Zhenjie Ren
email: ren@cmap.polytechnique.fr

Xiaolu Tan
email: tan@ceremade.dauphine.fr

On the convergence of monotone schemes for path-dependent PDE *

Keywords: Numerical analysis, monotone schemes, viscosity solution, path-dependent PDE

. These conditions are satisfied, to the best of our knowledge, by all classical monotone numerical schemes in the context of stochastic control theory. In particular, the paper provides a unified approach to prove the convergence of numerical schemes for non-Markovian stochastic control problems, second order BSDEs, stochastic differential games etc.

Introduction

In their seminal work [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF], Barles and Souganidis proved a convergence theorem for monotone numerical schemes for viscosity solutions of fully nonlinear PDEs. Assuming that a strong comparison principle holds true for viscosity solutions of a PDE, they show that for all numerical schemes satisfying the three properties, "monotonicity", "consistency" and "stability", the numerical solutions converge locally uniformly to the unique viscosity solution of the PDE as the discretization parameters converge to zero. They mainly use the stability of viscosity solutions of PDEs and the local compactness of the state space. Due to their result, one only needs to check some local properties of a numerical scheme in order to get a global convergence result. Also, their our main convergence theorem, and thus can be applied in the PPDE context. Finally, we complete the proof of the main theorem in Section 6.

Preliminaries

Throughout this paper let T > 0 be a given finite maturity, Ω := {ω ∈ C([0, T ]; R d ) : ω 0 = 0} the set of continuous paths starting from the origin, and Θ := [0, T ] × Ω. We denote by B the canonical process on Ω, F = {F t , 0 ≤ t ≤ T } the canonical filtration, T the set of all F-stopping times taking values in [0, T ], and P 0 the Wiener measure on Ω. Moreover, let T + denote the subset of τ ∈ T taking values in (0, T ], and for h ∈ T , let T h and T + h be the subset of τ ∈ T taking values in [0, h] and (0, h], respectively. Following Dupire [START_REF] Dupire | Functional Itô calculus[END_REF], we introduce the following pseudo-distance on Θ: for all (t, ω), (t ′ , ω ′ ) ∈ Θ,

ω t := sup 0≤s≤t |ω s |, d (t, ω), (t ′ , ω ′ ) := |t -t ′ | + ω t∧• -ω ′ t ′ ∧• T .
Let E be a metric space, we say a process X : Θ → E is in C 0 (Θ, E) whenever t → X t (ω) is continuous for all ω ∈ Ω. Similarly, L 0 (F, E) and L 0 (F, E) denote the set of all F-measurable random variables and F-progressively measurable processes, respectively. We remark that C 0 (Θ, E) ⊂ L 0 (F, E), and when E = R, we shall omit it in these notations. We also denote by BUC(Θ) the set of all functions bounded and uniformly continuous with respect to d.

For any A ∈ F T , ξ ∈ L 0 (F T , E), X ∈ L 0 (F, E), and (t, ω) ∈ Θ, define respectively the shifted set, the shifted random variable and the shifted process by A t,ω := {ω ′ ∈ Ω : ω ⊗ t ω ′ ∈ A}, ξ t,ω (ω ′ ) := ξ(ω ⊗ t ω ′ ), X t,ω s (ω ′ ) := X(t + s, ω ⊗ t ω ′ )

where ω ⊗ t ω ′ is the concatenated path defined as (ω ⊗ t ω ′ ) s := ω s 1 [0,t] (s) + (ω t + ω ′ s-t )1 (t,T ] (s), 0 ≤ s ≤ T.

Following the standard arguments with monotone class theorem, we have the following results.

Lemma 2.1. Let (t, ω) ∈ Θ and s ∈ [t, T ]. Then A t,ω ∈ F s-t for all A ∈ F s , ξ t,ω ∈ L 0 (F s-t , E) for all ξ ∈ L 0 (F s , E), X t,ω ∈ L 0 (F, E) for all X ∈ L 0 (F, E), and τ t,ω -t ∈ T s-t for all τ ∈ T s .

Next, let us introduce the nonlinear expectation. As in [START_REF] Ekren | Viscosity Solutions of Fully Nonlinear Path Dependent PDEs: Part I[END_REF], we fix a constant L > 0 throughout the paper, and denote by P the collection of all continuous semimartingale measures P on Ω whose drift and diffusion coefficients are bounded by L. More precisely, a probability measure P ∈ P if under P, the canonical process B is a semimartingale with natural decomposition B = A P + M P , where A P is a process of finite variation, M P is a continuous martingale with quadratic variation M P , such that A P and M P are absolutely continuous in t, and µ P ∞ , a 3 Convergence of monotone schemes for PPDE

We consider the following PPDE -∂ t u(t, ω) -G •, u, ∂ ω u, ∂ 2 ωω u (t, ω) = 0, for all (t, ω) ∈ [0, T ) × Ω, (3.1) with the terminal condition u(T, •) = ξ.

Definition of PPDE

As in the survey of Ren, Touzi and Zhang [START_REF] Ren | An Overview of Viscosity Solutions of Path Dependent PDEs, Stochastic Analysis and Applications[END_REF], one may define viscosity solution of path dependent PDE by using the jets. For α ∈ R, β ∈ R d , γ ∈ S d , denote:

φ α,β,γ (t, x) := αt + β • x + 1 2 γ : (xx T ) for all (t, x) ∈ R + × R d ,
where

A 1 : A 2 := Tr[A 1 A 2 ]. Then define the semijets of a function u ∈ BUC(Θ) at (t, ω) ∈ [0, T ) × Ω: J u(t, ω) := (α, β, γ) : u(t, ω) = max τ ∈Th δ E[u t,ω τ -φ α,β,γ τ ], for some δ > 0 , J u(t, ω) := (α, β, γ) : u(t, ω) = min τ ∈Th δ E[u t,ω τ -φ α,β,γ τ ], for some δ > 0 , where h δ (ω ′ ) := δ ∧ inf{s ≥ 0 : |ω ′ s | ≥ δ} ∈ T + and φ α,β,γ t := φ α,β,γ (t, B t ). Definition 3.1. Let u ∈ BUC(Θ).
(i) u is a P-viscosity subsolution (resp. supersolution) of the path dependent PDE

(3.1), if at any point (t, ω) ∈ [0, T ) × Ω it holds for all (α, β, γ) ∈ J u(t, ω) (resp. J u(t, ω)) that -α -G(t, ω, u(t, ω), β, γ) ≤ (resp. ≥) 0.
(ii) u is a P-viscosity solution of the path dependent PDE (3.1), if u is both a Pviscosity subsolution and a P-viscosity supersolution of (3.1).

There are equivalent definitions of viscosity solution of path dependent PDE, for example in [START_REF] Ren | An Overview of Viscosity Solutions of Path Dependent PDEs, Stochastic Analysis and Applications[END_REF] we may find the definition in which one uses smooth test functions in the time-path space Θ. Here we are going to introduce another equivalent definition using constant localization and test functions in C 1,2 0 (R + × R d ), i.e. the class of all C 1,2 scalar functions ϕ of which the partial derivatives ∂ t ϕ, ∂ x ϕ, ∂ 2 xx ϕ are of compact support. Consider the set of test functions:

Au(t, ω) := ϕ ∈ C 1,2 0 (R + × R d ) : (u t,ω -ϕ) 0 = max τ ∈T δ E (u t,ω -ϕ) τ , for some δ > 0 , Au(t, ω) := ϕ ∈ C 1,2 0 (R + × R d ) : (u t,ω -ϕ) 0 = min τ ∈T δ E (u t,ω -ϕ) τ , for some δ > 0 ,
where ϕ t = ϕ(t, B t ).

L t,ω ϕ 0 := -∂ t ϕ 0 -G(t, ω, u(t, ω), ∂ x ϕ 0 , ∂ 2 xx ϕ 0 ) ≤ (resp. ≥) 0. (3.2)
We will report the proof of the above proposition in Section 6. The next lemma is proved in Section 4.4 of [START_REF] Ekren | Viscosity Solutions of Fully Nonlinear Path Dependent PDEs: Part II[END_REF]. Lemma 3.3. Let u ∈ BUC(Θ) be a P-viscosity subsolution of PPDE (3.1). Then for the constant L ∈ R, ũ(t, ω) := e -Lt u(t, ω) is a P-viscosity subsolution of the PPDE

-∂ t ũ(t, ω) -Lũ(t, ω) -e -Lt G •, e Lt ũ, e Lt ∂ ω ũ, e Lt ∂ 2 ωω ũ (t, ω) = 0.
The similar result holds for supersolutions.

Since we only consider nonlinearity G(t, ω, y, z, γ) uniformly Lipschtiz in y, it follows from the previous lemma that without loss of generality we may assume that G is non-decreasing in y.

Remark 3.4 (Examples of path-dependent PDE). One of the motivations of the PPDE theory is to characterize the value functions of non-Markovian stochastic control problems.

In particular, let consider a 2BSDE (Cheridito, Soner, Touzi and Victoir [START_REF] Cheridito | Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs[END_REF], Soner, Touzi and Zhang [START_REF] Soner | Wellposedness of second order backward SDEs[END_REF]) with generator F : Θ × R × R d × K → R and the controlled generating process with diffusion coefficient σ : Θ × K → S d , where K is some set in which the control processes take values. Then the solution of the 2BSDE corresponds to a PPDE with the nonlinearity:

G(t, ω, y, z, γ) := sup k∈K 1 2 σ 2 (t, ω, k) : γ + F (t, ω, y, σ(t, ω, k)z, k) . (3.3)
Another example is the application of PPDEs in the stochastic differential games (see e.g. Pham and Zhang [START_REF] Pham | Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation[END_REF]), where the nonlinearity of PPDE turns to be of the form:

G(t, ω, y, z, γ) := sup k 1 ∈K 1 inf k 2 ∈K 2 1 2 σ 2 (t, k 1 , k 2 ) : γ + F (t, ω, y, σ(t, k 1 , k 2 )z, k 1 , k 2 ) . (3.4)
We refer to Section 4 of Ekren, Touzi and Zhang [START_REF] Ekren | Viscosity Solutions of Fully Nonlinear Path Dependent PDEs: Part I[END_REF] for more details.

Main results

Definition 3.5. Let {U i , i ≥ 1} be a sequence of independent random variables defined on a probability space ( Ω, F, P). Every U i follows the uniform distribution on [0, 1]. Let h > 0, K be a subset of a metric space, Φ h : K × [0, 1] → R be a Borel measurable function such that for all ν ∈ K we have

| Ẽ Φ h (v, U ) | ≤ Lh, Var Φ h (ν, U ) ≤ Lh and Ẽ Φ h (ν, U ) 3 ≤ Lh 3/2 . (3.5)
Denote the filtration F := { Fi , i ∈ N}, where Fn := σ{U i , i ≤ n}. Let K = L 0 ( F, K) denote the collection of all F-adapted control processes taking values in K. For all ν ∈ K, we define

X h,ν (i+1)h = X h,ν ih + Φ h (ν ih , U i ). (3.6)
Further, we denote by X h,ν : [0, T ] × Ω → Ω the linear interpolation of the discrete process X h,ν ih , i ∈ N such that X h,ν ih = X h,ν ih for all i. Finally, for any function ϕ ∈ L 0 (F), we define the nonlinear expectation:

E h [ϕ] := inf ν∈U Ẽ ϕ X h,ν and E h [ϕ] := sup ν∈U Ẽ ϕ X h,ν . (3.7)
We next introduce the numerical schemes T.

Let (t, ω) ∈ [0, T ) × Ω and 0 < h ≤ T -t, T t,ω
h be a family of functions from L 0 (F t+h ) to R. We then define

u h (t, ω) := T t,ω h u h t+h ,
and assume that T satisfies the following conditions.

