x) =X

i∈Z^d

|xi|²mi,k+1 =E(|Xk+1|²),

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from which, using (4.5) and(H)(ii), E(|Xk+1|²) = P

i∈Z^dE |Xk+1|²

Xk =xi

mi,k,

= _2r¹ Pr

`=1

P

i∈Z^d

h

Φ^`,+_i,k[m]

2+

Φ^`,−_i,k[m]

2i

mi,k+O(ρ²_n),

= E

h

Xk+hnb[m](Xk, tk) +√

rhnσ[m](Xk, tk)Zk

2i

+O(ρ²_n),

= E

|Xk|²+h²_n|b[m](Xk, tk)|²+rhnPr

`=1|σ`[m](Xk, tk)|²+ 2hnXk·b[m](Xk, tk) +O(ρ²_n),

≤ (1 +Ch_n)E(|Xk|²) +Ch_n+O(ρ²_n),

where Z_k is anr-valued random variable, independent of X_k, satisfying (3.12) and C is independent of n. Iterating, we get

R

R^d|x|²dm(tk+1)(x) ≤ (1 +Chn)^hn^T E(|X0|²) + (Chn+O(ρ²_n))PN

k=0(1 +Chn)^k+O(ρ²_n)

≤ e^CTE(|X₀|²) +_h^T

n

Ch_n+O(ρ²_n)

e^CT +O(ρ²_n)

≤ e^CTE(|X0|²) +

CT +O_ρ2 n

h_n

e^CT,

from which the result follows.

Now, we prove a consistency property of the chainXⁿ in the spirit of Kushner [40]. For all 0≤k≤ Nn−1 let us define δkXⁿ :=X_k+1ⁿ −X_kⁿ, Y_kⁿ :=δkXⁿ−E(δkXⁿ|X_kⁿ).

Lemma 4.1. For allk= 0, . . . , N_n−1 we have that E(δkXⁿ|X_kⁿ) = hnb[mⁿ](X_kⁿ, tk), E(|Y_kⁿ|²|X_kⁿ) = hnPr

`=1|σ`[mⁿ](X_kⁿ, tk)|²+O(ρ²_n).

Proof. By definition ofp^n,k_i

k,i_k+1 we have E(δ_kXⁿ|X_kⁿ=x_i_k) = P

i_k+1 x_i_k+1−x_i_k p^n,k_i

k,ik+1

= _2r¹ Pr

`=1

I[id−xi_k](Φ^`,+_i

k,k[mⁿ]) +I[id−xi_k](Φ^`,−_i

k,k[mⁿ])

= hnb[mⁿ](xi_k, tk),

where id(x) =xand the last equality follows from the fact that I[id−xi_k](y) =y−xi_k for ally ∈R^d. Analogously,

E(|Y_kⁿ|²|X_kⁿ =x_i_k) =X

i_k+1

x_i_k+1−x_i_k−E(δ_kXⁿ|X_kⁿ=x_i_k)2

p^n,k_i

k,ik+1. Using (4.3) and the definition ofp^n,k_i

k,i_k+1 again we get that E(|Y_kⁿ|²|Xk=x_i_k) =h_n

d

X

`=1

|σ`[mⁿ](x_i_k, t_k)|²+O(ρ²_n),

from which the result follows.

Now, we prove thatC:={mⁿ ; n∈N} satisfies (2.2).

Proposition 4.2. Suppose that ρ²_n =O(hn). Then, there exists a constantC >0 such that

(4.8) sup

n∈N

d₂(mⁿ(t), mⁿ(s))≤C|t−s|¹² ∀t, s∈[0, T].

In particular, sinced₁≤d₂, we have thatC satisfies (2.2).

Proof. The proof is divided into two steps:

Step 1: We first show that for givenNn there exists a constantC, independent ofn, such that (4.9) d2(mⁿ(tk), mⁿ(t⁰_k))≤Cp

|k−k⁰|hn ∀k, k⁰ = 0, . . . , Nn.

