Antoine Lejay
email: antoine.lejay@univ-lorraine.fr

The Girsanov theorem without (so much) stochastic analysis

Keywords: Girsanov theorem, Lie-Trotter-Kato formula, Feynman-Kac formula, Stochastic differential equation, Euler scheme, splitting scheme, flow, heat equation, Cameron-Martin theorem

In this pedagogical note, we construct the semi-group associated to a stochastic differential equation with a constant diffusion and a Lipschitz drift by composing over small times the semi-groups generated respectively by the Brownian motion and the drift part. Similarly to the interpretation of the Feynman-Kac formula through the Trotter-Kato-Lie formula in which the exponential term appears naturally, we construct by doing so an approximation of the exponential weight of the Girsanov theorem. As this approach only relies on the basic properties of the Gaussian distribution, it provides an alternative explanation of the form of the Girsanov weights without referring to a change of measure nor on stochastic calculus.

Introduction

This pedagogical paper aims at presenting the Girsanov theorem -a change of measure for the Brownian motion -using the point of view of operator analysis. We start from the sole knowledge of the Brownian distribution and its main properties. By doing so, stochastic calculus is avoided excepted for identifying the limit. Therefore, in a simplified context, we give an alternative proof of a result which is usually stated and proved using stochastic analysis and measure theory.

In 1944, R.H. Cameron and W.T. Martin proved the celebrated theorem on the change of the Wiener measure. They later extend it [START_REF] Cameron | The transformation of Wiener integrals by nonlinear transformations[END_REF][START_REF] Cameron | Transformations of Wiener integrals under a general class of linear transformations[END_REF].

measures and their exponential nature gives rise to the theory of Doléans-Dade martingales [START_REF] Doléans-Dade | Quelques applications de la formule de changement de variables pour les semimartingales[END_REF][START_REF] Maisonneuve | Quelques martingales remarquables associées à une martingale continue[END_REF].

These theorems provide us with measures which are equivalent to the Wiener's one. The converse is also true [START_REF] Orey | Conditions for the absolute continuity of two diffusions[END_REF]: Absolute continuity of Wiener or diffusions measures can only be reached by adding terms of bounded variation [START_REF] Orey | Conditions for the absolute continuity of two diffusions[END_REF][START_REF] Skorohod | On the densities of probability measures in functional spaces[END_REF].

The Cameron-Martin and Girsanov theorems have a profound meaning as well as a deep impact on modern stochastic calculus. For example, for example, they are one of the cornerstones of Malliavin calculus [START_REF] Bell | The Malliavin calculus[END_REF][START_REF] Malliavin | Stochastic analysis[END_REF], likewise a major tool in filtering, statistics of diffusion processes [START_REF] Liptser | Statistics of random processes[END_REF][START_REF] Liptser | Statistics of random processes[END_REF], mathematical finance [START_REF] Jeanblanc | Mathematical methods for financial markets[END_REF], ... The Girsanov theorem has also been extended to some Gaussian processes, including the fractional Brownian motion [START_REF] Erraoui | Canonical representation for Gaussian processes[END_REF][START_REF] Hult | Approximating some Volterra type stochastic integrals with applications to parameter estimation[END_REF].

The Feynman-Kac formula related the Brownian motion with some PDEs. It involves a probabilistic representation with an exponential weight (see Section 9.1). The Feynman-Kac formula could be proved by many ways, including stochastic calculus from one side and the Trotter-Kato-Lie formula on the other side (see [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]Chap. 3,Sect. 5] or [START_REF] Chernoff | Product formulas, nonlinear semigroups, and addition of unbounded operators[END_REF] for a nice introduction to this subject, and [START_REF] Goldstein | Remarks on the Feynman-Kac formula[END_REF] for a proof of the Feynman-Kac formula with this procedure).

To illustrate the latter approach, let us consider three linear matrix-valued equations Ẋ = A𝑋, Ẏ = B𝑌, Ṙ = (A + B)𝑅 with 𝑋 0 = 𝑌 0 = 𝑅 0 = Id, where A and B are 𝑑 × 𝑑-matrices. These equations are easily solved by 𝑋 𝑡 = exp(𝑡A), 𝑌 𝑡 = exp(𝑡B) and 𝑅 𝑡 = exp(𝑡(A + B)).

The solutions 𝑋, 𝑌 and 𝑅 satisfy the semi-group property: 𝑋 𝑡+𝑠 = 𝑋 𝑡 𝑋 𝑠 . If AB = BA, then 𝑅 𝑡 = 𝑌 𝑡 𝑋 𝑡 . This is no longer true in general. However, as shown first by S. Lie [START_REF] Lie | Theorie der Transformationsgruppen[END_REF], the solution 𝑅 could be constructed from 𝑋 and 𝑌 by the following limit procedure:

𝑅 𝑡 = exp(𝑡(A + B)) = lim 𝑛→∞ 𝑋 𝑡/𝑛 𝑌 𝑡/𝑛 𝑋 𝑡/𝑛 𝑌 𝑡/𝑛 • • • 𝑋 𝑡/𝑛 𝑌 𝑡/𝑛 = lim 𝑛→∞ (𝑒 A𝑡/𝑛 𝑒 B𝑡/𝑛 ) 𝑛 .
H.F. Trotter [START_REF] Trotter | On the product of semi-groups of operators[END_REF] and T. Kato [START_REF] Kato | Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups[END_REF] have shown that this could be generalized for large families of linear unbounded operators A and B. In this case, (𝑋 𝑡 ) 𝑡≥0 and (𝑌 𝑡 ) 𝑡≥0 are families on linear operators with the semi-groups property.

On the space of continuous, bounded functions C b (R 𝑑 , R), consider the (scaled) Laplace operator A = 1 2 △ and let B be defined by B𝑓 (𝑥) = 𝑈 (𝑥)𝑓 (𝑥) for any 𝑓 ∈ C(R 𝑑 , R), where the continuous function 𝑈 is called a potential. For a Brownian motion B, 𝑋 𝑡 = E[𝑓 (𝑥 + B 𝑡 )] whereas 𝑌 𝑡 𝑓 (𝑥) = exp(𝑡𝑈 (𝑥))𝑓 (𝑥) for any 𝑓 ∈ C b (R 𝑑 , R), 𝑥 ∈ R 𝑑 and 𝑡 ≥ 0.

With the Trotter-Kato-Lie formula, we compose over short times the semigroup 𝑋 of the Laplace operator with the one 𝑌 of the potential term. Using the Markov property of the Brownian motion, for any bounded, measurable function 𝑓 ,

𝑅 𝑡 𝑓 (𝑥) = lim 𝑛→∞ E [ exp ( 𝑡 𝑛 𝑛-1 ∑ 𝑖=0 𝑈 (𝑥 + B 𝑖𝑡/𝑛 ) ) 𝑓 (𝑥 + B 𝑡 ) ] . ( 1
)
With the right integrability conditions on 𝑈 , we obtain in the limit an exponential representation of the Feynman-Kac formula [START_REF] Kac | On distributions of certain Wiener functionals[END_REF], that is

𝑅 𝑡 𝑓 (𝑥) = E [ exp (∫ 𝑡 0 𝑈 (𝑥 + B 𝑠 ) d𝑠 ) 𝑓 (𝑥 + B 𝑡 ) ]
, which gives a probabilistic representation to the PDE

𝜕 𝑡 𝑅 𝑡 𝑓 (𝑥) = 1 2
△𝑅 𝑡 𝑓 (𝑥) + 𝑈 (𝑥)𝑅 𝑡 𝑓 (𝑥) with 𝑅 0 𝑓 (𝑥) = 𝑓 (𝑥).

The seminal derivation of the Feynman-Kac formula by M. Kac in [START_REF] Kac | On distributions of certain Wiener functionals[END_REF] used an approximation of the Brownian motion by a random walk, which leads to an expression close to [START_REF] Akahori | An algebraic approach to the Cameron-Martin-Maruyama-Girsanov formula[END_REF].

What happens now if we use for B the first-order differential operator 𝑏∇• for a function 𝑏? This means that we consider giving a probabilistic representation of the semi-group (𝑅 𝑡 ) 𝑡≥0 related to the PDE

𝜕 𝑡 𝑅 𝑡 𝑓 (𝑥) = 1 2 △𝑅 𝑡 𝑓 (𝑥) + 𝑏(𝑥)∇𝑅 𝑡 𝑓 (𝑥) with 𝑅 0 𝑓 (𝑥) = 𝑓 (𝑥).
Of course, a probabilistic representation is derived by letting X 𝑡 be the solution of the SDE

X 𝑡 = 𝑥 + B 𝑡 + ∫ 𝑡 0 𝑏(X 𝑠 ) d𝑠 and 𝑅 𝑡 𝑓 (𝑥) = E[𝑓 (X 𝑡 )].
With the Girsanov theorem (Theorem 2), 𝑅 𝑡 𝑓 (𝑥) = E[Z 𝑡 𝑓 (𝑥 + B 𝑡 )] for the Girsanov weight Z given by ( 5) below.

The Feynman-Kac formula is commonly understood as a byproduct of the Trotter-Kato-Lie formula [START_REF] Gzyl | The Feynman-Kac formula and the Hamilton-Jacobi equation[END_REF], at least in the community of mathematical physics in relation with the Schrödinger equation [START_REF] Nelson | Feynman integrals and the Schrödinger equation[END_REF][START_REF] Simon | Schrödinger semigroups[END_REF]. The Trotter-Kato-Lie formula also offers a simple interpretation, leading to splitting procedures that provide explicit construction of numerical schemes (see e.g., [START_REF] Blanes | Splitting and composition methods in the numerical integration of differential equations[END_REF][START_REF] Hairer | Geometric numerical integration[END_REF], among others).