Assumption 3.6. (i)Consistency: for every

(t, ω) ∈ [0, T )×Ω and ϕ ∈ C 1,2 0 (R + ×R d ), lim (t ′ ,ω ′ ,h)→(t,0,0) ϕ(t ′ , ω ⊗ t ω ′ ) -T t ′ ,ω⊗tω ′ h ϕ(t ′ + h, •) h = L t,ω ϕ 0 .
(ii) Monotonicity: there exists a nonlinear expectation E h as in Definition 3.5 such that, for any ϕ, ψ ∈ L 0 (F t+h ), it holds that

T t,ω h [ϕ] -T t,ω h [ψ] ≥ inf 0≤α≤L E h e αh (ϕ -ψ) t,ω -hρ(h). (3.8)
(iii) Stability: u h is uniformly bounded and uniformly continuous in (t, ω), uniformly on h.

Our main theorem is the following convergence result of the monotone scheme for PPDE (3.1). Theorem 3.7. Assume that

• PPDE (3.1) is parabolic, i.e. G(t, ω, y, z, γ) is nondecreasing in γ,
• the nonlinearity G of PPDE (3.1) and the terminal condition ξ are continuous in all arguments, and G(t, ω, y, z, γ) is uniformly Lipschitz in y,

• the comparison principle of viscosity solutions of (3.1) holds, i.e. if u, v ∈ BUC(Θ) are P-viscosity subsolution and supersolution of PPDE (3.1), respectively, and u(T, •) ≤ v(T, •), then u ≤ v on Θ.

If the numerical scheme T satisfies Assumption 3.6, then PPDE (3.1) admits a unique bounded viscosity solution u, and u h → u locally uniformly, as h → 0. (3.9)

Remark 3.8. A comparison result of viscosity solutions of fully nonlinear PPDEs is proved in Ekren, Touzi and Zhang [START_REF] Ekren | Viscosity Solutions of Fully Nonlinear Path Dependent PDEs: Part II[END_REF] for PPDE (3.1) under certain conditions. Further, in the case of semilinear PPDEs, a comparison result is proved in Ren, Touzi and Zhang [START_REF] Ren | Comparison of Viscosity Solutions of Semi-linear Path-Dependent PDEs[END_REF] under very general assumptions.

Remark 3.9 (Comparison with Zhang and Zhuo [START_REF] Zhang | Monotone schemes for fully nonlinear parabolic path dependent PDEs[END_REF]). Let us compare our Assumption 3.6 with that in [START_REF] Zhang | Monotone schemes for fully nonlinear parabolic path dependent PDEs[END_REF]. Our condition (i) is weaker and thus easier to verify comparing to that in [START_REF] Zhang | Monotone schemes for fully nonlinear parabolic path dependent PDEs[END_REF]. The essential difference is between our condition (ii) and theirs.

Our condition (ii), although stated in a complicated way, is satisfied by all (to the best of our knowledge) classical monotone scheme in PDE context. Moreover, by the interpretation of the finite difference scheme for stochastic control problem as controlled Markov chains (see Kushner and Dupuis [START_REF] Kushner | Numerical methods for stochastic control problems in continuous time[END_REF]), this condition is consistent with the classical one in [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF].

Comparison with Barles and Souganidis's theorem When a PPDE degenerates to be a PDE:

Lu(t, x) := -∂ t u(t, x) -G 0 (•, u, ∂ x u, ∂ 2 xx u)(t, x) = 0, on [0, T ) × R d ,( 3
(T, •) ≤ v(T, •), then u ≤ v on [0, T ] × R d .
For any t ∈ [t, T ) and h ∈ (0, T -t], let T t,x h be an operator on the set of bounded functions defined on

R d . For n ≥ 1, denote h := T n < T -t, t i = ih, i = 0, 1, • • • , n, let the numerical solution be define by u h (T, x) := g(x), u h (t, x) := T t,x h [u h (t + h, •)], t ∈ [0, T ), i = n, • • • , 1. Assumption 3.11. (i) Consistency: for any (t, x) ∈ [0, T ) × R d and any smooth function ϕ ∈ C 1,2 ([0, T ) × R d ), lim (t ′ ,x ′ ,h,c)→(t,x,0,0) (c + ϕ)(t ′ , x ′ ) -T t ′ ,x ′ h (c + ϕ)(t ′ + h, •) h = Lϕ(t, x).
(ii) Monotonicity:

T t,x h [ϕ] ≤ T t,x h [ψ] whenever ϕ ≤ ψ. (iii) Stability: u h is bounded uniformly in h whenever g is bounded. (iv) Boundary condition: lim (t ′ ,x ′ ,h)→(T,x,0) u h (t ′ , x ′ ) = g(x) for any x ∈ R d .
We now recall the convergence theorem of the monotone scheme, deduced from Barles and Souganidis [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF] in this context of the parabolic PDE (3.10). Theorem 3.12. Let the generator function G 0 in (3.10) and the terminal condition g satisfy Assumption 3.10, and the numerical scheme T t,x h satisfy Assumption 3.11. Then the parabolic PDE (3.10) has a unique bounded viscosity solution and u h converges to u locally uniformly as h → 0. Remark 3.13 (Comparison with Assumption 3.6). (i) First, Assumption 3.6 (i) reduces exactly to be Assumption 3.11 (i) when the PPDE degenerates to a PDE.

(ii) Assumption 3.6 (ii) is stronger than Assumption 3.11 (ii). It is clear that the bound (3.8) of the difference of numerical solutions, provided by the discrete sublinear expectation, implies the monotonicity condition in Assumption 3.11 in the PDE case. In their book of numerical methods for stochastic control problem, Kushner and Dupuis [START_REF] Kushner | Numerical methods for stochastic control problems in continuous time[END_REF] studied the classical finite-difference scheme, and highlighted that the monotonicity condition is in fact equivalent to a controlled Markov chain interpretation, where the increments of the Markov chain satisfy (3.5). Our formulation of the monotonicity in Assumption 3.6 (ii) is consistent with this spirit. In particular, for concrete numerical schemes, the two monotonicity formulations demand exactly the same conditions on the coefficients. Moreover, it is satisfied by all classical monotone scheme, to the best of our knowledge, in the context of stochastic control theory. See also our review in Section 4.

(iii) The stability condition in Assumption 3.6 (iii) is also stronger than Assumption 3.11 (iii). Nevertheless, in the classical numerical analysis for parabolic PDE (3.10), in order to check Assumption 3.11 (iv), one needs (explicitly or implicitly) to prove a uniform continuity property of numerical solutions uniformly on the discretization parameter, which leads to the same condition as in Assumption 3.6 (iii). See also our review in Section 4.

Examples of monotone schemes

We discuss here some classical monotone numerical schemes in the stochastic control context, and provide some sufficient conditions Assumption 3.6 to hold true. Let us first add some assumptions on the functions G and ξ for PPDE (3.1).

Assumption 4.1. The terminal condition ξ is Lipschitz in ω, G is increasing in γ, and G is Lipschitz in (y, z, γ): i.e. there is some constant C such that for all (t, ω) ∈ Θ and (y, z, γ), (y

′ , z ′ , γ ′ ) ∈ R × R d × S d , G(t, ω, y, z, γ) -G(t, ω, y, z ′ , γ ′ ) ≤ C y -y ′ + z -z ′ + γ -γ ′ .
In this section, we denote

t k := hk for h = ∆t > 0. Given x = (x t 0 , x t 1 , • • • , x t k ) a sequence of points in R d , we denote by x ∈ Ω the linear interpolation of x such that x t i = x t i for all i. Further, for (t, ω) ∈ Θ, h > 0 and z ∈ R d , we define a path (ω ⊗ h t z) := ω ⊗ t z h , where z h s := s h z, for 0 ≤ s ≤ h; z, for s > h.
Let E be some normed vector space, then for maps ψ : Θ → E, we introduce the norm |ψ| 0 and |ψ| 1 by

|ψ| 0 := sup (t,ω)∈Θ |ψ(t, ω)| and |ψ| 1 := sup (t,ω) =(t ′ ,ω ′ ) |ψ(t, ω) -ψ(t ′ , ω ′ )| (|ω t∧• -ω ′ t ′ ∧• | + |t -t ′ | 1/2 .

Finite difference scheme

For simplicity, we assume that the state space is of dimension one (d = 1). Let ∆x > 0 be the space discretization size. For every (t, ω) ∈ Θ, h > 0 and F t+h -measurable random variable ψ : Ω → R, we define the discrete derivatives

D h ψ(t, ω) := D 0 h ψ, D 1 h ψ, D 2 h ψ (t, ω),
where

D 0 h ψ(t, ω) := ψ(ω t∧• ), D 1 h ψ(t, ω) := ψ(ω⊗ h t ∆x)-ψ(ωt∧•) ∆x , and D 2 h ψ(t, ω) := ψ ω⊗ h t ∆x -2ψ(ωt∧•)+ψ ω⊗ h t (-∆x) ∆x 2 .
Then an explicit finite difference scheme is given by

T t,ω h [u h t+h ] := u h (t + h, ω t∧• ) + hG t, ω, D h u h t+h (t, ω) . (4.1)
Proposition 4.2. Suppose that Assumption 4.1 holds true and G is Lipschitz in ω, i.e. there is a constant C such that for all ω, ω ′ ∈ Ω and all (t, y, z, γ)

∈ [0, T ]×R×R d ×S d , G(t, ω, y, z, γ) -G(t, ω ′ , y, z, γ) ≤ C ω t∧• -ω ′ t∧• .
Assume in addition the CFL (Courant-Friedrichs-Lewy) condition, i.e.

ε ≤ ∆t|∇ γ G| 0 ∆x 2 ≤ 1 2 -ε, (4.2)
and that ∇ γ G ≥ ε for some small constant ε > 0. Then Assumption 3.6 holds true for finite difference scheme (4.1). In particular, the numerical solution u h is 1 2 -Hölder in t and Lipschitz in ω, uniformly on h.

Proof. We will check each condition in Assumption 3.6. For the simplicity of presentation, we assume that G is independent of y. Clearly, the argument still works if G is Lipschitz in y.

two constants, and ζ a,b be a random variable defined on a probability space ( Ω, F , P ) such that

P(ζ a,b = 0) = 1 -b ∆t ∆x -2a ∆t ∆x 2 , P(ζ a,b = ∆x) = b ∆t ∆x + a ∆t ∆x 2 and P(ζ a,b = -∆x) = a ∆t ∆x 2 .
The law of ζ a,b is well defined for ∆t = h small enough, because every term above is positive and the sum of all terms equals to 1 under condition (4.2). Further, we have

Ẽ ζ a,b = bh, Var ζ a,b = ah, and Ẽ |ζ a,b | 3 ≤ |∆x| 3 ≤ Ch 3/2 , (4.3)
where the last terms follows by ∆x ≈ h

1 2 . Then let F h (a, b, •) : R → [0, 1] be the distribution function of ζ a,b and Φ h (a, b, •) : [0, 1] → R be the generalized inverse function of F h (a, b, •), i.e. Φ h (a, b, x) := inf{y : F h (a, b, y) > x}. (4.4)
In view of (4.3), the monotonicity condition of Assumption 3.6 (ii) holds true.