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We assume, without loss of generality, thatk⁰= 0. For notational convenience, we omit the superscript non the sequencesX_kⁿ,δkXⁿ andY_kⁿ. By the definition ofd2 we have

(4.10) d2(mⁿ(tk), mⁿ₀)≤

E(|Xk−X0|²)

1 2. We have that

(4.11) E |Xk−X0|²

=E

k−1

X

p=0

(Yp+E(δpX|Xp))

2

≤2E

k−1

X

p=0

Yp

2

+ 2E

k−1

X

p=0

E(δpX|Xp)

2

. Now, for 0≤r < l≤k−1 conditioning onFl:=σ(X0, . . . , Xl) and using that, by the Markov property, E(δlX|Fl) =E(δlX|Xl) we get

E(Y_l·Y_r) =E[(δ_lX−E(δ_lX|Xl))·Y_r] =E(E

(δ_lX−E(δ_lX|Xl)) Fl

·Y_r) = 0, and so, by Lemma 4.1,

(4.12)

E

Pk−1 p=0Yp

2

=Pk−1

p=0E(E(|Yp|²|Xp))

=hnPk−1 p=0

Pr

`=1E(|σ`[mⁿ](Xp, tp)|²) +O kρ²_n

≤Chnk 1 + sup_p=0,...,NE|Xp|²

+O kρ²_n . On the other hand, using Lemma 4.1 again,

k−1

X

p=0

E(δ_pX|Xp)

2

≤Ch²_nk

k−1

X

p=0

(1 +|Xp|²), and so

(4.13) E

k−1

X

p=0

E(δ_pX|X_p)

2

≤Ch²_nk²

1 + sup

p=0,...,NE|X_p|²

.

By Proposition 4.1, (4.12), (4.13), (4.11) and our assumptionρ²_n =O(h_n), we get the existence ofC >0 such that (4.9) holds true.

Step 2: proof of (4.8): Let 0≤s < t≤T andk⁰,ksuch thats∈[tk⁰, tk⁰+1[ andt∈[tk, tk+1[. Then, by the triangular inequality

(4.14) d₂(mⁿ(t), mⁿ(s))≤d₂(mⁿ(t), mⁿ(t_k)) +d₂(mⁿ(t_k), mⁿ(t_k⁰₊₁)) +d₂(mⁿ(t_k⁰₊₁), mⁿ(s)).

By the dual representation ofd²₂(·,·) (see [59, Theorem 1.3]), this function is convex inP2(R^d)× P2(R^d).

Thus, relations (3.17) and (4.9) imply that d²₂(mⁿ(t), mⁿ(tk))≤

t−tk

h_n

d²₂(mⁿ(tk+1), mⁿ(tk))≤C²(t−tk), from which

(4.15) d2(mⁿ(t), mⁿ(tk))≤C(t−tk)¹². Analogously,

(4.16) d2(mⁿ(tk⁰+1), mⁿ(s))≤C(tk⁰+1−s)

1 2.

Relations (4.14), (4.15), (4.16) and the Cauchy-Schwarz inequality imply the existence ofC > 0, independent ofn, such that

d2(mⁿ(t), mⁿ(s))≤C|t−s|¹².

Relation (4.8) follows.

For notational convenience, for allϕ∈C₀^∞(R^d) let us set Lb,σ,ϕ[µ](x, t) :=¹₂X

i,j

ai,j[µ](x, t)∂_x²_i_,x_jϕ(x) +b[µ](x, t)· ∇ϕ(x) ∀(µ, x, t)∈C([0, T];P1(R^d))×R^d×[0, T].

Now, we have all the elements to prove our main convergence result.

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Theorem 4.1. Assume (H)and that ρ²_n =o(hn). Then, every limit pointm∈C([0, T];P1(R^d))of mⁿ (there exists at least one)solves (F P K). In particular, under assumption (H), (F KP)admits at least one solution.