Splitting schemes have been proposed to solve SDE3 . Surprisingly enough, we found no trace where the Girsanov theorem is presented as a by-product of the Trotter-Kato-Lie formula 4 Yet its probabilistic interpretation is very simple: we combine Brownian evolution over a short time with a transport equation through the ODE Ẋ = 𝑏(X) over a short time, and so on... A heuristic argument shows how the exponential weight appears. For a continuous, bounded function 𝑓 , the semi-group solution (𝑃 𝑡 ) 𝑡≥0 such that 𝜕 𝑡 𝑃 𝑡 𝑓 (𝑥) =

1 2 △𝑃 𝑡 𝑓 (𝑥) is given by 𝑃 𝑡 𝑓 (𝑥) = E[𝑓 (𝑥 + B 𝑡 )]. The one (𝑄 𝑡 ) 𝑡≥0 that gives the solution to 𝜕 𝑡 𝑄 𝑡 𝑓 (𝑥) = 𝑏(𝑥)∇𝑄 𝑡 𝑓 (𝑥) is for 𝑓 ∈ C 1 b (R 𝑑 , R) 𝑄 𝑡 𝑓 (𝑥) = 𝑓 (Y 𝑡 (𝑥)) = 𝑓 (𝑥) + 𝑡𝑏(𝑥)∇𝑓 (𝑥) + O(𝑡 2 ) with Ẏ𝑡 = 𝑏(Y 𝑡 ) and Y 0 = 𝑥. (2)
Thus, in short time,

𝑃 𝑡 𝑄 𝑡 𝑓 (𝑥) = E[𝑓 (𝑥 + B 𝑡 ) + 𝑡𝑏(𝑥 + B 𝑡 )∇𝑓 (𝑥 + B 𝑡 )] + O(𝑡 2 ) = E[𝑓 (𝑥 + B 𝑡 ) + 𝑡𝑏(𝑥)∇𝑓 (𝑥 + B 𝑡 )] + E[𝑡(𝑏(𝑥 + B 𝑡 ) -𝑏(𝑥)∇𝑓 (𝑥 + B 𝑡 ))] + O(𝑡 2 ). ( 3
)
Since the Gaussian density 𝑝(𝑡, 𝑥) = exp(-𝑥 2 /2𝑡)/ √ 2𝜋𝑡 satisfies 𝜕 𝑥 𝑝(𝑡, 𝑥) = -𝑥 𝑡 𝑝(𝑡, 𝑥), an integration by parts implies that5

E[∇𝑓 (𝑥 + B 𝑡 )] = ∫ 𝑝(𝑡, 𝑦)𝜕 𝑦 𝑓 (𝑥 + 𝑦) d𝑦 = ∫ 𝑦 𝑡 𝑝(𝑡, 𝑦)𝑓 (𝑥 + 𝑦) d𝑦 = E [ B 𝑡 𝑡 𝑓 (𝑥 + B 𝑡 ) ] so that E[𝑡𝑏(𝑥)∇𝑓 (𝑥 + B 𝑡 )] = E [B 𝑡 𝑏(𝑥)𝑓 (𝑥 + B 𝑡 )] .
Since 𝑏 is Lipschitz, the term involving 𝑏(𝑥 + B 𝑡 ) -𝑏(𝑥) in (3) is considered as a higher order term, roughly of order O(𝑡 3/2 ) since B 𝑡 is roughly of order O( √ 𝑡) and it is multiplied by 𝑡. Thus,

𝑃 𝑡 𝑄 𝑡 𝑓 (𝑥) = E [𝑓 (𝑥 + B 𝑡 ) + B 𝑡 𝑏(𝑥)𝑓 (𝑥 + B 𝑡 )] + O(𝑡 3/2 ).
Again owing to the regularity of the Brownian motion since (B 2 𝑡 -𝑡) 𝑡≥0 is a martingale, we get

𝑃 𝑡 𝑄 𝑡 𝑓 (𝑥) = E [ exp ( 𝑏(𝑥)B 𝑡 - 𝑡 2 𝑏(𝑥) 2 ) 𝑓 (𝑥 + B 𝑡 ) ] + O(𝑡 3/2 )
to take into account the Taylor expansion of the exponential at order 2. The exponential weight is an approximation of the Girsanov weight in short time so that a similar analysis may be performed to show that

𝑃 𝑡 𝑄 𝑡 𝑓 (𝑥) = E[Z 𝑡 𝑓 (𝑥 + B 𝑡 )] + O(𝑡 3/2 ).
We also see that limit of our approach, as it requires both 𝑓 to be C 1 b (R 𝑑 , R) and 𝑏 to be Lipschitz. Such a restriction does not hold with the classical approach.

Using the above set of ideas, we then aim at proving rigorously the Girsanov theorem in a restricted setting, by considering only the solution of the stochastic differential equation

X 𝑡 = 𝑥 + B 𝑡 + ∫ 𝑡 0 𝑏(X 𝑠 ) d𝑠 for a drift 𝑏 ∈ C 1 b (R 𝑑 , R 𝑑
) and a 𝑑-dimensional Brownian motion on a probability space (Ω, ℱ, P). As a rule, we try to stuck on functional analysis arguments, and not on stochastic calculus to see how far we could go. When X is the canonical process 6 , we show that for any

𝑓 ∈ C 1 b (R 𝑑 , R),  E[𝑓 (X 𝑡 )] = E[Z 𝑡 𝑓 (X 𝑡 )] (4)
with Z 𝑡 = exp

(∫ 𝑡 0 𝑏(X 𝑠 ) dB 𝑠 - 1 2 ∫ 𝑡 0 |𝑏(X 𝑠 )| 2 d𝑠 ) (5) = 1 + ∫ 𝑡 0 Z 𝑠 𝑏(X 𝑠 ) dB 𝑠 , ( 6
)
where ̂ E is the expectation of the distribution ̂ P of 𝑋, and P the one of the Brownian motion. It is only at the last stage that stochastic calculus is used, to give the expression for Z by passing to the limit. All the other arguments come from functional analysis or simple computations on the Gaussian density.

In the course of events, we obtain an upper bound for the weak rate of convergence of the Euler scheme with a drift coefficient of class C 1 b (R 𝑑 , R 𝑑 ), as well as some insight on the exponential nature of the Girsanov weight.

We actually gives two derivations. One is based on the Euler scheme and could be used without reference to the splitting procedure. It was used, with a different formulation, by G. Maruyama in 1954 to show the differentiability of the transition densities of SDE [START_REF] Maruyama | On the transition probability functions of the Markov process[END_REF]. This approach now seems to be part of the "folklore". The second one is an application of the Trotter-Kato-Lie using the heuristic given above. Both procedures complement each others and gives two alternative discretization of the Girsanov weights.

Our hypotheses are a priori more restrictive than the full Girsanov theorem. Actually,

• Using the Markov property of Z and X, it is possible to show that

E[Z 𝑡 2 𝑓 (X 𝑡 2 )𝑓 (X 𝑡 1 )] = E [ E X𝑡 1 [Z 𝑡 2 ,𝑡 1 𝑓 (X 𝑡 2 -𝑡 1 )]Z 𝑡 1 𝑓 (X 𝑡 1 ) ] = ̂ E[𝑓 (X 𝑡 2 )𝑓 (X 𝑡 1 )]
for 0 ≤ 𝑡 1 ≤ 𝑡 2 , and so on. A limiting argument shows that the transform is valid for functionals of the Brownian motion, not only for the marginal distribution.

• The restriction that 𝑓 ∈ C 1 b (R 𝑑 , R) in ( 4) is easily removed by using a sequence of smooth approximations of a function 𝑓 which is only continuous.

• The Girsanov theorem assumes nothing on the regularity of the drift and requires only integrability condition. Here, the hypothesis that 𝑏 is C 1 b (R 𝑑 , R 𝑑 ) (or at least Lipschitz continuous) is crucial for our analysis. Yet any drift could be reached by regularizing the drift both in the PDE and the expression of the change of measure. However, non-anticipative functional drifts cannot be treated by this approach. We comment this further in Section 10.

• We use the independence of the increments of the Brownian motion as well as the explicit expression of its density (the heat or Gaussian kernel) and its derivative. Considering a diffusion coefficient 𝜎 which is smooth enough, we could apply to approach to

X 𝑡 = 𝑥 + ∫ 𝑡 0 𝜎(X 𝑠 ) dB 𝑠 ,
where the Markov property is used, as well as Gaussian controls over the density of the process and its derivatives, as well as expressions over the derivative of X 𝑡 using Malliavin calculus. It requires fine stochastic and analytic tools while our aim is to be as basic as possible, we use only the Brownian motion seen as a process with independent, Gaussian increments. • Time could be added in 𝑏, provided that 𝑏 is uniformly Lipschitz continuous in space with respect to the time and uniformly bounded. Finally, let us mention an alternative derivation of the Girsanov theorem that relies mostly on algebraic manipulations on the exponential weight [START_REF] Akahori | An algebraic approach to the Cameron-Martin-Maruyama-Girsanov formula[END_REF]. This approach also avoids stochastic calculus as much as they can. Yet more sophisticated tools than ours are used.

Outline. The semi-groups are introduced in Section 2. The main estimates on elementary steps are given in Section 3. The convergence of the semi-groups associated to the Euler scheme is shown in Section 4, while the ones for the splitting procedure is shown in Section 5. The limits of the weights are identified in Section 7, leading to our proof of the Girsanov theorem. The infinitesimal generator of the SDE with drift is identified in Section 8 by the sole use of functional analysis. At last, to complement our idea to avoid stochastic calculus, we present in Section 9 a representation of the Girsanov weights that does not involve stochastic integrals for some special form the of drift. Finally, we discuss in Section 10 the difference between SDEs and ODEs. The article ends with two appendices.

Brownian motion and flows

For a dimension 𝑑, let B be a 𝑑-dimensional Brownian motion on a probability space (Ω, ℱ, P). The drift 𝑏 we consider belongs to C 1 b (R 𝑑 , R), the space of bounded, continuous functions with a bounded, continuous first order derivatives.

We set for convenience ℬ 𝑡,𝑠 (𝑥, 𝜔) = 𝑥 + B 𝑡 (𝜔) -B 𝑠 (𝜔) for any 𝑥 ∈ R 𝑑 , 𝑡 ≥ 𝑠 ≥ 0 and 𝜔 ∈ Ω. This family (B 𝑡,𝑥 (𝑥, 𝜔)) 𝑡≥𝑠≥0 is called a random dynamical system. Clearly,

ℬ 𝑡,𝑠 (ℬ 𝑠,𝑟 (𝑥, 𝜔), 𝜔) = ℬ 𝑡,𝑟 (𝑥, 𝜔) for any 𝑡 ≥ 𝑠 ≥ 𝑟 ≥ 0 and any 𝑥 ∈ R 𝑑 . ( 7
)
This property ( 7) is called the flow property.