(iii) To prove Assumption 3.6 (iii), we will prove that there is a constant C independent of h such that

u h (t, ω) -u h (t ′ , ω ′ ) ≤ C ω t∧• -ω ′ t ′ ∧• + |t ′ -t| , ∀(t, ω), (t ′ , ω ′ ) ∈ Θ.(4.5)
Let us first prove that u h is Lipschtiz in ω. Denote

L h t := sup (t,ω),(t ′ ,ω ′ )∈Θ u h (t, ω) -u h (t, ω ′ ) ω t∧• -ω ′ t∧• 1 { ωt∧•-ω ′ t∧• >0} .
By direct computation, we have

u h (t, ω) -u h (t, ω ′ ) = hG ω ω t∧• -ω ′ t∧• + DG h u h t+h (t, ω) -DG h u h t+h (t, ω ′ ), (4.6)
where

DG h u h t+h := (1 + hG y )D 0 h + hG z D 1 h + hG γ D 2 h u h t+h ,
with G y , G z and G γ uniformly bounded by L. Then there is a constant C independent of h such that

L h t ≤ (1 + Ch)L h t+h + Ch.
Notice that the terminal condition ξ is Lipschitz, it follows by the discrete Gronwall inequality, we have L h t ≤ Ce CT for a constant C independent of h. Hence, there is a constant C ′ independent of h such that

u h (t, ω) -u h (t, ω ′ ) ≤ C ′ ω t∧• -ω ′ t∧• , ∀t ∈ [0, T ], ω, ω ′ ∈ Ω. (4.7)
We next consider the regularity of u h in t. Let t := ih and t ′ := jh > t. Note that

u h (t, ω) = u h (t + h, ω t∧• ) + hG(t, ω, 0, 0, 0) + h G(t, ω, D h u h t+h (t, ω)) -G(t, ω, 0, 0, 0) .
By a direct computation, we have

u h (t, ω) = Ẽ j-1 k=i G(t k , ω ⊗ t X h , 0, 0, 0) h + u h (t ′ , ω ⊗ t X h ) , (4.8)
where X h is a discrete process defined as X h 0 := 0,

X h t k+1 := X h t k + Φ h (∇ γ G, ∇ z G, U k+1 ),
with Φ h be given by (4.4), and X h is the linear interpolation of X h . Define

A h 0 := 0, A h t k := k-1 i=0 Ẽ Φ h (∇ γ G, ∇ z G, U i+1 ) Fi , and M h := X h -A h .
Clearly, M h is a martingale and A h is a predictable process. Further, it follows from the property of Φ h in (4.3) that

Ẽ A h t k+1 -A h t k ≤ Lh and Var M h t k+1 -M h t k ≤ Lh.
Then by (4.8), we have

|u(t, ω) -u(t ′ , ω t∧• )| ≤ C(t ′ -t) + C Ẽ sup i≤k≤j M t k . (4.9)
Further, by Doob's inequality, it follows that

Ẽ sup i≤k≤j M h t k ≤ Ẽ sup i≤k≤j M h t k 2 ≤ 2 Ẽ (M h t j ) 2 ≤ C t j -t i .
Finally, combining the above estimation with (4.7) and (4.9), we obtain (4.5).

Remark 4.3. We here assume that the PPDE is non-degenerate (∇ γ G ≥ ε > 0). When ∇ γ G = 0 and ∇ z G ≥ 0, the scheme is still monotone. When ∇ γ G = 0 and ∇ z G ≤ 0, it is possible to redefine the first order discrete derivative by

D 1 h ψ(t, ω) := ψ(ω t∧• ) -ψ(ω ⊗ h t (-∆x)) ∆x
to obtain a monotone scheme. Remark 4.4. In the multidimensional case, ∇ γ G is a matrix. If ∇ γ G is diagonal dominated, then following Kushner and Dupuis [START_REF] Kushner | Numerical methods for stochastic control problems in continuous time[END_REF], it is easy to construct a monotone scheme under similar CFL condition (4.2). When ∇ γ G is not diagonal dominated, it is possible to use the generalized finite difference scheme proposed by Bonnans, Ottenwaelter and Zidani [START_REF] Bonnans | A fast algorithm for the two dimensional HJB equation of stochastic control[END_REF].

4.2

The trinomial tree scheme of Guo-Zhang-Zhuo [START_REF] Guo | A Monotone Scheme for High Dimensional Fully Nonlinear PDEs[END_REF] We consider the PPDE of the form (3.1). Let σ 0 be some symmetric d × d matrix, denote

F (t, ω, y, z, γ) := G(t, ω, y, z, γ) - 1 2 σ 2 0 : γ, Gγ := σ -1 0 G γ σ -1 0 . Let ζ = (ζ 1 , • • • , ζ d
) a random vector defined on a probability space ( Ω, F , P) such that

ζ i , i = 1, • • • , d are i.i.d and P(ζ i = 1 √ p ) = p 2 , P(ζ i = - 1 √ p ) = p 2 , P(ζ i = 0) = 1 -p, with p ∈ (0, 1).
For every

F t+h -measurable function ψ : Ω → R, let us define D i h ψ(t, ω) := Ẽ ψ ω ⊗ h t ( √ hσ 0 ζ) K i (ζ) with K 0 := 1, K 1 := σ -1 0 ζ √ h , K 2 := σ -1 0 [(1 -p)ζζ T -(1 -3p)Diag[ζζ T ] -2pI d ]σ -1 0 (1 -p)h ,
where for any matrix γ = [γ i,j ] 1≤i,j≤d ∈ S d , Diag[γ] denotes the diagonal matrix whose (i, i)-th component is γ ii . Then the numerical scheme is defined as

T t,ω h [u h (t + h, •)] := D 0 h u h (t, ω) + hF •, D h u h t+h (t, ω). (4.10)
Proposition 4.5. Let Assumptions 4.1 hold true and G is Lipschitz in ω. Suppose in addition that Assumption 3.3 in Guo, Zhang and Zhuo [START_REF] Guo | A Monotone Scheme for High Dimensional Fully Nonlinear PDEs[END_REF] holds true (where we replace their notation Gγ by ∇ γ G in our context). Then the trinomial tree scheme (4.10) satisfies Assumption 3.6.

Proof. The consistency and monotonicity condition in Assumption 3.6 (i) and (ii) can be justified by almost the same argument as in [START_REF] Guo | A Monotone Scheme for High Dimensional Fully Nonlinear PDEs[END_REF]. Similarly to the finite difference scheme, the monotonicity in sense of Barles and Souganidis [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF] implies the interpretation of the controlled discrete processes of the numerical scheme, which implies the monotonicity condition (3.8) in our context. Further, using the same argument as in Proposition 4.2, it is easy to show that

u h (t, ω) -u h (t ′ , ω ′ ) ≤ C ω t∧• -ω ′ t ′ ∧• + |t ′ -t| , ∀(t, ω), (t ′ , ω ′ ) ∈ Θ,
for some constant C independent of h, which implies in particular (iii) of Assumption 3.6.

Remark 4.6. As a PPDE degenerates to be a classical PDE, the conditions in Proposition 4.5 turns to be exactly the same conditions in Theorem 3.10 of [START_REF] Guo | A Monotone Scheme for High Dimensional Fully Nonlinear PDEs[END_REF].

4.3

The probabilistic scheme of Fahim-Touzi-Warin [START_REF] Fahim | A probabilistic numerical method for fully nonlinear parabolic PDEs[END_REF] We consider PPDE (3.1) in which G is in the form of

G(t, ω, y, z, γ) = µ(t, ω) • z - 1 2 σσ T (t, ω) : γ -F (t, ω, y, z, γ).
Before introducing the numerical scheme, we first define a random vector

X (t,ω) h := µ(t, ω)h + σ(t, ω)W h ,
where W h ∼ N (0, hI d ) is a Gaussian vector. For every bounded function ψ ∈ L 0 (F t+h ), we define

D h ψ(t, ω) := E ψ(ω ⊗ t X (t,ω) • )H h (t, ω) ,
where

H h (t, ω) = (H h 0 , H h 1 , H h 2 ) T with H h 0 := 1, H h 1 := (σ T (t, ω)) -1 W h h , H h 2 := (σ T (t, ω)) -1 W h W T h -hI d h 2 σ -1 (t, ω).
Then the probabilistic scheme is given by

T t,ω h [u h (t + h, •)] := E u h (t + h, X (t,ω) ) + hF •, D h u h t+h (t, ω). (4.11)
Remark 4.7. The probabilistic scheme in [START_REF] Fahim | A probabilistic numerical method for fully nonlinear parabolic PDEs[END_REF] is inspired by the second order BSDE theory of Cheridito, Soner, Touzi and Victoir [START_REF] Cheridito | Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs[END_REF], and extends the classical numerical scheme of BSDE (see e.g. Bouchard and Touzi [START_REF] Bouchard | Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations[END_REF], Zhang [START_REF] Zhang | A numerical scheme for backward stochastic differential equations[END_REF]). In practice, one can use the simulation-regression method to estimate the conditional expectation in the above scheme (see e.g. Gobet, Lemor and Warin [START_REF] Gobet | A regression-based Monte-Carlo method to solve backward stochastic differential equations[END_REF]). We refer to Guyon and Henry-Labordère [START_REF] Guyon | Uncertain Volatility Model: A Monte-Carlo Approach[END_REF] for more details on the use of the scheme, to Tan [START_REF] Tan | A splitting method for fully nonlinear degenerate parabolic PDEs[END_REF] for an extension to a degenerate case, and to Tan [START_REF] Tan | Discrete-time probabilistic approximation of path-dependent stochastic control problems[END_REF] for an extension to path-dependent control problems. (ii) F is elliptic and dominated by the diffusion term of X, that is,

∇ γ F ≤ σσ T , on Ω × R × R d × S d . (4.12) (iii) ∇ p F ∈ Image(∇ γ F ) and (∇ p F ) T (∇ γ F ) -1 ∇ p F 0 < ∞.
(iv) |µ| 1 , |σ| 1 < ∞ and σ is invertible and ξ is bounded Lipschitz.

Proposition 4.9. Suppose that Assumption 4.8 holds true. Then the probabilistic numerical scheme (4.11) satisfies Assumption 3.6.

Proof. (i) Assumption 3.6 (i) is obviously satisfied in view of Lemma 3.11 of [START_REF] Fahim | A probabilistic numerical method for fully nonlinear parabolic PDEs[END_REF].

(ii) Further, using probabilistic interpretation of this scheme in Tan [30, Section 3.2], we may verify (ii) of Assumption 3.6. See also the estimation given by Lemma 3.1 of [START_REF] Tan | Discrete-time probabilistic approximation of path-dependent stochastic control problems[END_REF].

(iii)For (iii)of Assumption 3.6, we shall prove that the numerical solution u h is Lipschitz in ω and 1/2-Hölder in t. In [START_REF] Fahim | A probabilistic numerical method for fully nonlinear parabolic PDEs[END_REF], the authors proved this property in the case of PDEs. Their arguments for the Lipschitz continuity in ω can be easily adapted to this pathdependent case. For the regularity of u h in t, they used a regularization technique, which seems impossible to be adapted to the path-dependent case. However, we can still use similar arguments as in Proposition 4.2, i.e. use the discrete-time controlled semimartingale interpretation, to prove the Hölder property of u h in t.

Remark 4.10. As a PPDE degenerates to be a PDE, the conditions in Assumption 4.8 reduce exactly the same conditions as in [START_REF] Fahim | A probabilistic numerical method for fully nonlinear parabolic PDEs[END_REF] (see their Theorem 3.6).

The semi-Lagrangian scheme

For the semi-Lagrangian scheme, we shall consider the PPDE (3.1) of the Bellman-Issac type, i.e. the function G is in the form of

G(t, ω, y, z, γ) = inf k 1 ∈K 1 sup k 2 ∈K 2 1 2 a k 1 ,k 2 (•) : γ + b k 1 ,k 2 (•) • z + c k 1 ,k 2 (•)y + f k 1 ,k 2 (•) (t, ω),
where K 1 and K 2 are some sets, (a

k 1 ,k 2 , b k 1 ,k 2 , c k 1 ,k 2 , f k 1 ,k 2 ) are functionals defined on Θ.
Let ζ be a random vector satisfying

E ζ = 0, Var ζ = I d and E ζ 3 < ∞. (4.13)
Then the semi-Lagragian scheme is defined as Then the semi-Lagrangian scheme (4.14) for the Bellman-Issac path-dependent equation satisfies Assumption 3.6.