Proof. By Proposition 4.1, Proposition 4.2 and Lemma 2.1, with C ={mⁿ ; n∈ N}, the sequence mⁿ has at least one limit point m. We use the same superscript n to index a subsequence mⁿ converging to min C([0, T];P1(R^d)) and we need to show that msatisfies (2.4). Let t∈]0, T] and, wihtout loss of generality, consider a sequencetn⁰ =n⁰hn such thatt∈]tn⁰, tn⁰+1]. Then, for everyϕ∈C₀^∞(R^d)

(4.17)

Z

R^d

ϕ(x)dmⁿ(t_n⁰)(x) = Z

R^d

ϕ(x)dmⁿ(0)(x) +

n⁰−1

X

k=0

Z

R^d

ϕ(x)d [mⁿ(t_k+1)−mⁿ(t_k)] (x).

For allk= 0, . . . , n⁰−1 we have that

(4.18) R

R^dϕ(x)dmⁿ(tk+1)(x) =P

i∈Z^dϕ(xi)mⁿ_i,k+1

=P

j∈Z^dmⁿ_j,k_2r¹ Pr

`=1

P

i∈Z^dϕ(xi)h

βi(Φ^`,+_j,k[mⁿ]) +βi(Φ^`,−_j,k[mⁿ])i

=P

`=1

h

I[ϕ](Φ^`,+_j,k[mⁿ]) +I[ϕ](Φ^`,−_j,k[mⁿ])i

=P

`=1

h

ϕ(Φ^`,+_j,k[mⁿ]) +ϕ(Φ^`,−_j,k[mⁿ])i

+O(ρ²_n)

=P

j∈Z^dmⁿ_j,k[ϕ(xj) +hnLb,σ,ϕ[mⁿ](xj, tk)] +O ρ²_n+h²_n

=R

R^d[ϕ(x) +hnLb,σ,ϕ[mⁿ](x, tk)] dmⁿ(tk)(x) +O ρ²_n+h²_n ,

where we have used a fourth order Taylor expansion for the terms ϕ(Φ^`,+_j,k[mⁿ]) and ϕ(Φ^`,−_j,k[mⁿ]). As a consequence, (4.17) yields

(4.19) Z

R^d

ϕ(x)dmⁿ(tn⁰)(x) = Z

R^d

ϕ(x)dmⁿ(0)(x) +hn n⁰−1

X

k=0

Z

R^d

Lb,σ,ϕ[mⁿ](x, tk)dmⁿ(tk)(x) +O ρ²n

hn

+hn

.

Assumption (H)(i) implies the existence of a modulus of continuity ¯ω1, independent ofk, such that Z

R^d

Lb,σ,ϕ[mⁿ](x, tk)dmⁿ(tk)(x) = Z

R^d

Lb,σ,ϕ[m](x, tk)dm(tk)(x) + ¯ω1(d1(mⁿ, m)).

Thus, (4.20)

R

R^dϕ(x)dmⁿ(tn⁰)(x) = R

R^dϕ(x)dmⁿ(0)(x) +hnPn⁰−1 k=0

R

R^dLb,σ,ϕ[m](x, tk)dm(tk)(x) +O_ρ2

n

h_n +h_n+ ¯ω₁(d₁(mⁿ, m)) .

Sincem∈C([0, T];P1(R^d)), (H)(i) implies that the second term in the r.h.s. of (4.20) converges to the Riemman integral

Z t 0

Z

R^d

Lb,σ,ϕ[m](x, s)dm(s)(x).

Thus, passing to the limit in (4.20), we get that (2.4) holds true.

Remark 4.1. In particular, Theorem 4.1 yields a Peano type existence result for(F P K). We point out that more general existence results for the (F P K) equation are proven in the articles [49, 50], by using purely analytical techniques.

As we will see in Section 5, in some applications we do not know explicitly the coefficientsb and σ.

In this case, we need to approximate them numerically in order to obtain an implementable scheme. Let (ρn, hn) be as in Theorem 4.1 and consider a family of coefficients

b_n:C([0, T];P1(R^d))×R^d×[0, T]→R^d,

(σ_n)_i,j:C([0, T];P1(R^d))×R^d×[0, T]→R, i= 1, . . . , d, j= 1, . . . , r,