For any 𝑠 ≥ 0, let us define (𝒳 𝑡,𝑠 (𝑥, 𝜔)) 𝑡≥𝑠 as the unique solution to

𝒳 𝑡,𝑠 (𝑥, 𝜔) = 𝑥 + B 𝑡 (𝜔) -B 𝑠 (𝜔) + ∫ 𝑡 𝑠 𝑏(𝒳 𝑟,𝑠 (𝑥, 𝜔)) d𝑟 = ℬ 𝑡,𝑠 (𝑥, 𝜔) + ∫ 𝑡 𝑠 𝑏(𝒳 𝑟,𝑠 (𝑥, 𝜔)) d𝑟, 𝑡 ≥ 𝑠. ( 8
)
Actually, there is no need of a theory of stochastic differential equation for this, so that 𝒳

) 9
For any 𝑠 ≥ 0, 𝑥 ∈ R 𝑑 and 𝜔 ∈ Ω, Eq. ( 9) has a unique solution, since 𝛽 is bounded in time and Lipschitz continuous in space. Then, (8) has necessarily a unique solution as the map which transforms 𝒳 to 𝒰 is one-to-one.

To simplify our notations, we drop from now any reference to the event 𝜔 ∈ Ω, which is implicit.

From the uniqueness of the solution to (8), the family (𝒳 𝑡,𝑠 ) 𝑡≥𝑠≥0 satisfies the flow property:

𝒳 𝑡,𝑠 (𝒳 𝑠,𝑟 (𝑥)) = 𝒳 𝑡,𝑠 (𝑥) for 0 ≤ 𝑟 ≤ 𝑠 ≤ 𝑡.
For 𝑥 ∈ R 𝑑 , we also consider the solution to the one-dimensional ODE

𝒴 𝑡,𝑟 (𝑥) = 𝑥 + ∫ 𝑡 𝑟 𝑏(𝒴 𝑠,𝑟 (𝑥)) d𝑠, 𝑡 ≥ 𝑟.
This family also satisfies the flow property: 𝒴 𝑡,𝑟 (𝑥) = 𝒴 𝑡,𝑠 (𝒴 𝑠,𝑟 (𝑥)) for any 0 ≤ 𝑟 ≤ 𝑠 ≤ 𝑡 and any 𝑥 ∈ R 𝑑 . As 𝑏 is time-homogeneous, 𝒴 𝑡,𝑠 (𝑥) = 𝒴 𝑡-𝑠 (𝑥) for any 0 ≤ 𝑠 ≤ 𝑡.

Let us introduce several families of linear operators on C

b (R 𝑑 , R) to C b (R 𝑑 , R) defined by 𝑋 𝑠,𝑡 𝑔(𝑥) = E[𝑔(𝒳 𝑡,𝑠 (𝑥))], 𝑃 𝑠,𝑡 𝑔(𝑥) = E[𝑔(ℬ 𝑡,𝑠 (𝑥))], 𝑄 𝑠,𝑡 𝑔(𝑥) = 𝑔(𝒴 𝑡,𝑠 (𝑥)), 𝑉 𝑠,𝑡 𝑔(𝑥) = E[(1 + 𝑏(𝑥)(B 𝑡 -B 𝑠 ))𝑔(ℬ 𝑡,𝑠 (𝑥))], 𝐸 𝑠,𝑡 𝑔(𝑥) = E[𝑔(ℬ 𝑡,𝑠 (𝑥) + (𝑡 -𝑠)𝑏(𝑥))], for 𝑔 ∈ C b (R 𝑑 , R) and any 0 ≤ 𝑠 ≤ 𝑡.
The time indices of 𝑋 have been inverted with respect to the ones of 𝒳 . The same convention holds for the other operators. The reason is the following: as the increments B 𝑡 -B 𝑠 and B 𝑠 -B 𝑟 are independent,

E[𝑔(𝒳 𝑡,𝑠 (𝒳 𝑟,𝑠 (𝑥))] = E[𝑋 𝑠,𝑡 𝑔(𝒳 𝑟,𝑠 (𝑥))] = 𝑋 𝑟,𝑠 𝑋 𝑠,𝑡 𝑔(𝑥).
Acting the same for the other operators,

𝑋 𝑟,𝑠 𝑋 𝑠,𝑡 = 𝑋 𝑟,𝑡 , 𝑄 𝑟,𝑠 𝑄 𝑠,𝑡 = 𝑄 𝑟,𝑡 and 𝑃 𝑟,𝑠 𝑃 𝑠,𝑡 = 𝑃 𝑟,𝑡 . ( 10
)
Besides, 𝒳 𝑠,𝑠 (𝑥) = 𝑥 so that 𝑋 𝑠,𝑠 = Id. The same holds for the pairs (𝒴, 𝑄) and (ℬ, 𝑃 ). This means that (𝑋 𝑟,𝑡 ) 0≤𝑟≤𝑡 , (𝑄 𝑟,𝑡 ) 0≤𝑟≤𝑡 and (𝑃 𝑟,𝑡 ) 0≤𝑟≤𝑡 are nonhomogeneous semi-groups 7 . Their infinitesimal generators are

B := 𝑏∇ for (𝑄 𝑟,𝑡 ) 0≤𝑟≤𝑡 , L := 1 2 △ for (𝑃 𝑟,𝑡 ) 0≤𝑟≤𝑡 and A := L + B for (𝑋 𝑟,𝑡 ) 0≤𝑟≤𝑡
with the appropriate domains (see Section 8).

Some estimates

The next lemma is central for our analysis. Its proof is immediate so that we skip it.

Lemma 1. For bounded, linear operators 𝐴 𝑖 and 𝐵 𝑖 , 𝑖 = 1, . . . , 𝑛,

𝑛 ∏ 𝑖=1 (𝐴 𝑖 + 𝐵 𝑖 ) - 𝑛 ∏ 𝑖=1 𝐴 𝑖 = 𝑛 ∑ 𝑗=1 ⎛ ⎝ 𝑗-1 ∏ 𝑖=1 (𝐴 𝑖 + 𝐵 𝑖 ) ⎞ ⎠ 𝐵 𝑗 ⎛ ⎝ 𝑛 ∏ 𝑖=𝑗+1 𝐴 𝑖 ⎞ ⎠ .
Actually, such a lemma just follows from algebraic manipulations which are valid in any Banach algebra. In the context of semi-group, it can be seen as a discrete version of the perturbation formula [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] which is central for the analysis of semigroups. This perturbation is also related to the so-called parametrix method for constructing densities (see e.g., [START_REF] Bally | A probabilistic interpretation of the parametrix method[END_REF] for an interpretation in the stochastic context).

Differentiability of the flows

We recall here a classical result about the differentability of the flow. A formal proof is given for the sake of clarity.

To simplify the notations, we set The control over the derivative of the flow is then transferred as a control on the semi-groups.

𝛽 = ‖∇𝑏‖ ∞ . Proposition 2.

Corollary 1. For any 𝑡

≥ 𝑠 ≥ 0 and 𝑔 ∈ C 1 b (R 𝑑 , R), ‖∇𝑋 𝑠,𝑡 𝑔‖ ∞ ≤ exp(𝛽(𝑡 -𝑠))‖∇𝑔‖ ∞ .
Proof. For any 0 ≤ 𝑠 ≤ 𝑡, the chain rule implies that

∇𝑋 𝑠,𝑡 𝑔(𝑥) = ∇E[𝑔(𝒳 𝑡,𝑠 (𝑥))] = E[∇𝒳 𝑡,𝑠 (𝑥)∇𝑔(𝒳 𝑡,𝑠 (𝑥))].
The result stems from [START_REF] Cameron | Transformations of Wiener integrals under translations[END_REF].

The heat semi-group

Being associated to the heat equation, we call (𝑃 𝑠,𝑡 ) 0≤𝑠≤𝑡 the heat semi-group (see App. B.1).

Lemma 2. For any 0 ≤ 𝑠 ≤ 𝑡,

‖𝑃 𝑠,𝑡 𝑔‖ ∞ ≤ ‖𝑔‖ ∞ , 𝑔 ∈ C b (R 𝑑 , R) and ‖∇𝑃 𝑠,𝑡 𝑔‖ ∞ ≤ ‖∇𝑔‖ ∞ , 𝑔 ∈ C 1 b (R 𝑑 , R). ( 12
)
In addition, for

𝑔 ∈ C b (R 𝑑 , R) and 𝑖 = 1, . . . , 𝑑, 𝜕 𝑥 𝑖 𝑃 𝑠,𝑡 𝑔(𝑥) = 𝜕 𝑥 𝑖 E[𝑔(ℬ 𝑡,𝑠 (𝑥))] = E [ B 𝑖 𝑡 -B 𝑖 𝑠 𝑡 -𝑠 𝑔(ℬ 𝑡,𝑠 (𝑥)) ] . ( 13
)
For 𝜇 ∈ R, 𝜎 > 0, let us denote by 𝒩 (𝜇, 𝜎) the Gaussian distribution of mean 𝜈 and variance 𝜎.

Proof. Inequalities (12) are immediate. For [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]

, since 𝑥 + B 𝑡 -B 𝑠 ∼ 𝒩 (𝑥, 𝑡 -𝑠), 𝑃 𝑠,𝑡 𝑔(𝑥) = ∫ R 𝑑 1 (2𝜋(𝑡 -𝑠)) 𝑑/2 exp ( -|𝑦 -𝑥| 2 2(𝑡 -𝑠) ) 𝑔(𝑥 + 𝑦) d𝑦.
Since

𝜕 𝑥 𝑖 exp ( -|𝑦 -𝑥| 2 2(𝑡 -𝑠) ) = (𝑦 𝑖 -𝑥 𝑖 ) 𝑡 -𝑠 exp ( -|𝑦 -𝑥| 2 2(𝑡 -𝑠) ) , ( 14
)
we obtain the integration by part formula [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF].