T t,ω h [u h (t + h, •)] := inf k 1 ∈K 1 sup k 2 ∈K 2 u h t + h, ω ⊗ t σ k 1 ,k 2 (t, ω)ζ √ h + b k 1 ,k 2 (t, ω)h + u h t + h, ω)c k 1 ,k 2 (t, ω)h + f k 1 ,k 2 (t,
Proof. (i) The consistency condition (Assumption 3.6 (i)) is easy to check. (ii) Let E be a set, e : K 1 × K 2 → E be an arbitrary mapping, and ψ, ϕ : E → R be two bounded functions. Note that inf

k 1 ∈K 1 sup k 2 ∈K 2 ψ(e(k 1 , k 2 )) -inf k 1 ∈K 1 sup k 2 ∈K 2 ϕ(e(k 1 , k 2 )) ≤ sup k 1 ∈K 1 ,k 2 ∈K 2 (ψ -ϕ)(e(k 1 , k 2 )). (4.15)
Notice that R d is isomorphic to R, we can always consider the random vector σ k 1 ,k 2 ζ √ h+ b k 1 ,k 2 h as a one-dimensional random variable. By consider the inverse function of its distribution function, then there is a family Φ

h (k 1 , k 2 , •) such that Φ h (k 1 , k 2 , U ) ∼ σ k 1 ,k 2 ζ √ h + b k 1 ,k 2 h in law with U ∼ U ([0, 1]), for all (k 1 , k 2 ) ∈ K 1 × K 2 .
Then it follows from (4.15) that the monotonicity condition in Assumption 3.6 (ii) holds true with Φ h (k 1 , k 2 , •) and K = K 1 × K 2 . (iii) Finally, by the same arguments as in Proposition 4.2, we can easily deduce that u h is Lipschitz in ω and 1/2-Hölder in t, uniformly on h, and hence complete the proof for the stability condition in Assumption 3.6. Remark 4.12. Solutions of path dependent Bellman-Issac equations can characterize value functions of stochastic differential games (see e.g. Pham and Zhang [START_REF] Pham | Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation[END_REF]). Remark 4.13. (i) For Bellman-Issac PDE, Debrabant and Jakobsen [START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF] studied the semi-Lagrangian scheme with a random variable ζ following a discrete distribution, together with an interpolation technique for the implementation.

(ii) For Bellman equation (PDE), Kharroubi, Langrené and Pham [START_REF] Kharroubi | Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps[END_REF] propose a semi-Lagrangian type numerical scheme with ζ ∼ N (0, 1), and provide a simulationregression technique for the implementation. It is worth of mentioning that [START_REF] Kharroubi | Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps[END_REF] provides a convergence rate for the scheme, while we only prove in this paper a general convergence theorem as in Barles and Souganidis [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF].

Numerical examples

In this section, we provide two toy examples of numerical implementation in lowdimensional case. For more numerical examples (in high-dimensional case), we would like to refer to [START_REF] Fahim | A probabilistic numerical method for fully nonlinear parabolic PDEs[END_REF][START_REF] Guo | A Monotone Scheme for High Dimensional Fully Nonlinear PDEs[END_REF][START_REF] Guyon | Uncertain Volatility Model: A Monte-Carlo Approach[END_REF][START_REF] Kharroubi | Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps[END_REF][START_REF] Tan | A splitting method for fully nonlinear degenerate parabolic PDEs[END_REF], etc.

First numerical example For a first numerical example, we consider the PPDE

-∂ t u -min µ∈[µ,µ] µ∂ ω u -max a∈[a,a] a 2 ∂ 2 ωω u = f (t, ω, ω), u(T, ω) = g(ω T , ωT ).
(5.1)

where d = 1, ωt := t 0 ω s ds, f : [0, T ] × R × R → R and g : R × R → R are two functions.
The above PPDE (5.1) is motivated by a stochastic differential game:

u 0 = inf µ≤µt≤µ sup a≤at≤a E T 0 f (t, X µ,a t , X µ,a )dt + g(X µ,a T , X µ,a T ) ,
where X µ,σ is controlled diffusion such that

X µ,a t = t 0 µ s ds + t 0
√ a s dW s , with W a Brownian motion, and X µ,a t = t 0 X µ,a s ds (see e.g. Pham and Zhang [START_REF] Pham | Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation[END_REF] for more details). We choose the terminal condition g(x, y) = cos(x + y) and the function f (t, x, y) = (x -µ) sin(x -y)

-+ (x + µ) sin(x -y)

+ + a 2 cos(x -y) + - a 2 cos(x -y) -,
so that the solution of PPDE (5.1) is given explicitly by u(t, ω) = cos(ω t + ωt ), which serves as a reference value for the numerical examples. This idea is borrowed from Guo, Zhang and Zhuo [START_REF] Guo | A Monotone Scheme for High Dimensional Fully Nonlinear PDEs[END_REF]. For numerical test, we implemented the finite difference scheme in Section 4.1 and the probabilistic scheme (of Fahim, Touzi and Warin [START_REF] Fahim | A probabilistic numerical method for fully nonlinear parabolic PDEs[END_REF]) in Section 4.3. The results are reported in Figure 1.

Second numerical example

The second example of PPDE we considered is given by

-∂ t u -max a≤a≤a 1 2 a∂ 2 ωω u -f (t, u, ∂ ω u, a) = 0, (5.2)
where f (t, y, z, a)

= 1 2 ( √ az + b/ √ a) -2 -zb -b 2 /2a, (5.3)
which is taken from Matoussi, Possamai and Zhou [START_REF] Matoussi | Robust utility maximization in nondominated models with 2BSDE: the uncertain volatility model[END_REF]. The above equation is motivated by solving a robust utility maximization problem using 2BSDE, which can be instead characterized by a PPDE (see e.g. (3.3)). We consider the terminal condition Step length

u(T, ω) = K 1 + (ω T -K 1 ) + -(ω T -K 2 ) + , ω T := T 0 ω s ds.

Error

Numerical Error

Finite difference scheme Probabilistic scheme

Figure 1: For PPDE (5.1), we choose µ = -0.2, µ = 0.2, a = 0.04, a = 0.09, T = 1 and ω 0 = ω 0 = 0. Then the reference solution is given by u(0, 0) = cos(0) = 1. We compute the error between the reference solution and the numerical solutions, w.r.t. difference time step length ∆t.

Then the solution of PPDE (5.2) can also be characterized by the PDE, by adding an associated variable y,

-∂ t v -x∂ y v -max a≤a≤a 1 2 a∂ 2 xx v -f (t, v, ∂ x v, a) = 0, (5.4)
v(T, x, y) = K 1 + (y -K 1 ) + -(y -K 2 ) + . (5.5)
We implemented the finite difference scheme (Section 4.1) and the probabilistic scheme (Section 4.3) for PPDE (5.2). For reference, we implemented the classical finite difference scheme of PDE (5.4). We also notice that the generator in PPDE (5.2) is in fact not Lipschitz but quadratic in z, however, the convergence of the numerical solutions can be still observed, see Figure 2.

Proofs

Preliminary results

In preparation of the proof of Theorem 3.7, we prove the following lemmas. . We provide all the numerical solutions w.r.t. difference time step length ∆t. It seems that the faire value is closed to 0.129. For finite-difference scheme, when ∆t is greater than 0.025, we need to use a coarser space-discretization to ensure the monotonicity (similar to the classical CFL condition), which makes a big difference to the numerical solutions for the case ∆t < 0.25. However, the convergence as ∆t → 0 is still obvious. Lemma 6.1 (Fatou's Lemma). Assume that the random variables X n ∈ C 0 (F) are bounded. Then we have

lim n→∞ E[X n ] ≥ E lim n→∞ X n
Proof. In order to prove the Fatou lemma, it is enough to show the monotone convergence theorem, i.e. given a sequence {X n : n ∈ N} of increasing random variables, we have

lim n→∞ E[X n ] = E[ lim n→∞ X n ]. (6.1)
Since X n ∈ C 0 (F) for each n, it follows from Theorem 31 in [6] that (6.1) holds true.

Recall the nonlinear expectation E h defined in (3.7).

Lemma 6.2. Let ϕ : Ω → R be bounded uniformly continuous. Then there exists a modulus continuity ρ : R + → R + which depends only on the continuity modulus of ϕ and |ϕ| 0 , such that

E[ϕ] ≤ E h [ϕ] + ρ(h).
Proof. Denote ρ ′ : R + → R + as a continuity modulus of ϕ. Let ν ∈ K and X h,ν be defined by (3.6) and X h,ν its linear interpolation on [0, T ]. Then under the condition (3.5), it follows from Lemma 4.8 of Tan [START_REF] Tan | Discrete-time probabilistic approximation of path-dependent stochastic control problems[END_REF] (see also Dolinsky [START_REF] Dolinsky | Numerical Schemes for G-Expectations[END_REF]) that we can construct a process X h,ν and another process X in the same probability space ( Ω, F , P), such that the image measure of X lies in P, and for some constant C independent of h,

P X h,ν -X ≥ h 1/8 ≤ Ch 1/8 . Let ρ(h) := ρ ′ (h 1/8 ) + 2 ϕ ∞ h 1/8 , then it follows that E[ϕ] ≤ Ẽ[ϕ(X)] ≤ Ẽ ϕ( X h,ν ) + ρ(h),
which concludes the proof by the arbitrariness of ν ∈ K.

Lemma 6.3. Let ϕ : Ω → R be lower semicontinuous and bounded from below, then it holds for all (t, ω) ∈ Θ that

lim h→0 E h [ϕ] ≥ E[ϕ].
In particular, by defining h := inf{t ≥ 0 : |B t | ≥ x} for some x > 0, we have

lim h→0 E h [1 {h≤δ} ] ≤ E[1 {h≤δ} ] for any δ > 0.
Proof. Define the approximation for the function ϕ:

ϕ n (ω) := inf ω ′ ∈Ω ϕ(ω ′ ) + n ω -ω ′ .
Clearly, for each n ∈ N, function ϕ n is Lipschitz continuous, and ϕ n ↑ ϕ. By Lemma 6.2, we obtain that

lim h→0 E h [ϕ] ≥ lim h→0 E h [ϕ n ] ≥ E[ϕ n ], for all n ∈ N.
Since ϕ n ↑ ϕ, by Fatou's lemma we have

lim n→∞ E[ϕ n ] ≥ E[ϕ]. Therefore lim h→0 E h [ϕ] ≥ E[ϕ]. (6.2)
Then we easily get the symmetric result for upper semicontinuous function ψ, i.e.

lim h→0 E h [ψ] ≤ E[ϕ].
To conclude, it remains to prove that the function ω -→

|B 1 t | ≥ x d
By the definition of P above (2.1), for all P ∈ P, the canonical process B admits the canonical decomposition B = A P + M P , where

A P = (A 1 , • • • , A d ) is a finite variation process and M = (M 1 , • • • , M d ) is a P-martingale. Moreover, for each i = 1, • • • , d, P max t∈[0,δ] |B i t | ≥ x d = Q max t∈[0,δ] |A i t + M i t | ≥ x d ≤ Q max t∈[0,δ] |M i t | ≥ x d -Lδ .
Further, by the time-change for martingales (see e.g. Theorem 4.6 on page 174 of [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]), there is a scalar Brownian motion W defined on a probability space (Ω, F, P) such that

P max t∈[0,δ] |M i t | ≥ x d -Lδ = P max t∈[0,δ] |W <M 1 >t | ≥ x d -Lδ ≤ P max t∈[0,L 2 δ] |W t | ≥ x d -Lδ = 4P W 1 ≥ x/d -Lδ L √ δ Since η := x/d-Lδ L √ δ
= -2 ln εδ 4d > 1 when δ is small enough, we have

4P W 1 ≥ η ≤ 4e -η 2 2 = εδ d .
We then conclude that sup P∈P P[h δ ≤ δ] ≤ εδ.