Remark 1. An immediate consequence of ( 13) is that

‖∇𝑃 𝑠,𝑡 𝑔‖ ∞ ≤ ‖𝑔‖ ∞ √ 𝑡 -𝑠 , ∀0 < 𝑠 < 𝑡, ∀𝑔 ∈ C b (R 𝑑 , R). (15)

The transport semi-group

The semi-group (𝑄 𝑠,𝑡 ) 0≤𝑠≤𝑡 is associated to a transport equation (see App. B.2).

Lemma 3. For any 0 ≤ 𝑠 ≤ 𝑡,

‖𝑄 𝑠,𝑡 𝑔‖ ∞ ≤ ‖𝑔‖ ∞ for 𝑔 ∈ C b (R 𝑑 , R). ( 16
)
In addition,

‖∇𝑄 𝑠,𝑡 𝑔‖ ∞ ≤ ‖∇𝑔‖ ∞ exp((𝑡 -𝑠)𝛽), ∀𝑡 ≥ 0 for 𝑔 ∈ C 1 b (R 𝑑 , R). ( 17
)
Proof. Inequality ( 16) is immediate and (17) follows from Proposition 2.

For 𝑔 ∈ C 1 b (R 𝑑 , R),

The Euler scheme

We give a first convergence result which is related to the Euler scheme.

Let us set

ℰ 𝑡,𝑠 (𝑥) = 𝑥 + B 𝑡 -B 𝑠 + 𝑏(𝑥)(𝑡 -𝑠) = ℬ 𝑡,𝑠 (𝑥) + 𝑏(𝑥)(𝑡 -𝑠).
This is one step of the Euler scheme, in the sense that for 𝑡 𝑖 = 𝑖𝑇 /𝑛, 𝑖 = 0, . . . , 𝑛,

𝜉 𝑖 (𝑥) = ℰ 𝑡 𝑖 ,𝑡 𝑖-1 ∘ • • • ∘ ℰ 𝑡 1 ,0 (𝑥)
and the 𝜉 𝑖 (𝑥)'s are easily recursively computed by

𝜉 0 (𝑥) = 𝑥 and 𝜉 𝑖+1 (𝑥) = ℰ 𝑡 𝑖+1 ,𝑡 𝑖 (𝜉 𝑖 (𝑥)) = 𝜉 𝑖 + B 𝑡 𝑖+1 -B 𝑡 𝑖 + 𝑏(𝜉 𝑖 (𝑥)) 𝑇 𝑛 for 𝑖 = 0, . . . , 𝑛 -1.
The Euler scheme provides a simple way to approximate the flow 𝒳 𝑇,0 (𝑥) as

ℰ 𝑇,(𝑛-1)𝑇 /𝑛 ∘ • • • ∘ ℰ 𝑇 /𝑛,0 (𝑥) ---→ 𝑛→∞ 𝒳 𝑇,0 (𝑥) almost surely. (20)
We give an insight of the proof in Appendix A. In Proposition 3 below, an immediate consequence of the next lemma, we provide a weak rate of convergence for the Euler scheme.

Lemma 4. For 𝑔 ∈ C 1 b (R 𝑑 , R), ‖𝐸 𝑠,𝑡 𝑔 -𝑋 𝑠,𝑡 𝑔‖ ∞ ≤ ‖∇𝑔‖ ∞ 𝐾(𝑡 -𝑠) 3/2 . Proof. Actually, 𝒳 𝑡,𝑠 (𝑥) -ℰ 𝑠,𝑡 (𝑥) = ∫ 𝑡 𝑠 (𝑏(𝒳 𝑟,𝑠 (𝑥)) -𝑏(𝑥)) d𝑟 so that with 𝛽 = ‖∇𝑏‖ ∞ , |𝒳 𝑡,𝑠 (𝑥) -ℰ 𝑡,𝑠 (𝑥)| ≤ 𝛽 ∫ 𝑡 𝑠 ⃒ ⃒ ⃒ ⃒ B 𝑟 -B 𝑠 - ∫ 𝑟 𝑠 𝑏(𝒳 𝑢,𝑠 (𝑥)) d𝑢 ⃒ ⃒ ⃒ ⃒ d𝑟. Then for 𝑔 ∈ C 1 (R 𝑑 ; R), ‖𝐸 𝑠,𝑡 𝑔 -𝑋 𝑠,𝑡 𝑔‖ ∞ ≤ ‖∇𝑔‖ ∞ 𝛽 ∫ 𝑡 𝑠 E[|B 𝑟 -B 𝑠 |] d𝑟 + ‖∇𝑔‖ ∞ (𝑡 -𝑠) 2 2 𝛽‖𝑏‖ ∞ . But E[|B 𝑟 -B 𝑠 |] ≤ √ 𝑟 -𝑠. Hence the result. For 𝑔 ∈ C 1 b (R 𝑑 , R), let us denote ‖𝑔‖ ⋆ = max{‖𝑔‖ ∞ , ‖∇𝑔‖ ∞ }.
For a bounded linear operator 𝐴 :

C 1 b (R 𝑑 , R) → C 1 b (R 𝑑 , R), let us set ‖𝐴‖ ∞→∞ = sup 𝑔∈C b ,‖𝑔‖∞=1
‖𝐴𝑔‖ ∞ and ‖𝐴‖ ⋆ = sup

𝑔∈C 1 b ,‖𝑔‖⋆=1 ‖𝐴𝑔‖ ⋆ . ( 21
)
Proposition 3 (Weak rate of convergence of the Euler scheme). For any

𝑔 ∈ C 1 b (R 𝑑 , R), ⃦ ⃦ ⃦ ⃦ ⃦ ( 𝑛-1 ∏ 𝑖=0 𝐸 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 ) 𝑔 -𝑋 0,𝑇 𝑔 ⃦ ⃦ ⃦ ⃦ ⃦ ∞ ≤ ‖𝑔‖ ⋆ 𝐶 √ 𝑛 , ( 22
)
for a constant 𝐶 that depends only on ‖𝑏‖ ⋆ and 𝑇 .

Proof. Writing 𝐷 𝑠,𝑡 = 𝐸 𝑠,𝑡 -𝑋 𝑠,𝑡 and using Lemma 1,

𝐸 0,1/𝑛 • • • 𝐸 (𝑛-1)𝑇 /𝑛,𝑇 𝑔 -𝑋 0,1/𝑛 • • • 𝑋 (𝑛-1)𝑇 /𝑛,𝑇 𝑔 = 𝑛-1 ∑ 𝑗=0 ⎛ ⎝ 𝑛-1 ∏ 𝑖=𝑗+1 𝐸 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 ⎞ ⎠ 𝐷 (𝑗-1)𝑇 /𝑛,𝑗𝑇 /𝑛 ⎛ ⎝ 𝑗-1 ∏ 𝑖=0 𝑋 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 ⎞ ⎠ 𝑔.
With Lemma 4, the product property ( 10) of (𝑋 𝑠,𝑡 ) and Corollary 1,

‖𝐷 (𝑗-1)𝑇 /𝑛,𝑗𝑇 /𝑛 𝑋 0,(𝑗-1)𝑇 /𝑛 𝑔‖ ∞ ≤ ‖∇𝑔‖ ∞ 𝐾 𝑇 3/2 𝑛 3/2 .
In addition, since ‖𝐸 𝑠,𝑡 ‖ ∞→∞ ≤ 1 and ‖𝐸 𝑟,𝑠 𝐸 𝑠,𝑡 ‖ ∞→∞ ≤ ‖𝐸 𝑟,𝑠 ‖ ∞→∞ ‖𝐸 𝑠,𝑡 ‖ ∞→∞ , on get easily [START_REF] Gzyl | The Feynman-Kac formula and the Hamilton-Jacobi equation[END_REF] from the above inequality.

Remark 2. The weak rate of convergence of the Euler scheme is generally established for smoother coefficients (e.g. 𝑏 ∈ C 4 (R 𝑑 , R 𝑑 )) to achieve a rate 1 [START_REF] Talay | Discrétisation d'une équation différentielle stochastique et calcul approché d'espérances de fonctionnelles de la solution[END_REF]. In [START_REF] Mikulevičius | Weak Euler approximation for Itô diffusion and jump processes[END_REF][START_REF] Mikulevičius | Rate of convergence of the Euler approximation for diffusion processes[END_REF], it is shown that for 𝛼-Hölder continuous coefficients with 𝛼 < 2, the order of convergence is 𝛼/2. This approach excludes the integer values of 𝛼, and the terminal condition is required to be (2+𝛼)-Hölder continuous. With our regularity condition on the drift, we complete this result for 𝛼 = 1 when the diffusivity is constant. This result can also be recovered with the results in [START_REF] Kohatsu-Higa | Weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with nonregular drift[END_REF].

Remark 3. For a closely related approach with semi-groups, the article [START_REF] Bally | Approximation of Markov semigroups in total variation distance[END_REF] provides us with some conditions in a more general context to exhibit the rate of convergence of Euler schemes.

An intermediary convergence result

We prove that the products of the operators 𝑉 𝑠,𝑡 over finer and finer partitions of [0, 𝑇 ] converges to the operator 𝑋 0,𝑇 . This result is crucial to study the convergence of the products of 𝑄 𝑠,𝑡 𝑃 𝑠,𝑡 in the next section.

Proposition 4. For any

𝑔 ∈ C 1 b (R 𝑑 , R), ⃦ ⃦ ⃦ ⃦ ⃦ ( 𝑛-1 ∏ 𝑖=0 𝑉 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 ) 𝑔 -𝑋 0,𝑇 𝑔 ⃦ ⃦ ⃦ ⃦ ⃦ ∞ ≤ ‖𝑔‖ ⋆ 𝐶 √ 𝑛 ,
for a constant 𝐶 that depends only on ‖𝑏‖ ⋆ and 𝑇 .