Proof of Proposition 3.2

We only discuss the case of subsolution. The result about the supersolution follows similarly.

1. We first prove the only if part. Let (t, ω) ∈ [0, T ) × Ω and (α, β, γ) ∈ J u(t, ω) with a localizing time h δ . Clearly, there is a function ϕ ∈ C 1,2 0 (R + × R d ) such that ϕ = φ α,β,γ on the set [0, δ] × {x ∈ R d : |x| ≤ x(δ)}, where x(•) is defined as in Lemma 6.4. Thus,

(ϕ -u) 0 = max τ ∈T hδ,x E[(ϕ -u) τ ],
where hδ,x := δ ∧ h δ,x with h δ,x be defined as in Lemma 6.4. We have

(ϕ -u) 0 ≥ E[(ϕ -u) δ ] -E[(ϕ -u) δ -(ϕ -u) hδ,x ]. (6.4)
For the second term on the right hand side of (6.4), we have

E[(ϕ -u) δ -(ϕ -u) hδ,x ] ≤ E |(ϕ -u) δ -(ϕ -u) hδ,x |; h δ,x ≤ δ ≤ C sup P∈P P[h δ,x ≤ δ].
Take ε > 0. By Lemma 6.4, there is a constant C(ε) > 0 such that for all δ < C(ε) we have sup P∈P P[h δ,x ≤ δ] < εδ 2C . Then it follows from (6.4) that

(ϕ -u) 0 > E[(ϕ -u) δ ] - εδ 2 .
We next consider the optimal stopping problem:

Y t (ω) = sup τ ∈T δ-t E[(ϕ -u) t,ω τ -ετ ].
According to Ekren, Touzi and Zhang [START_REF] Ekren | Optimal Stopping under Nonlinear Expectation[END_REF], τ * := inf{t : Y t = ϕ t -u t -εt} is an optimal stopping rule. Suppose that we always have hδ,x ≤ τ * ≤ δ. Then we obtain that

E[(ϕ -u) τ * -ετ * ] ≤ E[(ϕ -u) δ -εδ] + E[(ϕ -u) τ * -(ϕ -u) δ -ε(τ * -δ)] ≤ E[(ϕ -u) δ -εδ] + E[|(ϕ -u) τ * -(ϕ -u) δ -ε(τ * -δ)|; h δ,x ≤ δ] ≤ E[(ϕ -u) δ -εδ] + C sup P∈P P[h δ,x ≤ δ] ≤ E[(ϕ -u) δ ] - εδ 2 < (ϕ -u) 0 .
However, this is in contradiction with the optimality of τ * . Therefore, there is ω * such that t * := τ * (ω * ) < hδ,x (ω * ) and

(ϕ -u) t * (ω * ) = max τ ∈T δ-t * E[(ϕ -u) t * ,ω * τ -ετ ].
So we have

-∂ t ϕ + ε -G(•, u, ∂ x ϕ, ∂ 2 xx ϕ) (t * , ω * ) ≤ 0.
By letting δ → 0 and then ε → 0, we obtain

-∂ t ϕ -G(•, u, ∂ x ϕ, ∂ 2 xx ϕ) (0, 0) ≤ 0. Finally, since α = ∂ t ϕ 0 , β = ∂ x ϕ 0 , γ = ∂ 2 xx ϕ 0 , this provides that -α-G(0, u 0 , β, γ) ≤ 0. 2.
For the if part, one may apply the same argument as in Proposition 3.11 in [START_REF] Ren | An Overview of Viscosity Solutions of Path Dependent PDEs, Stochastic Analysis and Applications[END_REF]. For completeness, we provide the full argument. Let (t, ω) ∈ [0, T )×Ω and ϕ ∈ Au(t, ω) with a localizing time δ ∈ R + . Without loss of generality, we assume that (t, ω) = (0, 0) and (ϕ -u) 0 = 0. Denote α := ∂ t ϕ 0 , β := ∂ x ϕ 0 , and γ := ∂ 2 xx ϕ 0 . (6.5)

For any ε > 0, since ϕ is smooth, by otherwise choosing a stopping time h δ ′ < δ we may assume

|∂ t ϕ t -α| ≤ ε, |∂ x ϕ t -β| ≤ ε, |∂ 2 xx ϕ t -γ| ≤ 2ε, 0 ≤ t ≤ h δ ′ . Denote α ε := α + [1 + 2L]ε. Then, for all τ ∈ T h δ ′ , E (u -φ αε,β,γ ) τ -u 0 = E (u -u 0 -φ αε,β,γ ) τ ≤ E (u -ϕ) τ + E (ϕ-ϕ 0 -φ αε,β,γ ) τ ≤ E τ 0 (∂ t ϕ s -α ε )ds + (∂ x ϕ s -β) • dB s + 1 2 (∂ 2 xx ϕ s -γ) : d B s .
where the last inequality is due to the Itô's formula. Note that, for any µ ∞ , a ∞ ≤ L, we have

E Qµ,σ τ 0 (∂ t ϕ s -α ε )ds + (∂ x ϕ s -β) • dB s + 1 2 (∂ 2 xx ϕ s -γ) : d B s = E Qµ,σ τ 0 ∂ t ϕ s -α + (∂ x ϕ s -β) • µ s + 1 2 (∂ 2 xx ϕ s -γ) : a s ds -[1 + 2L]ετ ≤ 0.
By the arbitrariness of µ, σ, we see that

E (u -φ αε,β,γ ) τ -u 0 ≤ 0.
That is, (α ε , β) ∈ J u 0 . Since u is a P-viscosity subsolution, it follows that

-α ε -G(0, 0, u 0 , β, γ) ≤ 0.
Let ε → 0, then the desired result follows.

Proof of Theorem 3.7

We first introduce two functions:

u(t, ω) = lim h→0 u h (t, ω) and u(t, ω) = lim h→0 u h (t, ω). (6.6)
Note that u, u inherit the uniform modulus of continuity of u h , so u, u ∈ BUC(Θ). It is also clear that u ≤ u and u T = u T . Then it is enough to prove that u is a P-viscosity supersolution and u is a P-viscosity subsolution, so that by the comparison principle we may obtain u ≤ u, to conclude the proof of Theorem 3.7.

Proposition 6.5. The functions u and u defined in (6.6) are P-viscosity supersolution and subsolution, respectively.

Proof. We only prove the result for u. The corresponding result for u can be proved similarly.

1. Without loss of generality, we only verify the viscosity supersolution property at the point (0, 0). Let function ϕ ∈ Au(0, 0), and by adding a constant to ϕ, we assume that u(0, 0) > ϕ(0, 0), so that

0 < η := (u -ϕ) 0 = min τ ∈T δ E[(u -ϕ) τ ],
for some δ > 0. (6.7)

Assume that u and ϕ are both bounded by a constant M ≥ 0. Take a subsequence still named as u h such that u 0 = lim h→0 u h 0 . Now fix a constant ε > 0, and denote ϕ ε (t, x) = ϕ(t, x) -εt. By Lemma 6.4, there is a constant C(ε) ∈ (0, 1/L) such that for all 0 < δ < C(ε), we have

sup P∈P P[h δ,x ≤ δ] ≤ ε 32(2M + ε) δ. (6.8)
Since u h is uniformly continuous uniformly in h, by considering δ small enough we may assume that u h -ϕ ε > 0 on [0, hδ,x ], where hδ,x := δ ∧ h δ,x . It follows from (6.7) that

(u -ϕ ε ) 0 ≤ E [(u -ϕ) δ ] = E[(u -ϕ ε ) δ ] -εδ. (6.9)
In Step 2 we will show that

E[(u -ϕ ε ) δ ] ≤ lim h→0 E h [(u h -ϕ ε ) δ ]. (6.10)
It follows that for h sufficiently small .11) Then by the optimal stopping argument in Step 3, we may find (t

(u h -ϕ ε ) 0 ≤ E h (u h -ϕ ε ) δ - 3εδ 4 . ( 6
* , ω * ) ∈ Θ such that hδ,x (ω * ) ∧ (δ -h) > t * ∈ ∆ h and (u h -ϕ ε ) t * ,ω * 0 = min τ ∈T h δ-t * ,β∈B h E h [β τ (u h -ϕ ε ) t * ,ω * τ ], (6.12)
where ∆ h := {kh : k ∈ N}, T h δ-t * := {τ ∈ T δ-t * : τ takes values in ∆ h } and B h is the collection of all processes β defined by β t := e [t/h]-1 i=0 α i h for some F ih -measurable α i taking value in [0, L]. In particular, (6.12) implies that

(u h -ϕ ε )(t * , ω * ) ≤ inf 0≤α≤L E h [e αh (u h -ϕ ε ) t * ,ω * h ] By (ii) of Assumption 3.6, we obtain (u h -ϕ ε )(t * , ω * ) ≤ T t * ,ω * h [u h ] -T t * ,ω * h [ϕ ε ] + hρ(h). Since u h (t * , ω * ) = T t * ,ω * h [u h ], it follows that T t * ,ω * h [ϕ ε ] -ϕ ε (t * , ω * ) h ≤ ρ(h).
Further, by (i) of Assumption 3.6, letting h → 0, we obtain

-∂ t ϕ(t * , ω * t * ) + ε -G(•, ϕ, ∂ x ϕ, ∂ 2 xx ϕ)(t * , ω * t * ) ≥ 0. (6.13)
We next let δ → 0. Since t * < hδ,x (ω * ), we have (t * , ω * ) → 0 as δ → 0. Therefore, it follows from (6.13) that -∂ t ϕ 0 + ε -G(0, u 0 -η, ∂ x ϕ 0 , ∂ 2 xx ϕ 0 ) ≥ 0.

Finally, we can conclude the proof by letting ε → 0 and then η → 0.

2.

For the simplification of notation, we denote X := (u -ϕ ε ) δ and X h := (u h -ϕ ε ) δ . It follows from (iii) of Assumption 3.6 that {X h : h > 0} is uniformly bounded and uniformly continuous uniformly in h, and note that X = lim h→0 X h . By Lemma 6.1 and 6.2, we obtain that

lim h→0 E h [X h ] ≥ lim h→0 E[X h ] + lim h→0 E h [X h ] -E [X h ] ≥ E[X] + lim h→0 inf ℓ>0 E h [X ℓ ] -E[X ℓ ] ≥ E[X] + lim h→0 ρ(h) = E[X].
3. We consider the mixed control and optimal stopping problem in finite discretetime: Recall that Z h > 0 on [0, h δ,x ] for h small enough. Then since τ * ≤ δ, we have On the other hand, we have

Y h t (
E h [Z h τ * ] ≤ E h [Z h τ * ; h δ,x > δ] + E h [Z h τ * ; h δ,x ≤ δ] = inf β∈B E h [β τ * Z h τ * ; h δ,x > δ] + E h [Z h τ * ; h δ,
I 2 := E h Z h τ * ∨(δ-h) -Z h δ ≥ -E h (ρ u + ρ ϕ )(h + 2 B (δ-h)∧• -B δ∧• ) -εh,
where ρ u , ρ ϕ are module of continuity of function u, ϕ, and are chosen to be bounded and continuous. Again by Lemma 6.3, we have for h sufficiently small that

E h (ρ u + ρ ϕ )(h + 2 B (δ-h)∧• -B δ∧• ) < E (ρ u + ρ ϕ )(h+ 2 B (δ-h)∧• -B δ∧• ) + εδ 8 = E (ρ u + ρ ϕ )(h + 2 B h∧• + εδ 8 ,
It follows that lim h→0 E (ρ u + ρ ϕ )(h + 2 B h∧• = 0 and therefore

I 2 > - εδ 4
, for h sufficiently small.