Proof. From the Newton formula,

𝑔(ℰ 𝑡,𝑠 (𝑥)) = 𝑔(ℬ 𝑡,𝑠 (𝑥)) + (𝑡 -𝑠) ∫ 1 0 𝑏(𝑥)∇𝑔(ℬ 𝑡,𝑠 (𝑥) + 𝜏 (𝑡 -𝑠)𝑏(𝑥)) d𝜏, so that |𝐸 𝑠,𝑡 𝑔(𝑥) -𝑉 𝑠,𝑡 𝑔(𝑥)| ≤ |E[𝑔(ℰ 𝑡,𝑠 (𝑥)] -E[𝑔(ℬ 𝑡,𝑠 (𝑥))] -E[(𝑡 -𝑠)𝑏(𝑥)∇𝑔(ℬ 𝑡,𝑠 (𝑥))]| ≤ (𝑡 -𝑠) ∫ 1 0 ‖𝑏‖ ∞ ⃒ ⃒ ⃒ ⃒ E[∇𝑔(ℬ 𝑡,𝑠 (𝑥) + 𝜏 (𝑡 -𝑠)𝑏(𝑥)) -∇𝑔(ℬ 𝑡,𝑠 (𝑥))] d𝜏 ⃒ ⃒ ⃒ ⃒ .
Again with an integration by parts on the density of the normal distribution, for 𝐺 ∼ 𝒩 (0, 1),

E[∇𝜑(𝜇 + 𝜎𝐺)] = 1 √ 2𝜋 ∫ exp ( -|𝑦| 2 2 ) ∇𝜑(𝜇 + 𝜎𝑦) d𝑦 = E [ 𝐺 𝜎 𝜑(𝜇 + 𝜎𝐺) ]
, from which we obtain for 𝜏 ∈ [0, 1],

E[∇𝑔(ℬ 𝑡,𝑠 (𝑥) + 𝜏 (𝑡 -𝑠)𝑏(𝑥))] = E [ B 𝑡 -B 𝑠 𝑡 -𝑠 𝑔(ℬ 𝑡,𝑥 (𝑥) + 𝜏 (𝑡 -𝑠)𝑏(𝑥) ]
.

Thus,

|𝐸 𝑠,𝑡 𝑔(𝑥) -𝑉 𝑠,𝑡 𝑔(𝑥)| ≤ ‖𝑏‖ 2 ∞ 2 E[|B 𝑡 -B 𝑠 |]‖∇𝑔‖ ∞ (𝑡 -𝑠) ≤ 𝐶‖∇𝑔‖ ∞ (𝑡 -𝑠) 3/2 .
With Lemma 4,

|𝑋 𝑠,𝑡 𝑔(𝑥) -𝑉 𝑠,𝑡 𝑔(𝑥)| ≤ 𝐶 ′ (𝑡 -𝑠) 3/2 ‖∇𝑔‖ ∞ .
The result follows from the same argument as in the proof of Proposition 3.

The splitting procedure

The composition 𝑄 𝑠,𝑡 𝑃 𝑠,𝑡 corresponds to a splitting (or composition): first, we follow the flow generated ℬ starting from 𝑥 and then the one generated by 𝒴 •,𝑠 starting at ℬ 𝑡,𝑠 (𝑥). The convergence of the products of 𝑄 and 𝑃 over finer and finer partitions is the spirit of the Trotter-Kato-Lie approach to construct the semi-group 𝑋 (see [START_REF] Chernoff | Product formulas, nonlinear semigroups, and addition of unbounded operators[END_REF][START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]).

The main point is that if

𝑔 is C 1 b (R, R), then 𝑄 𝑠,𝑡 𝑔(𝑥) = 𝑔(𝒴 𝑡,𝑠 (𝑥)) = 𝑔(𝑥) + ∫ 𝑡 𝑠 𝑏∇𝑔(𝒴 𝑟,𝑠 (𝑥)) d𝑟 = 𝑔(𝑥) + (𝑡 -𝑠)𝑏∇𝑔(𝑥) + • • • .
We use this Taylor development to obtain some control over the product 𝑄 𝑠,𝑡 𝑃 𝑠,𝑡 , together with an integration by parts on the Brownian density.

Proposition 5. For any 𝑔

∈ C 1 b (R 𝑑 , R), ⃦ ⃦ ⃦ ⃦ ⃦ ( 𝑛-1 ∏ 𝑖=0 𝑄 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 𝑃 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 ) 𝑔 -𝑋 𝑇,0 𝑔 ⃦ ⃦ ⃦ ⃦ ⃦ ∞ ≤ ‖𝑔‖ ⋆ 𝐶 √ 𝑛 ,
for a constant 𝐶 that depends only on ‖𝑏‖ ⋆ and 𝑇 .

Proof. With ( 12) and ( 17),

‖∇(𝑄 𝑡 𝑃 𝑡 )𝑔‖ ∞ ≤ exp(𝑡‖∇𝑏‖ ∞ )‖∇𝑃 𝑡 𝑔‖ ∞ ≤ exp(𝑡𝛽)‖𝑔‖ ∞ .
From [START_REF] Erraoui | Canonical representation for Gaussian processes[END_REF],

𝑄 𝑠,𝑡 𝑃 𝑠,𝑡 𝑔(𝑥) = 𝑃 𝑠,𝑡 𝑔(𝑥) + (𝑡 -𝑠)𝑏(𝑥)∇𝑃 𝑠,𝑡 𝑔(𝑥) + 𝑅(𝑠, 𝑡, 𝑃 𝑠,𝑡 𝑔, 𝑥)

where the remainder term 𝑅(𝑠, 𝑡, •, 𝑥) is defined by [START_REF] Flandoli | Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations[END_REF]. With [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF], 𝑄 𝑠,𝑡 𝑃 𝑠,𝑡 𝑔(𝑥) = 𝑉 𝑡,𝑠 𝑔(𝑥) + 𝑅(𝑠, 𝑡, 𝑃 𝑠,𝑡 𝑔, 𝑥).

Since 𝑏 ∈ C 1 b (R 𝑑 , R), |𝒴 𝑡,𝑠 (𝑥) -𝑥| = ⃒ ⃒ ⃒ ⃒ ∫ 𝑡 𝑠 𝑏(𝒴 𝑟,𝑠 (𝑥)) d𝑟 ⃒ ⃒ ⃒ ⃒ ≤ ‖𝑏‖ ∞ (𝑡 -𝑠). ( 23
)
Again from [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF],

𝑅(𝑠, 𝑡, 𝑃 𝑠,𝑡 𝑔, 𝑥)

= ∫ 𝑡 𝑠 E [ B 𝑡 -B 𝑠 𝑡 -𝑠 • ( 𝑏(𝒴 𝑟,𝑠 (𝑥))𝑔(𝒴 𝑟,𝑠 (𝑥) + B 𝑡 -B 𝑠 ) -𝑏(𝑥)𝑔(ℬ 𝑡,𝑠 (𝑥))) ) d𝑟 ]
so that with [START_REF] Hairer | Geometric numerical integration[END_REF],

‖𝑅(𝑠, 𝑡, 𝑃 𝑠,𝑡 𝑔, •)‖ ∞ ≤ E [ |B 𝑡 -B 𝑠 | 𝑡 -𝑠 ] ∫ 𝑡 𝑠 (‖∇𝑏‖ ∞ ‖𝑔‖ ∞ + ‖𝑏‖ ∞ ‖∇𝑔‖ ∞ )𝑟 d𝑟 ≤ 2‖𝑏‖ ⋆ ‖𝑔‖ ⋆ (𝑡 -𝑠) 3/2 .
This proves that

⃦ ⃦ ⃦ ⃦ ⃦ ( 𝑛-1 ∏ 𝑖=0 𝑄 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 𝑃 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 )
𝑔 -

( 𝑛-1 ∏ 𝑖=0 𝑉 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 ) 𝑔 ⃦ ⃦ ⃦ ⃦ ⃦ ∞ ≤ ‖𝑔‖ ⋆ 𝐶 √ 𝑛 ,
and the result follows from Proposition 4.

The weights and their limits: the Girsanov theorem

The expression of the weight will be obtained by two ways: The first one involves the expression of the Euler scheme. For this, we need however a change of measure which is easily deduced from the explicit expression of the Gaussian density. The second one involves the expressions of 𝑉 in the splitting procedure. With the discrete time approximation, both expressions leads to different expressions for the weights, whose limit is however the same. The expression obtained by the Euler scheme is a discretization of the stochastic integral in the exponential weight given by ( 5). The ones obtained using the splitting is a discretization of the SDE (6) the exponential weights solve.

We have now all the elements to prove the Girsanov theorem with our approach.

Theorem 3 (Girsanov, simplified form). There exists a stochastic process (Z 𝑡 (𝑥)) 𝑡∈[0,𝑇 ]

given by Z 𝑡 (𝑥) = exp

(∫ 𝑡 0 𝑏(𝑥 + B 𝑠 ) dB 𝑠 - 1 2 ∫ 𝑡 0 |𝑏(𝑥 + B 𝑠 )| 2 d𝑠 ) (24)
or Z 𝑡 (𝑥) = 1 +

∫ 𝑡 0 Z 𝑠 (𝑥)𝑏(𝑥 + B 𝑠 ) dB 𝑠 ( 25
)
such that E[𝑔(𝒳 𝑇,0 (𝑥))] = E[Z 𝑇 (𝑥)𝑔(ℬ 𝑇,0 (𝑥))] for any bounded, measurable function 𝑔.

The weight (Z 𝑡 (𝑥)) 𝑡≥0 will be identified as a limit, and stochastic analysis is needed for this. For this, we will rewrite the probabilistic representation of the operators given by both the Euler scheme and the splitting scheme as involving some exponential term, and then combine them. The expression ( 24) is related to the representation provided by the Euler scheme while (25) stems from the splitting scheme.

The proof of this theorem is then obtained by combining Propositions 7 and 8 below with Remark 5.

Exponential representation of the one-step Euler scheme

The following exponential representation is a direct consequence of the expression of the density of the Gaussian distribution. Lemma 5. For any 𝑥 ∈ R 𝑑 and 0 ≤ 𝑠 ≤ 𝑡,

E[𝑔(ℰ 𝑡,𝑠 (𝑥))] = 𝐸 𝑡,𝑠 𝑔(𝑥) = E [E 𝑠,𝑡 (𝑥)𝑔(ℬ 𝑡,𝑠 (𝑥))] with E 𝑠,𝑡 (𝑥) = exp ( 𝑏(𝑥) • (B 𝑡 -B 𝑠 ) - (𝑡 -𝑠) 2 |𝑏(𝑥)| 2 )
.

Proof. When 𝐺 ∼ 𝒩 (0, 𝜎 2 Id 𝑑×𝑑 ) and 𝐻 ∼ 𝒩 (𝜇, 𝜎 2 Id 𝑑×𝑑 ), then for any measurable, bounded function 𝜑 : R 𝑑 → R,

E[𝜑(𝜇 + 𝐺)] = E[𝜑(𝐻)] = ∫ 𝜑(𝑥) exp ( -|𝑥-𝜇| 2 2𝜎 2 ) (2𝜋𝜎 2 ) 𝑑/2 d𝑥 = ∫ 𝜑(𝑥) exp ( 𝑥 • 𝜇 𝜎 2 - |𝜇| 2 2𝜎 2 ) exp ( -|𝑥| 2 2𝜎 2 ) (2𝜋𝜎 2 ) 𝑑/2 d𝑥 = E [Ψ(𝐺, 𝜇)𝜑(𝐺)] with Ψ(𝐺, 𝜇) = exp ( 𝐺 • 𝜇 𝜎 2 - |𝜇| 2 2𝜎 2 )
.

With

𝜑(𝑦) = 𝑔(𝑥 + 𝑦), 𝐺 = B 𝑡 -B 𝑠 ∼ 𝒩 (0, (𝑡 -𝑠)Id 𝑑×𝑑 )
and 𝐻 = (𝑡 -𝑠)𝑏(𝑥) + B 𝑡 -B 𝑠 ∼ 𝒩 ((𝑡 -𝑠)𝑏(𝑥), (𝑡 -𝑠)Id 𝑑×𝑑 ), this yields the result.

The weights

Let us set for 𝑥 ∈ R 𝑑 and 0 ≤ 𝑟 ≤ 𝑡,

V 𝑟,𝑡 (𝑥) = 1 + 𝑏(𝑥)(B 𝑡 -B 𝑟 ) so that 𝑉 𝑟,𝑡 𝑔(𝑥) = E[V 𝑟,𝑡 (𝑥)𝑔(ℬ 𝑡,𝑟 (𝑥))].
Proposition 6. For any 𝑥 ∈ R 𝑑 , 𝑇 > 0 and 𝑛 ∈ N,

𝑛-1 ∏ 𝑖=0 𝐸 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 𝑔(𝑥) = E [ Z 𝐸 𝑛,𝑛 (𝑥)𝑔(ℬ 𝑡,0 (𝑥)) ] (26)
and By iterating this computation, this leads to [START_REF] Jeanblanc | Mathematical methods for financial markets[END_REF].

𝑛-1 ∏ 𝑖=0 𝑉 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 𝑔(𝑥) = E [ Z 𝑉 𝑛,𝑛 (𝑥)𝑔(ℬ 𝑡,0 (𝑥)) ] with Z 𝐸 𝑘,𝑛 (𝑥) = 𝑘-1 ∏ 𝑖=0 E 𝑖𝑇 /

Uniform integrability of the weights

For To be able to pass to the limit, we prove that the family weights are uniformly integrable martingales. Proof. Using the independence of the increment and the property of the Laplace transform of the Gaussian distribution,

E [ ( exp ( 𝑏(𝑥 + B 𝑖𝑇 /𝑛 )Δ 𝑛 𝑖 B - 𝑇 2𝑛 |𝑏(𝑥 + B 𝑖𝑇 /𝑛 )| 2 )) 2 ⃒ ⃒ ⃒ ⃒ ⃒ ℱ 𝑛 𝑖 ] ≤ exp ( ‖𝑏‖ ∞ 𝑇 𝑛 )
, so that, by computing iteratively the conditional expectations, sup

𝑘=1,...,𝑛 𝑛≥1 E[(Z 𝐸 𝑘,𝑛 (𝑥)) 2 ] ≤ exp(‖𝑏‖ ∞ 𝑇 ).
Similarly,

E [ ( 1 + 𝑏(𝑥 + B 𝑖𝑇 /𝑛 )Δ 𝑛 𝑖 B ) 2 ⃒ ⃒ ⃒ ⃒ ℱ 𝑛 𝑖 ] ≤ 1 + 𝑇 𝑛 ‖𝑏‖ ∞ ≤ exp ( ‖𝑏‖ ∞ 𝑇 𝑛 ) , so that sup 𝑘=1,...,𝑛 𝑛≥1 E[(Z 𝑉 𝑘,𝑛 (𝑥)) 2 ] ≤ exp(‖𝑏‖ ∞ 𝑇 ).
This is sufficient to prove and uniform integrability.

Identification of the limit of the weights: stochastic calculus at last

At this stage, we need some stochastic calculus in order to identify the limit of Z 𝐸 𝑘,𝑛 (𝑥) and Z 𝑉 𝑘,𝑛 (𝑥). Let us rewrite Z 𝐸 𝑘,𝑛 (𝑥) as the iterative family by setting Z 𝐸 0,𝑛 (𝑥) = 1 and

𝑍 𝐸 𝑘+1,𝑛 (𝑥) = Z 𝐸 𝑘,𝑛 (𝑥) exp ( 𝑏(𝑥 + B 𝑡 𝑛 𝑘 )Δ 𝑛 𝑘 B - 𝑇 2𝑛 |𝑏(𝑥 + B 𝑡 𝑛 𝑘 )| 2 ) = exp ( 𝑘 ∑ 𝑖=0 𝑏(𝑥 + B 𝑡 𝑛 𝑖 )Δ 𝑛 𝑖 B - 𝑇 2𝑛 𝑘 ∑ 𝑖=0 |𝑏(𝑥 + B 𝑡 𝑛 𝑖 )| 2 )
for 𝑘 = 0, . . . , 𝑛 -1.

The next result is an immediate consequence of the definition of a stochastic integral. We recover the traditional expression for the Girsanov weight.

Proposition 7. When 𝑛 → ∞ and 𝑘 𝑛 → ∞ with 𝑘 𝑛 /𝑛 ---→

𝑛→∞ 𝑡 for some 𝑡 ∈ [0, 𝑇 ], Z 𝐸 𝑘𝑛,𝑛 (𝑥) converges in probability to Z 𝑡 (𝑥) given by [START_REF] Hult | Approximating some Volterra type stochastic integrals with applications to parameter estimation[END_REF]. An application of the Itô formula yields the well known fact that Z 𝑡 given by ( 24) is solution to the Stochastic Differential Equation [START_REF] Jeanblanc | The Feynman-Kac formula and decomposition of Brownian paths[END_REF].

On the other hand, writing Z 𝑉 𝑘,𝑛 (𝑥) as an iterative family yields that

𝑍 𝑉 𝑘+1,𝑛 (𝑥) = Z 𝑉 𝑘,𝑛 (1 + 𝑏(𝑥 + B 𝑡 𝑛 𝑘 )Δ 𝑛 𝑘 B) for 𝑘 = 0, . . . , 𝑛 -1
with Z 𝑉 0,𝑛 = 1. This could immediately be rewritten as the discrete analogue of ( 25) by

𝑍 𝑉 𝑘+1,𝑛 (𝑥) = 1 + 𝑘 ∑ 𝑖=0 Z 𝑉 𝑖,𝑛 (𝑥)𝑏(𝑥 + B 𝑡 𝑛 𝑖 )Δ 𝑛 𝑖 B for 𝑘 = 0, . . . , 𝑛 -1.
The convergence of Z 𝑉 𝑘,𝑛 (𝑥) is less immediate than the one of Z 𝐸 𝑘,𝑛 (𝑥). Hopefully, it can be dealt with the results of D. Duffie and P. Protter [START_REF] Duffie | From Discrete-to Continuous-Time Finance: Weak Convergence of the Financial Gain Process[END_REF] (see also [START_REF] Kurtz | Weak convergence of stochastic integrals and differential equations[END_REF]Example 8.7,p. 33]).

Proposition 8. When 𝑛 → ∞ and 𝑘 𝑛 → ∞ with 𝑘 𝑛 /𝑛 ---→ 𝑛→∞ 𝑡 for some 𝑡 ∈ [0, 𝑇 ], Z 𝑉 𝑘𝑛,𝑛 (𝑥) converges in distribution to Z 𝑡 (𝑥)
given by [START_REF] Jeanblanc | The Feynman-Kac formula and decomposition of Brownian paths[END_REF] or equivalently [START_REF] Hult | Approximating some Volterra type stochastic integrals with applications to parameter estimation[END_REF].

8 The infinitesimal generator of the semi-group (𝑋 𝑡 ) 𝑡≥0

We compute now the infinitesimal generator of (𝑋 𝑡 ) 𝑡≥0 , still using only functional analysis. For the details, we refer for example to the book [START_REF] Chernoff | Product formulas, nonlinear semigroups, and addition of unbounded operators[END_REF][START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]. The semi-groups (𝑋 𝑠,𝑡 ) 0≤𝑠≤𝑡 , (𝑃 𝑠,𝑡 ) 0≤𝑠≤𝑡 and (𝑄 𝑠,𝑡 ) 0≤𝑠≤𝑡 are indeed homogeneous in time, as actually 𝑋 𝑠,𝑡 = 𝑋 0,𝑡-𝑠 , 𝑃 𝑠,𝑡 = 𝑃 0,𝑡-𝑠 and 𝑄 𝑠,𝑡 = 𝑄 0,𝑡-𝑠 . We now set 𝑋 𝑡 = 𝑋 0,𝑡-𝑠 , 𝑃 𝑡 = 𝑃 0,𝑡-𝑠 and 𝑄 𝑡 = 𝑄 0,𝑡-𝑠 .

Definition 1 (Strongly continuous semi-group). Let B be a Banach space with a norm ‖•‖. A semi-group (𝐹 𝑡 ) 𝑡≥0 on B is said to be strongly continuous if for any 𝑓 ∈ B, 𝑇 𝑓 𝑓 is continuous.