Finally, by (6.17) and (6.11) we have

Y h 0 ≥ E h [Z h δ ] - εδ 4 + I 1 + I 2 > E h [Z h δ ] - 3εδ 4 ≥ Z h 0 ,
which contradicts the definition of Y in (6.14). Therefore, (6.16) is wrong, i.e. there is ω * such that t * := τ * (ω * ) < hδ,x (ω * ) ∧ (δ -h). Further, since Y t * (ω * ) = Z h t * (ω * ), we obtain (6.12).

Conclusion

We provide a convergence theorem of monotone numerical schemes for a class of parabolic PPDE, which generalizes the classical convergence theorem of Barles and Souganidis [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF]. In contrast to the formulation of [START_REF] Zhang | Monotone schemes for fully nonlinear parabolic path dependent PDEs[END_REF], our conditions are satisfied by all classical monotone numerical scheme for PDEs, to the best of our knowledge. Moreover, our results permit to deduce some numerical schemes for path-dependent stochastic differential game problems and the second order BSDEs whose generator depends on z (see (3.3), (3.4)), which are new in literatures.

Other numerical schemes, such as the branching process scheme of Henry-Labordère, Tan and Touzi [START_REF] Henry-Labordère | A numerical algorithm for a class of BSDE via branching process[END_REF], are possible for some PPDE, but it is not analyzed by the monotone scheme arguments.

Assumption 4 . 8 .

 48 (i) The nonlinearity F is Lipschtiz w.r.t. (ω, u, z, γ) uniformly in t and |F (•, •, 0, 0, 0)| 0 < ∞.

Figure 2 :

 2 Figure2: For PPDE (5.1), we choose K 1 = -0.2, K 2 = 0.2, a = 0.04, a = 0.09, b = 0.05 and T = 1. We provide all the numerical solutions w.r.t. difference time step length ∆t. It seems that the faire value is closed to 0.129. For finite-difference scheme, when ∆t is greater than 0.025, we need to use a coarser space-discretization to ensure the monotonicity (similar to the classical CFL condition), which makes a big difference to the numerical solutions for the case ∆t < 0.25. However, the convergence as ∆t → 0 is still obvious.

  Since the function ϕ : ω → max t∈[0,δ] |B t (ω)| is continuous, the set {h ≤ δ} is closed. Consequently, the function 1 {h≤δ} is upper semicontinuous. Lemma 6.4. For any δ > 0 and ε > 0, define x(δ) = Ld √ δ √ δ + -2 ln εδ 4d and h δ,x = inf{t ≥ 0 : |B t | ≥ x}. Then, for δ small enough we have

			sup P∈P	P[h δ ≤ δ] ≤ εδ.	(6.3)
	Proof. Note that
	sup P∈P	P[h δ ≤ δ] = sup P∈P	P max t∈[0,δ]	|B t | ≥ x ≤ d sup P∈P	P max t∈[0,δ]
	tinuous. Note that				1 {h(ω)≤δ} is upper semicon-
		{h ≤ δ} = { max t∈[0,δ]	|B t | ≥ x}

  ω) := inf τ ∈T h δ-t ,β∈B E h [β τ (Z h ) t,ω τ ],whereZ h t := (u h -ϕ ε ) t , t ∈ ∆ h .

	(6.14)
	By standard argument, we have
	Y h 0 = inf

β∈B E h [β τ * Z h τ * ], where τ * := inf{t ∈ ∆ h : Y h t = Z h t }.

  Lδ )(2M + ε)E h [1 {h δ,x ≤δ} ].Further, we obtain from Lemma 6.3 that for h small enough it holdsE h [1 {h δ,x ≤d} ] < E[1 {h δ,x ≤d} ] + Lδ )(2M + ε)E[1 {h δ,x ≤δ} ] + Further, it follows from (6.8) that (1 + e Lδ )(2M + ε)E[1 {h δ,x ≤δ} ] ≤ εδ 8 . Therefore,Further it follows from (6.8) and (6.15) thatI 1 := E h [Z h τ * -Z h τ * ∨(δ-h) ] = E h [Z h τ * -Z h δ-h ; τ * < δ -h] ≥ -(4M + 2ε)E h [1 {h δ,x ≤δ} ] > -(4M + 2ε)E[1 {h δ,x ≤δ} ] -

					εδ 8	≥ -	εδ 4	.
		εδ 8(4M + 2ε)	,	(6.15)
	So we get
	E h [Z h τ εδ 8	.
	Y h 0 ≥ E h [Z h τ * ] -	εδ 4	.
	Suppose that

x ≤ δ] ≤ inf β∈B E h [β τ * Z h τ * ] + sup β∈B E[β τ * |Z h τ * |; h δ,x ≤ δ] + E h [|Z h τ * |; h δ,x ≤ δ] ≤ Y h 0 + (1 + e Lδ )E h [|Z h τ * |; h δ,x ≤ δ] ≤ Y h 0 + (1 + e * ] ≤ Y h 0 + (1 + e hδ,x (ω) ∧ (δ -h) ≤ τ * (ω) ≤ δ, for all ω.

(6.16)

Note that

E h [Z h τ * ] ≥ E h [Z h δ ] + E h [Z h τ * -Z h τ * ∨(δ-h) ] + E h [Z h τ * ∨(δ-h) -Z h δ ]. (

6

.17)

(i) The consistency condition (Assumption 3.6 (i)) is obviously satisfied by (4.1) as in the no path-dependent case.(ii) For the monotonicity in Assumption 3.6 (ii), let us consider two different bounded functions ϕ and ψ. Denote φ := ϕ -ψ, then by direct computation,T t,ω h [ϕ] -T t,ω h [ψ] = φ(ω 0 ) + h G y D 0,t h φ + G z D 1,t h φ + G γ D 2,t h φ ,where G y , G z and G γ is some function depending on (t, ω) and (ϕ, ψ), but uniformly bounded by the Lipschitz constant L of G. Let b ∈ [-L, L] and ε ≤ a ≤ |∇ γ G| 0 be

The authors gratefully acknowledge the financial support of the ERC 321111 Rofirm, the ANR Isotace, the Chairs Financial Risks (Risk Foundation, sponsored by Société Générale), Finance and Sustainable Development (IEF sponsored by EDF and CA), and the region of l'île de France.


G L Klein

G Pierre

M Graber
email: mgraber@univ-lr.fr

Inverse Gas Chromatography with film cell unit: an attractive alternative method to characterize surface properties of thin films

Keywords: Inverse gas chromatography, film sample, surface energy, contact angle

Inverse gas chromatography (IGC) is widely used for the characterization of surfaces. The present work describes a novel IGC tool, the recently developed film cell module, which measures monolithic thin solid film surface properties, whereas only samples in powder or fiber state or polymer-coated supports can be studied by classic IGC. The surface energy of four different solid supports was measured using both classic IGC with columns packed with samples in the powder state, and IGC with the new film cell module or the sessile drop technique, using samples in the film state. The total surface energy and its dispersive and specific components, were measured for Glass, Polyethylene, Polyamide and Polytetrafluoroethylene. Similar results were obtained for the four materials using the three different techniques. The main conclusion is that the new film cell module for IGC is an attractive alternative to the sessile drop technique as it gives very accurate and reproducible results for surface energy components, with significant savings in time and the possible control of sample humidity and temperature. This film cell module for IGC extends the application field of IGC to any thin solid film and can be used to study the effect of any surface treatment on surface energy.

Introduction

Surface energy measurements are very frequently used in material sciences to investigate wettability, adhesion characteristics, specific interactions with other molecules, cohesion and coating performance [START_REF] Good | Contact-angle, wetting and adhesion -A critical review[END_REF]. The most commonly used method to achieve such measurements is the Sessile Drop Technique using a goniometer which is relatively easy to perform and inexpensive. It consists of producing a drop of liquid on a solid and measuring the angle formed between the solid/liquid interface and the liquid/vapor interface, which is called the "contact angle" (CA) [START_REF]Contact Angle, Wettability and Adhesion[END_REF]. This angle, measured for a minimum of three types of liquids, and the known surface energy of the liquids are the parameters that are used to calculate the surface energy of the solid sample [START_REF] Van Oss | Interfacial Lifshitz-van der Waals and polar interactions in macroscopic systems[END_REF]. The main disadvantage of this method is that it is unable to reflect the totality of the surface energy properties, even if multiple droplets are deposited on various locations on the sample. CA gives reproducible results, but problems of reproducibility and accuracy may occur, in case of droplets that are not axially symmetric, or with surface-accessible pores, which can decrease the droplet volume by capillary action during measurement [START_REF] Chini | A method for measuring contact angle of asymetric and symetric drops[END_REF][START_REF] Wenzel | Surface roughness and contact angle[END_REF]. To overcome these disadvantages, alternative methods of measuring surface energy must be developed. Two of the most common alternative methods are capillary intrusion of liquid analytes into the sample and analyte adsorption onto a sample bed at infinite dilution using Inverse Gas Chromatography (IGC). Capillary intrusion and conventional IGC are both restricted to the samples in the powder or fiber state and also to polymers and polymer-coated supports, either packed in columns or deposited as a film on the inside of a column to create a capillary column.

Nevertheless, a significant advance has been made recently with the development of a new system called film cell module for IGC, which is convenient for flat and monolithic samples. That is why we chose to focus on this last method as an alternative to CA measurements.

IGC is a vapor adsorption technique, which consists in an inversion of conventional gas chromatography. Physicochemical characteristics in the stationary phase are studied by injecting specific well-characterized gaseous probes [START_REF] Conder | Physical measurements by gas chromatography[END_REF][START_REF] Hegedus | A review of inverse gas chromatography theory used in the thermodynamic analysis of pigment and polymer surfaces[END_REF][START_REF] Voelkel | Inverse gas chromatography: characterization of polymers, fibers, modified silicas, and surfacants[END_REF]. IGC is conventionally performed in columns containing the packed solid under investigation in the powder or fiber state. The interaction between these probes and the solid material forming the stationary phase is then investigated by determining the retention time for a given probe and used to calculate many physico-chemical properties, such as surface energies of solids [START_REF] Huang | Surface characterization of nylon 66 by inverse gas chromatography and contact angle[END_REF], enthalpy and entropy adsorption [START_REF] Santos | Characterisation of the surface Lewis acid-base properties of poly(butylene terephthalate) by inverse gas chromatography[END_REF], solubility parameter, crystallinity [START_REF] Santos | Analysis of Interactions in Multicomponent Polymeric Systems: The Key-role of Inverse Gas Chromatography[END_REF], surface heterogeneity [START_REF] Gavril | Investigation of the surface heterogeneity of solids from reversed-flow inverse gas chromatography[END_REF], nanorugosity [START_REF] Santos | Analysis of Interactions in Multicomponent Polymeric Systems: The Key-role of Inverse Gas Chromatography[END_REF], glass transition [START_REF] Santos | Analysis of Interactions in Multicomponent Polymeric Systems: The Key-role of Inverse Gas Chromatography[END_REF] and melting temperature [START_REF] Hegedus | A review of inverse gas chromatography theory used in the thermodynamic analysis of pigment and polymer surfaces[END_REF].

For surface energy determination, IGC presents many advantages compared to CA measurements: (i) the ability to quantify strong interaction occurring between the solid and the probes that cannot be characterized by contact angle measurement because of a contact angle close to zero [START_REF] Santos | Analysis of Interactions in Multicomponent Polymeric Systems: The Key-role of Inverse Gas Chromatography[END_REF], (ii) no problem of nanorugosity and surface heterogeneity [START_REF] Brendlé | A new topological index for molecular probes used in inverse gas chromatography for the surface nanorugosity evaluation, 1. Method of evaluation[END_REF][START_REF] Brendlé | A new topological index for molecular probes used in inverse gas chromatography, 2. Application for the evaluation of the solid surface specific interaction potential[END_REF], as the interactions of the probes are measured all along a wide solid surface and give mean values of interaction through the measure of the retention times, whereas CA measurements are restricted to the number of droplets deposited on the surface. Film cell module for IGC provides quite a large interacting area with gaseous probes: a relatively large rectangular sheet of flat sample (10 x 400 mm) is submitted to a gaseous flow carrying the probes, which are situated in a small groove all along the sample (iii) IGC is less time consuming, (iv) IGC is an accurate, versatile, reproducible method, with relatively easy sample preparation, (v) advanced IGC instruments have been developed with fully automated operation, humidity and temperature control, in-situ sample preconditioning; thus experiments may be carried out over appreciable temperature ranges, so that the temperature dependence of thermodynamic interactions can be determined.