Definition 2 (Infinitesimal generator). The infinitesimal generator of a semigroup (𝐹

𝑡 ) 𝑡≥0 on B is a linear operator F : Dom(F) ⊂ B → B such that Dom(F) = { 𝑓 ∈ B ⃒ ⃒ ⃒ ⃒ ⃒ lim ℎ→0 𝐹 ℎ 𝑓 -𝑓 ℎ exists } and F𝑓 = lim ℎ→0 (𝐹 ℎ 𝑓 -𝑓 )/ℎ.
When (F, Dom(F)) is closed for the graph norm ‖•‖ + ‖F•‖ and densely defined in B, then it determines the semi-group uniquely [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]Proposition 1.4,p. 51].

We consider that the underlying Banach space is B = C z (R 𝑑 , R), the space of continuous, bounded functions that vanish at infinity, and

‖•‖ = ‖•‖ ∞ . We denote by C 𝑘 c (R 𝑑 , R) ⊂ C z (R 𝑑 , R) the space of functions of class C 𝑘 (R 𝑑 , R) with compact support.
On B, we consider the heat operator L = 1 2 △, whose domain is the closure of C 2 c (R 𝑑 , R) for the graph norm. This is the infinitesimal generator of (𝑃 𝑡 ) 𝑡≥0 (see App. B.1).

We also consider B = 𝑏∇ whose domain is the closure of C 1 c (R 𝑑 , R) for the graph norm. The operator (B, Dom(B)) is the infinitesimal generator of (𝑄 𝑡 ) 𝑡≥0 (See App. B.2).

The next result is an "almost" direct consequence of the Trotter-Kato-Lie formula.

Proposition 9. The infinitesimal generator of the semi-group (𝑋

𝑡 ) 𝑡≥0 is A = L + B with Dom(A) = Dom(L).
Proof. To construct the semi-group (𝑋 𝑡 ) 𝑡≥0 , we apply Corollary 5.8 in [17, p. 227], which follows from the Chernoff product formula [17, Theorem 5.2, p. 220].

We show first that Dom(L) ⊂ Dom(B).

For any 𝜆 > 0, (𝜆-L) is invertible on B and Dom(L) = (𝜆-L) -1 (B). Moreover,

(𝜆 -L) -1 𝑓 (𝑥) = ∫ +∞ 0 𝑒 -𝜆𝑡 𝑃 𝑡 𝑓 (𝑥) d𝑡. ( 27
)
Using the control [START_REF] Doob | Conditional Brownian motion and the boundary limits of harmonic functions[END_REF], for any 𝑓 ∈ B,

‖∇(𝜆 -L) -1 𝑓 (𝑥)‖ ∞ ≤ 𝐶(𝜆)‖𝑓 ‖ ∞ (28)
with

𝐶(𝜆) = ∫ +∞ 0 𝑡 -1/2 𝑒 -𝜆𝑡 d𝑡. Thus, ‖B(𝜆 -L) -1 𝑓 ‖ ∞ ≤ ‖𝑏‖ ∞ 𝐶(𝜆)‖𝑓 ‖ ∞ .
It follows that B is well defined for any Dom(L) since for any 𝑔 ∈ Dom(L), there exists 𝑓 ∈ B such that 𝑔 = (𝜆 -L) -1 𝑓 . Both (𝑄 𝑡 ) 𝑡≥0 and (𝑃 𝑡 ) 𝑡≥0 are contraction semi-groups8 , since ‖𝑃 𝑡 𝑔‖ ∞ ≤ ‖𝑔‖ ∞ and ‖𝑄 𝑡 𝑔‖ ∞ ≤ ‖𝑔‖ ∞ for any 𝑔 ∈ B = C z (R 𝑑 , R). Thus, for any 𝑡 ≥ 0, 𝑃 𝑡 𝑄 𝑡 is a bounded operator on B with norm 1.

It remains to show that (𝜆 -L -B)(Dom(L)) is dense in B = C z (R 𝑑 , R) for some 𝜆 > 0. We actually show that (𝜆 -L -B) is one-to-one between Dom(L) to B.

For this, we consider finding the pairs (𝑓, 𝑔) ∈ Dom(L) × B such that

(𝜆 -L -B)𝑓 = 𝑔. (29)
We rewrite [START_REF] Kohatsu-Higa | Weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with nonregular drift[END_REF] as

𝑓 -(𝜆 -L) -1 B𝑓 = (𝜆 -L) -1 𝑔,
so that 𝑓 is sought as

𝑓 = K𝑔 = lim 𝑛→∞ K 𝑛 𝑔 with K 𝑛 𝑔 = 𝑛 ∑ 𝑘=0 ( (𝜆 -L) -1 B ) 𝑘 (𝜆 -L) -1 𝑔. ( 30
)
For any 𝑘 ≥ 1,

( (𝜆 -L) -1 B ) 𝑘 (𝜆 -L) -1 = (𝜆 -L) -1 ( B(𝜆 -L) -1
) 𝑘 .

Inequality [START_REF] Blanes | Splitting and composition methods in the numerical integration of differential equations[END_REF] proves that B(𝜆 -L) -1 is a bounded operator on B with constant ‖𝑏‖ ∞ 𝐶(𝜆).

When 𝜆 → ∞, 𝐶(𝜆) decreases to 0. We choose 𝜆 large enough so that ‖𝑏‖ ∞ 𝐶(𝜆) < 1. An immediate consequence of ( 27) is that (𝜆 -L) -1 is bounded by 1/𝜆. Thus, for any 𝑘 ≥ 1,

⃦ ⃦ ⃦ ⃦ ( (𝜆 -L) -1 B ) 𝑘 (𝜆 -L) -1 𝑔 ⃦ ⃦ ⃦ ⃦ ∞ ≤ 1 𝜆 ( ‖𝑏‖ ∞ 𝐶(𝜆) ) 𝑘 ‖𝑔‖ ∞ .
This means that the series K𝑔 defined in [START_REF] Kurtz | Weak convergence of stochastic integrals and differential equations[END_REF] converges in B. Moreover, it is easily checked that

(𝜆 -L -B)K 𝑛 (𝜆 -L) -1 𝑔 = -(B(𝜆 -L) -1 ) 𝑛+1 𝑔 + 𝑔.
With [START_REF] Kato | Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups[END_REF], under the condition that 𝐶(𝜆)‖𝑏‖ ∞ < 1, 𝑓 = K𝑔 solves (𝜆-L-B)𝑓 = 𝑔 in B. Since

BK 𝑛 𝑔 = 𝑛 ∑ 𝑘=0 B[(𝜆 -L) -1 B] 𝑘 (𝜆 -L) -1 𝑔 = 𝑛+1 ∑ 𝑘=1 [B(𝜆 -L) -1 ] 𝑘 𝑔,
it follows from (8) that when 𝐶(𝜆)‖𝑏‖ ∞ < 1, (BK 𝑛 𝑔) 𝑛≥1 forms a Cauchy sequence. Since (B, Dom(B)) is a closed operator, its limit is necessarily BK𝑔. Thus,

(𝜆 -L)K 𝑛 𝑔 = 𝑔 -(B(𝜆 -L) -1 ) 𝑛+1 𝑔 + BK 𝑛 𝑔.
Clearly, K 𝑛 𝑔 ∈ Dom(L). Since (L, Dom(L)) is also a closed operator, we obtain that by passing to the limit, K𝑔 ∈ Dom(L) and

(𝜆 -L)K𝑔 = 𝑔 + BK𝑔 so that 𝜆 -L -B is invertible from Dom(L) to B with inverse K which is bounded on B. This implies that (L + B) is itself a closed operator, with domain Dom(L). The latter domain is dense in B. The Chernoff product formula [17, Theorem 5.2, p. 220] then proves that the infinitesimal generator of (𝑋 𝑡 ) 𝑡≥0 is (L + B, Dom(L + B)).

A case where Itô formula could be avoided (or Feynman, Kac, Girsanov and Doob meet together)

Still to stuck to our rule to avoid stochastic calculus for the sake of play, we show that for special form of the drift, an already known probabilistic representation can be obtained by combining the previous results. The representation is the starting point of the so-called exact simulation method from A. Beskos and G.O. Roberts [START_REF] Beskos | Exact simulation of diffusions[END_REF] for simulating a Brownian motion with drift by performing an acceptance/rejection scheme on paths from the Brownian bridge.

The Feynman-Kac formula

The Feynman-Kac formula provides a probabilistic representation to the solution to

⎧ ⎨ ⎩ 𝜕 𝑡 𝑢(𝑡, 𝑥) = 1 2 △𝑢(𝑡, 𝑥) + 𝑈 (𝑥)𝑢(𝑡, 𝑥), 𝑢(0, 𝑥) = 𝑓 (𝑥). ( 31
)
Theorem 4 (M. Kac [START_REF] Kac | On distributions of certain Wiener functionals[END_REF]). Let 𝑈 be bounded 9 . Then the solution of (31) is solution to

𝑢(𝑡, 𝑥) = E [ exp (∫ 𝑡 0 𝑈 (𝑥 + B 𝑠 ) d𝑠 ) 𝑓 (𝑥 + B 𝑡 ) ] . ( 32
)
Proved first by M. Kac, it is also related to the Feynman path integral for solving the Schrödinger equation (See e.g. [START_REF] Nelson | Feynman integrals and the Schrödinger equation[END_REF][START_REF] Simon | Schrödinger semigroups[END_REF]). This formula has many applications, in mathematical physics of course, but also in analysis. For example, it provides an effective way to compute Laplace transforms of functionals of the Brownian motion -the original goal of M. Kac [START_REF] Kac | On distributions of certain Wiener functionals[END_REF] (for applications, see e.g. [START_REF] Jeanblanc | The Feynman-Kac formula and decomposition of Brownian paths[END_REF])or to perform some change of measures [START_REF] Roynette | Penalising Brownian paths[END_REF].

The Feynman-Kac formula could also be proved naturally through the Trotter-Kato-Lie formula (see e.g. [START_REF] Goldstein | Remarks on the Feynman-Kac formula[END_REF][START_REF] Takanobu | On the error estimate of the integral kernel for the Trotter product formula for Schrödinger operators[END_REF]).