The field of application of conventional IGC is wide. It concerns materials in the powder or fiber state and also polymers, that are either coated onto inert support and packed into columns or deposited as a uniform film on the inside of a column to create a capillary column [START_REF] Guillet | Studies of Polymer Structure and Interactions by Automated Inverse Gas Chromatography[END_REF]. It includes synthetic and biological polymers [START_REF] Voelkel | Inverse gas chromatography: characterization of polymers, fibers, modified silicas, and surfacants[END_REF], paper and other cellulosics, fillers and pigments, flavourings and perfumes, minerals and inorganic materials [START_REF] Papirer | Comparison of the surface properties of graphite, carbon black and fullerene samples, measured by inverse gas chromatography[END_REF], food products [START_REF] Boutboul | Influence of the nature and treatment of starch on aroma retention[END_REF], packaging and coatings, pharmaceuticals and medical products [START_REF] Jones | The use of inverse gas chromatography for the study of lactose and pharmaceutical materials used in dry powder inhalers[END_REF], building materials, cosmetics and ingredients, natural and artificial fibers [START_REF] Vega | Characterization of polyarylamide fibers by inverse gas chromatography[END_REF], supported catalysts and microporous material [START_REF] Marton | Lipase hydration in the gas phase: Sorption isotherm measurements and inverse gas chromatography[END_REF] and adsorbents [START_REF] Elizalde-Gonzalez | Gas chromatographic characterization of the adsorption properties of the natural adsorbent CACMM2[END_REF]. The new film cell module for IGC is able to extend these different application fields to monolithic thin solid films.

In the present work, we compare three approaches to determine the surface energy of solids, including the dispersive and specific components: (i) conventional IGC at infinite dilution (IGC-C) with solid samples in the powder state packed in a column as a fixed bed, (ii) two-dimension IGC with the film cell module at infinite dilution (IGC-FC) with solid samples in the film state and (iii) analysis of CA data using the Good-van Oss theory, with solid samples in the film state. The surface energetics of four different materials including polymers (Polyamide (PA), Polytetrafluoroehylene (PTFE), and Polyethylene (PE) and glass were determined. These materials all exist in both powder and film states and their surface energy was investigated by the three IGC-C methods for powders and by IGC-FC and CA for films.

To our knowledge, no study has yet been devoted to the comparison of surface energy obtained for the same material in the granular state using IGC-C columns and in the film state using IGC-FC, nor to the comparison of surface energy values obtained by IGC-FC and CA for samples in the film state.

Experimental

All materials (powders or films) were washed as below. The sample was immersed in PCC-54 (Fisher Scientifics) 2 % (v/v) for 10 minutes with an orbital agitation (100 rpm, Heidolph Rotamax 120). Then they were rinsed five times with sterile ultrapure water at 40°C for 5 minutes with the same orbital agitation. Each film sample was finally wiped with optical cleaning tissue (Whatman 105), and was dried at 40°C into an incubator. All samples were stored at room temperature.

For CA measurements, a minimum of 10 droplets were measured on each surface.

All experiments involving IGC analysis were performed in triplicate, so a standard deviation can be calculated.

Instrumentation and Reagents

CA were measured with a goniometer G40 (Krüss, Germany) at room temperature (23°C) with an accuracy of ± 2°C.

The chromatographic experiments were performed using an IGC 2000 (Surface Measurement Systems, London, UK). A technical drawing of the Film-cell module for IGC-FC (Surface Measurement Systems, London, UK) is presented in Figure 1. As an external bench-top unit made from stainless steel material, the film cell module provides an interacting area of 350 x 40 mm with a 0.3 mm groove.

The apolar probes (decane (C10), nonane (C9), octane (C8), heptane (C7) and hexane (C6)) and polar probes (dichloromethane, chloroform, ethyl acetate, toluene, diiodomethane and formamide) were supplied by Sigma with HPLC purity. They were used without further purifications. Ultrapure water was obtained via a Milli Q system (Millipore, France). The relevant characteristics of both amphotheric and polar probes, including the molecular crosssectional surface area, the acid-base character and the surface energy components are presented in Table I.

Contact angle (CA), IGC-C and IGC-FC measurements were performed on four different supports: Glass (Thermo, 1mm thick), Polyethylene (PE, Goodfellow 0.5mm thick), Polyamide-nylon 6 (PA, Goodfellow 0.5 mm thick) and Polytetrafluoroethylene (PTFE, Goodfellow 0.5 mm thick). For measuremenst by IGC-FC, the film materials was cut in 10 x 400 mm pieces, whereas IGC-C measurements in packed column were performed with the powders with particle size ranging from 100 to 150 µm of diameter. Packing is accomplished with the aid of a mechanical vibrator. For PE, PA and PTFE, both powder and film had the same chemical composition. Both powder or film glass samples were made in soda-lime-silica glass.

Methods

Determination of surface energy by CA

According to the GVOC (Good Van Oss Chaudhury) approach [START_REF] Van Oss | Interfacial Lifshitz-van der Waals and polar interactions in macroscopic systems[END_REF], CA were converted into surface energy components using the Young-van Oss equation (Eq.1), which ignores spreading pressure and highlights Lifshitz-van der Waals and Lewis acid/base surface free energy components.

( ) (

)

+ - - + + + = + L S L S LW L LW S t L γ γ γ γ γ γ θ γ 2 cos 1 (1)
Here, t γ , γ LW , γ + and γ -are the total surface energy, Lifshitz-van der Waals, electron-acceptor (or Lewis-acid) and electron-donor (or Lewis-base) components of the surface free energy respectively; θ is the CA and the subscripts L and S denote the liquid and solid samples, respectively. Equation 2 allowed accessing to the Lewis acid-base components of the surface energy:

- + = γ γ γ 2 AB (2)

Determination of surface energy by IGC

IGC was operated at "infinite dilution.

Determination of the net retention volume

The net retention volume V N of vapor probes is then calculated using Eq.3 [START_REF] Schultz | Surface properties of carbon-fibers determined by inverse gas chromatography[END_REF]:

F t j V N N × × = (ml) (3)
where t N is the net retention time, calculated using Eq. (4); F (ml.min -1 ) the carrier gas flow rate, at the sample temperature and j is the James and Martin compressibility factor, calculated using Eq. ( 5), taking into account the compression of the gas and the pressure drop upstream the column or module [START_REF] James | Gas-liquid partition chromatography: the separation and micro-estimation of volatile fatty acids from formic acid to dodecanoic acid[END_REF].

0 t t t R N - = (4)
where t R is the experimental retention time used by a probe to cross the column or the film cell and t 0 the dead time of the column or film cell, determined through the injection of methane which does not adsorb on the solid. where P i and P 0 are the inlet and outlet pressures respectively.

In practice, j is considered to be equal to 1 when using the film cell module.

Determination of the free energy of adsorption

The net retention volume V N of vapor probes is directly related to the variation of the free energy of adsorption ∆G ads , according to Eq.6

C V T R G N ads + × × - = ∆ ) ln ( (6)
where C is a constant depending on the choice of a reference state of the adsorbed probe and also on the total area of the solid accessible to the probe, R is the gas constant and T the absolute temperature. ∆G ads takes into account two kinds of interactions (Eq.7): dispersive interactions

G G G ∆ + ∆ = ∆ (7)
To obtain the dispersive as well as the specific parts of ∆G ads , different apolar and polar vapor molecules respectively are injected and their net retention volume V N determined.

Determination of the dispersive component of the surface energy

In case of apolar probes, like n-alkanes, which can only interact by dispersive interactions, V N is related to D ads G ∆ by Eq.8, which uses the relation of Fowkes [START_REF] Fowkes | Additivity of intermolecular forces at interfaces. 1. Determination of contribution to surface and interfacial tensions of dispersion forces in various liquids[END_REF]:

( ) ' ) ( 2 ln 2 1 2 1 C a N C V T R G D L D S A N D ads + × × - = + × × - = ∆ γ γ (8)
where N A is the Avogadro constant, R the gas constant, a the molecular cross-sectional area of probes adsorbing on the solid surfaces (m 2 ) and

Determination of the specific component of the surface energy

The experimental points for the polar probe molecules are located above the alkane line in the surface energy plot. The vertical distance between each point and the alkane straight line represents the specific contribution of the interaction, which is expressed as the specific free energy SP ads G ∆ . On the whole, this approach for acid-base calculations used in IGC is the van Oss concept, which provides acid and base numbers in the same units as the dispersive surface energy, according to Eq.9: A conditioning period of 12 hours was applied for each experiment to equilibrate the chromatographic column or the module with their sample in situ at constant conditions of temperature (40°C) and gas flow (helium, 10 mL/min) with a relative humidity (RH) of zero, except when the effect of RH on surface energy was studied. In this case RH was equal to values ranging from 10 to 50% in the carrier gas. The optimal quantity of probes to obtain an infinite dilution mode and sharp and symmetrical peaks were obtained at p/p 0 = 0.025, with p being the partial pressure of the solute in the gas phase and p 0 the saturation vapor pressure of the solute.

( ) ( )         × + × × × × = ∆ + - - + 2

Results

Comparison of the surface energy obtained for materials in either the granular state using IGC-C columns or in the film state using IGC-FC

The main aim of this paper is to investigate the use of a new system called film cell module for IGC, to study the surface energetics of flat and monolithic samples. In order to assess whether meaningful data were obtained using this new module, the surface energetics of the same materials in either the granular state or in the film state were assessed by using respectively IGC-C and IGC-FC.

The IGC-FC method allows direct access to the surface energy of solid films. However, the use of the film cell module, which is made of stainless steel (Figure 1), leads to a measurement bias because it is not inert. The empty module has a significant polar component for its surface energy that must be substracted by performing a blank. The dispersive component is, however, negligible. In contrast, IGC-C columns are made of inert glass and a blank is not necessary.

The results from both IGC-FC and IGC-C are summarized in Table II; the values of the total surface energy calculated from the IGC-FC experiments are in good agreement with those calculated from IGC-C for the different tested supports, except for PA. For this material, these values are equal to 43.8±1.8 mJ/m 2 from IGC-C and 48.5±2.3 mJ/m 2 from IGC-FC, respectively. The magnitude of all values determined here for all materials compared reasonably well with previously reported values in the literature [START_REF] Kinloch | Adhesion and Adhesives: Science and Technology[END_REF].

For all the surface energy components measured in IGC-FC the calculated values are not significantly different from those obtained in IGC-C (Table II), except for PA and for the electron-donor component of glass. For PA, both dispersive and specific components of surface energy obtained from IGC-FC were higher than the ones obtained from IGC-C. For glass, the electron-donor component from IGC-C was higher than from IGC-FC.

Comparison of the surface energy obtained by IGC-FC and CA for materials in the film state

CA is one of the most commonly used techniques in the characterization of surface energetics and wettability of materials in the film state. It was therefore used as a point of comparison with IGC-FC.As observed in Table II, the values of the total surface energy calculated from the CA experiments corroborate the surface energy trends established with IGC-FC and give somewhat lower absolute values for these numbers, in case of the three substrata with the lowest surface energy: PTFE, PE and PA. A GVOC approach was used for CA experiments, which allows the conversionof CA into LW S γ (Lifshitz-van der Waals), and + S γ and - S γ (Lewis acid and base) surface energy components. It is recognized that the Lifshitz-van der Waals contribution is primarily due to dispersion forces or London interactions, although small contributions resulting from the presence of permanent dipoles may also be accounted for in this term (induction or Debye and orientation or Keesom interactions) [START_REF] Steele | Surface Energy of Microcrystalline Cellulose Determined by Capillary Intrusion and Inverse Gas Chromatography[END_REF]. Therefore, it is relevant to compare values obtained from CA and values obtained from IGC-FC.

For PTFE, PE and PA, the D S γ obtained using IGC-FC were slightly superior to the LW S γ values obtained from CA : Interestingly, in the case of glass, a higher value for the total surface energy ( t S γ = 56.2 ± 0.3 mJ/m²) was calculated with CA measurements, compared to IGC-FC, for which a value of 47.9 ± 3.3 mJ/m² was obtained (Table II). This difference is statistically significant with a p-value of 0.00021 (calculated from Student test). It arose only from the

Effect of relative humidity on the surface energy of glass measured by IGC-FC

The influence of the relative humidity (RH) on the evolution of the specific component of the surface energy for glass was examined using IGC-FC.

The results for the total surface energy and its different components measured by IGC-FC as a function of RH are presented in Figure 2. On the right of Figure 2 are also presented the results for the total surface energy and its different components for glass measured by CA with atmospheric RH equal to 50%. For glass from IGC-FC measurements, it appears that the electron-donor component significantly increases with the RH. On the other hand, the dispersive and electron-acceptor components maintain similar values all along the humidity gradient. The values calculated for the electron-donor component increased from 11.0 mJ/m² to 66.79 mJ/m² in accord with the gradient of RH in the gaseous stream and from 0 to 50 %RH in IGC-FC, and showed a linear tendency between 10 and 50 %RH, with a slope equal to 15.3 (mJ/m 2 )/(%RH). The increase in the electron-donor component with RH may be due to the interaction of the probes with water molecules adsorbed on the surface and the formation of silanol (Si-O-H) sites by chemisorption. For glass from CA measurements taken at room temperature with atmospheric RH equal to 50%, it appears that the value of the electron-donor component is close to the one obtained with IGC-FC at 30 and 40%RH.

Discussion

As mentioned in the results part, the values obtained for surface energetics by IGC-FC corroborate those surface energy trends established with IGC-C and the magnitude of these numbers compare reasonably well. Some significant differences between the two methods exist however for the absolute values of both dispersive and specific components of surface energy found for PA and also for the electron-donor component of glass. These differences may arise from an effect of the geometry of the materials on the interpreted surface energetics, as shown previously for different works using IGC. For example, Guillet et al. reported differences for polystyrene between studies using pure polymer packed in a column and or capillary columns coated on the inside with polymer [START_REF] Guillet | Studies of Polymer Structure and Interactions by Automated Inverse Gas Chromatography[END_REF]. They found that specific retention volume values were slightly higher for an open column than for a packed column, possibly because of the higher specific surface area available in the open column. Jones et al. mention that milling increased the dispersive surface energy and surface acidity of lactose and several respiratory drugs. These effects could be ascribed to the introduction of surface structural defects or to the disruption of particle flaws exposing surfaces rich in hydroxyl groups in the case of lactose [START_REF] Jones | The use of inverse gas chromatography for the study of lactose and pharmaceutical materials used in dry powder inhalers[END_REF].

The differences between results from CA and IGC are well known and the discussion about this point is out of the scope of this paper. These differences arise from the vastly different energetics between the two systems, involving gas/condensed phase interaction for IGC and condensed phase/condensed phase interaction for CA. In the present study, another difference was introduced between the two systems through the difference in RH values during experiments: RH was not controlled during CA experiments and was equal to the ambient atmospheric RH which was 50% that day. This is one of the reasons explaining the important difference in the -S γ values found for glass using CA or IGC at RH equal to 0. Glass is by far the most hydrophilic material among the four studied. It is composed mainly of silica, and previous studies have shown that water can rupture siloxane (Si-O-Si) bonds via dissociative chemisorption, forming silanol (Si-O-H) sites, in particular at the first stage of humidification [START_REF] Sun | Effect of moisture on the surface free energy and acid-base properties of mineral oxides[END_REF][START_REF] Mahadevan | Dissociative Chemisorption of Water onto Silica Surfaces and Formation of Hydronium Ions[END_REF]. This chemical change occurring at the surface of glass also affects the specific component of the surface energy. For this reason, the effect of RH on the surface energy was studied for glass by IGC-FC. It appears that when increasing the RH from 0 to 50% in IGC, the difference in the -S γ values found for glass using CA or IGC-FC decreases. However, considering that temperature is changed from 23°C to 40°C between CA and IGC-FC experiments, the quantity of adsorbed water on glass for a same RH is higher in case of IGC-FC, undercutting the relevance of the exact comparison of results from CA and IGC-FC at the same RH.

Conclusion

The new film cell module for inverse Gas Chromatography experiments, presented in this work, enables to obtain meaningful data for the surface energetics of flat and monolithic samples, for solid materials with varying surface properties: PTFE, PE, PA and glass. In comparison with CA measurements, it offers the possibility to control the temperature and the RH perfectly during automated and rapid experiments, including in situ conditioning.

Thanks to these preliminary results, it can be concluded that IGC-FC appears as a viable method. However, more extensive study is needed to identify the origin of differences with standard IGC-C, since the differences in surface geometry may also have an influence on the interpreted surface energetics. IGC-FC extends the possibility of IGC to flat and monolithic samples. The analysis can be performed directly on two dimensions samples, without any prior preparation. The field of application of IGC-FC is large and includes the easy analysis of the influence of cleaning or painting on the surface energy parameters of solid surfaces, important in a wide variety of industries.

  London forces and specific interactions SP ads G ∆ , which consists mostly out in Lewis acid-base contributions. SP ads D ads ads

γ

  the dispersive component of the liquid probe surface energy and D S γ the dispersive component of the surface energy of the solid. If a series of alkanes is injected, D Sγ can be derived from the slope of the fitted line, which is called the "alkane line" in a plot of RTlnV N versus ( )

  component of the surface energy (mJ/mol), N A is the Avogadro constant, a the molecular cross-sectional area of adsorbates (m 2 ), + L γ and - L γ (mJ/m²) the electron acceptor (acid) and electron donor (base) parameters of the probe molecule, + S γ and - S γ (mJ/m²) the electron acceptor (acid) and electron donor (base) parameters of the surface. Then + S γ and - S γ can be calculated thanks to two couples of complementary polar probes: dichloromethane (DCM)/ethyl acetate (EA) and toluene (T)/chloroform (CF), where -S γ is equal to zero for DCM and CF and + S γ is equal to zero for EA and T. The specific or polar component of surface energy SP S γ (mJ/m²) can then be calculated from the + S γ and - S γ according to the Eq. 10. Finally, the total surface energy t Sγ is also accessible through the Eq. 11.

γ

  equal to 11.0 ± 2.6 mJ/m² with IGC-FC and 54.1 ± 2.6 mJ/m² with CA, whereas values on the other side were very similar.

Figure 1 :Figure 2 :

 12 Figure 1: Technical drawing of the film cell module for iGC (35x400x11 mm). The arrows show gas flow. 358 359

Table I :

 I Acid-base character and values of cross sectional area and surface energy components of amphoteric and polar probes used in contact angle and IGC experiments. Values from[START_REF] Hefer | Bitumen surface energy characterization by inverse gas chromatography[END_REF][START_REF] Della Volpe | Some reflections on acid-base solid surface free energy theories[END_REF][START_REF] Schultz | Surface properties of high-energy solids I. Determination of the dispersive component of the surface free energy of mica and its energy of adhesion to water and n-alkanes[END_REF][START_REF] Van Oss | Monopolar surfaces[END_REF][START_REF] Van Oss | Determination of contact angles and pore sizes of porous media by column and thin layer wicking[END_REF], DCM = dichloromethane, EA = ethyl acetate, T = toluene, CF = chloroform.

		Molecular
	Solvents -	cross-sectional
	probes	surface area "a"
		(m²)

Surface energy (mJ/m²) of the liquid probes

			total surface energy t L γ	dispersive component D L γ	electron acceptor parameter + L γ	electron donor parameter -L γ
	n-alkanes
	C6	5.15 x10 -19	18.4	18.4	0.0	0.0
	C7	5.73 x10 -19	20.3	20.3	0.0	0.0
	C8	6.30 x10 -19	21.3	21.3	0.0	0.0
	C9	6.92 x10 -19	22.7	22.7	0.0	0.0
	C10	7.44 x10 -19	23.9	23.9	0.0	0.0
	Contact angle
	water	/	72.8	21.8	25.5	25.5
	formamide	/	58.0	35.6	2.3	39.6
	diiodomethane	/	50.8	50.8	0.0	0.0
	iGC
	DCM	2.99 x10 -19	26.5	26.5	5.2	0.0
	EA	3.29 x10 -19	23.9	23.9	0.0	19.2
	T	4.20 x10 -19	28.5	28.5	0.0	2.3
	CF	3.51 x10 -19	27.2	27.2	3.8	0.0

Table II :

 II Calculated values of surface energy for five materials, PA = polyamide-6, PE = polyethylene, PTFE = polytetrafluoroethylene and Glass. a Materials at powder state. b Materials at film state. Standard deviations were calculated with n ≥ 10 measurements.

			LW S γ (mJ/m 2 )
	Materials	t S γ (mJ/m 2 )	(CA) γ (mJ/m 2 ) D S	SP S γ (mJ/m 2 )	+ S γ (mJ/m 2 )	-S γ (mJ/m 2 )
			(iGC)
	iGC columns a
	(iGC-C)
	PTFE	19.5 ± 2.3	19.1 ± 2.3	0.4 ± 0.0	0.1 ± 0.0	0.5 ± 0.1
	PE	32.2 ± 2.1	31.6 ± 2.1	0.6 ± 0.0	0.1 ± 0.0	0.9 ± 0.2
	PA	43.8 ± 1.8	41.9 ± 1.7	1.9 ± 0.2	0.2 ± 0.1	4.1 ± 2.5
	Glass	46.7 ± 1.6	39.8 ± 1.8	7.0 ± 0.3	0.5 ± 0.2	22.8 ± 1.8
	iGC film-cell b
	(iGC-FC)
	PTFE	21.1 ± 3.1	21.0 ± 3.2	0.1 ± 0.1	0.0 ± 0.0	0.5± 0.3
	PE	32.9 ± 1.5	31.0 ± 1.5	1.9 ± 0.6	0.3 ± 0.3	2.7 ± 1.2
	PA	48.5 ± 2.3	44.7 ± 2.6	3.8 ± 1.2	0.4 ± 0.3	8.9 ± 1.7
	Glass	47.9 ± 3.3	40.0 ± 2.2	7.8 ± 1.1	2.5 ± 2.1	11.0 ± 2.6
	Contact angle b
	(CA)
	PTFE	17.1 ± 0.9	15.9 ± 0.6	1.2 ± 0.3	0.2 ± 0.1	1.9 ± 0.5
	PE	32.5 ± 2.7	29.3 ± 1.5	3.2 ± 1.2	0.8 ± 0.3	3.4 ± 1.6
	PA	42.4 ± 0.9	40.4 ± 0.8	2.0 ± 0.2	0.1 ± 0.0	11.5 ± 1.5
	Glass	56.2 ± 0.3	39.4 ± 1.6	16.8 ± 1.9	1.3 ± 0.3	54.1 ± 0.2

Acknowledgment

Financial support for this work was received from the Conseil Général de Charente-Maritime through the Contrat de Projet Etat-Région (CPER) Littoral. Dr Majid Naderi and Pramod Kerai (Surface Measurement Systems Ltd., London, U.K.) are thanked for helpful advice about film cell use. The manuscript was corrected by a native English speaking scientific translator (http://traduction.lefevere-laoide.net).