With L = 1 2 △ and U be defined on the space of bounded functions by U𝑓 (𝑥) = 𝑈 (𝑥)𝑓 (𝑥), the semi-group (𝑈 𝑡 ) 𝑡≥0 of U is simply given by 𝑈 𝑡 𝑓 (𝑥) = exp(𝑈 (𝑥)𝑡)𝑓 (𝑥), 𝑡 ≥ 0, a fact which is easily verified. Thus, by a computation similar to the one of Proposition 6, for any 𝑡 ≥ 0,

(𝑈 𝑡/𝑛 𝑃 𝑡/𝑛 ) 𝑛 = E [ exp ( 𝑛-1 ∑ 𝑖=0 𝑈 (𝑥 + B 𝑖𝑡/𝑛 ) 𝑡 𝑛 ) 𝑓 (B 𝑡 ) ] ---→ 𝑛→∞ 𝑢(𝑡, 𝑥)
given by [START_REF] Lie | Theorie der Transformationsgruppen[END_REF].

On the other hand, the Trotter-Kato-Lie formula implies that 𝜕 𝑡 𝑣(𝑡, 𝑥) = (L + U)𝑣(𝑡, 𝑥) with 𝑣(0, 𝑥) = 𝑓 (𝑥) when 𝑣(𝑡, 𝑥) = lim 𝑛→∞ (𝑈 𝑡/𝑛 𝑃 𝑡/𝑛 ) 𝑛 𝑓 (𝑥).

The infinitesimal generator of a semi-group under a ℎ-transform

Given a semi-group (𝑃 𝑡 ) 𝑡>0 and a positive function 𝜑, one could naturally set

𝑃 𝜑 𝑡 𝑓 = 1 𝜑 𝑃 𝑡 (𝜑𝑓 ), 𝑡 ≥ 0.
The rationale is that for any measurable, bounded function 𝑓 ,

𝑃 𝜑 𝑡 𝑃 𝜑 𝑠 𝑓 = 1 𝜑 𝑃 𝑡 ( 𝜑 𝜑 𝑃 𝑠 (𝜑𝑓 ) ) = 1 𝜑 𝑃 𝑡+𝑠 (𝜑𝑓 ) =
P 𝜑 𝑓 = P𝑓 + 1 𝜑 ∇𝜑 • ∇𝑓 + 1 2𝜑 (△𝜑)𝑓.
Since 𝜑 is positive, let us write Φ = ln(𝜑) so that for 𝑓 ∈ C 2 b (R 𝑑 , R 𝑑 ),

P 𝜑 𝑓 = L𝑓 + V𝑓 + ∇Φ • ∇𝑓 + 1 2 (△Φ)𝑓 + 1 2 (∇Φ • ∇Φ)𝑓. Remark 6.
If 𝜑 is a harmonic function, then P 𝜑 is the infinitesimal generator of a diffusion process with drift ∇𝜑/𝜑. This is the spirit of the ℎ-transform or Doob's transform introduced by J.L. Doob [START_REF] Doob | Conditional Brownian motion and the boundary limits of harmonic functions[END_REF] for solving problem in potential analysis. This implies a large class of processes with conditioning (for example, to construct a Brownian bridge where the value at a given time is fixed) could be obtained through a process with a drift (Actually, the same computations hold when 𝜑 is also time dependent).

An alternative formulation for the Girsanov weights for some special form of the drift

Given a bounded function 𝜑, it is tempting but hopeless to look for a function 𝜑 such that ∇𝜑 = 𝜑𝑏 and △𝜑 = 0. We are however free to choose the potential 𝑈 . Let (𝑆 𝑡 ) 𝑡≥0 be the semi-group generated by

P = L + U with 𝑈 = -1 2 ∇Φ • ∇Φ + 1 2 △Φ with Φ = log 𝜑. Hence, 𝑆 𝜑 𝑡 𝑓 (𝑥) = 1 𝑒 Φ(𝑥) E 𝑥 [ exp ( - 1 2
∫ 𝑡 0 ∇Φ(B 𝑠 ) • ∇Φ(B 𝑠 ) d𝑠 + ∫ 𝑡 0 1 2 △Φ(B 𝑠 ) d𝑠 + Φ(B 𝑡 ) ) 𝑓 (B 𝑡 ) ] . ( 33
)
According to the above rules, the infinitesimal generator

S 𝜑 of 𝑆 𝜑 𝑡 is S 𝜑 𝑓 = L𝑓 + ∇Φ • ∇𝑓. Proposition 11. Assume that 𝑏 ∈ C 1 (R 𝑑 , R 𝑑 ) is a potential vector field, i.e., there exists Φ ∈ C 2 (R 𝑑 , R) such that 𝑏 = ∇Φ. Then for a bounded, measurable function 𝑓 , E[𝑓 (𝑋 𝑡 )] = 𝑆 𝜑 𝑡 𝑓 (𝑥)
where 𝑆 𝜑 𝑡 is given by [START_REF] Liptser | Statistics of random processes[END_REF]. Actually, (33) is not surprising. With the Itô's formula,

Φ(B 𝑡 ) -Φ(B 0 ) = ∫ 𝑡 0 ∇Φ(B 𝑠 ) dB 𝑠 - 1 2 ∫ 𝑡 0 △Φ(B 𝑠 ) d𝑠,
from which the classical representation of the exponential weight of the Girsanov theorem is easily obtained.

Complement: On the regularity of the drift and the difference between classical and stochastic analysis

With our approach, the drift was assumed to be Lipschitz continuous. The main reason for this condition is that from the very construction, the flow associated to the ODE Ẋ = 𝑏(X) needs to be defined to construct the semi-group (𝑄 𝑡 ) 𝑡>0 .

If 𝑏 is not Lipschitz continuous, the equation Ẋ = 𝑏(X) may have several solutions. A classical example is 𝑏(𝑥) = √ 𝑥. The regularity of 𝑏 could be weakened to define a flow or to consider a particular solution to Ẋ = 𝑏(X) but still a minimal regularity of 𝑏 should be enforced. For example, 𝑏 should belong to some Sobolev space [START_REF] Ambrosio | Lecture notes on the DiPerna-Lions theory in abstract measure spaces[END_REF][START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF].

On the other hand, the situation changes when SDE are considered. Striking results from A. Zvonkin [START_REF] Zvonkin | A transformation of the phase space of a diffusion process that removes the drift[END_REF] and afterwards A. Veretennikov [START_REF] Veretennikov | On strong solutions and explicit formulas for solutions of stochastic integral equations[END_REF] prove the existence of a unique strong solution to the SDE dX 𝑡 = 𝜎(X 𝑡 ) dB 𝑡 + 𝑏(X 𝑡 ) d𝑡 [START_REF] Liptser | Statistics of random processes[END_REF] whatever the regularity of 𝑏 provided that dX 𝑡 = 𝜎(X 𝑡 ) dB 𝑡 , 𝑡 ≥ 0

has a strong solution (of course, with suitable integrability conditions on 𝑏). The solution to (34) defines a flow of diffeomorphisms when 𝜎 is a positive constant and 𝑏 is Hölder continuous.

Taking 𝜎(𝑧) = 𝜖𝑧 for 𝜖 > 0 which is taken as small as possible proved the uniqueness of a solution to the SDE dX 𝑡 = 𝜖 dB 𝑡 + 𝑏(X 𝑡 ) d𝑡 seen as a noisy perturbation of the ODE Ẋ = 𝑏(X) even when the latter ODE has no single solution. This fact is discussed among other references in [START_REF] Bafico | Small random perturbations of Peano phenomena[END_REF][START_REF] Flandoli | Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations[END_REF].

For the Girsanov theorem seen as a change of measure, the regularity of 𝑏 plays no role. One has only to ensure that the exponential super-martingale Z 𝑡 = exp

( ∫ 𝑡 0 𝑏(X 𝑠 ) dB 𝑠 + 1 2 ∫ 𝑡 0 𝑏(X 𝑠 ) d𝑠 )
defining the weight is actually a martingale, and numerous conditions have been given on 𝑏 (See e.g. [START_REF] Girsanov | On transforming a class of stochastic processes by absolutely continuous substitution of measures[END_REF][START_REF] Liptser | Statistics of random processes[END_REF]). For this, the drift appears only in an integrated form. An immediate application is the existence of a weak solution to the (34) when [START_REF] Maisonneuve | Quelques martingales remarquables associées à une martingale continue[END_REF] has one, whatever the regularity of 𝑏 (of course, with the suitable integrability conditions on 𝑏, see e.g. [START_REF] Stroock | Multidimensional diffusion processes[END_REF]).

We then see that a difference holds by considering [START_REF] Maisonneuve | Quelques martingales remarquables associées à une martingale continue[END_REF] and Ẋ = 𝑏(X) separately through the Trotter-Kato-Lie, or by considering directly SDE of type [START_REF] Liptser | Statistics of random processes[END_REF], which draws a line of separation between stochastic and ordinary differential equations. This explains why our conditions on 𝑏 are stronger than the one required usually when invoking the Girsanov theorem.

A Almost sure convergence of the Euler scheme

The convergence property [START_REF] Girsanov | On transforming a class of stochastic processes by absolutely continuous substitution of measures[END_REF] |𝒳 𝑟,0 (𝑥) -𝒳 𝑡 𝑖 ,0 (𝑥)|.

For this, we have used the fact that 𝑏 is Lipschitz continuous and 1 + 𝑇 𝑛 ‖∇𝑏‖ ∞ ≤ exp(𝑇 ‖∇𝑏‖ ∞ /𝑛).

It can be proved that similarly to the Brownian path, each path of 𝒳 𝑡,0 (𝑥) is 𝛼-Hölder continuous for any 𝛼 < 1/2. This is a direct consequence of the Kolmogorov lemma on the regularity of stochastic processes. This proves that the right hand side of [START_REF] Malliavin | Stochastic analysis[END_REF] converges to 0 at rate 𝛼 < 1/2.

B The heat and the transport semi-group

The underlying Banach space is B = C z (R 𝑑 , R), the space of continuous, bounded functions that vanish at infinity. The norm on B is ‖𝑓 ‖ = sup 𝑥∈R 𝑑 |𝑓 (𝑥)|.

B.1 The heat semi-group

The heat semi-group is Multiplying the above equation by 𝑔(𝑥) ∈ C 2 c (R 𝑑 , R), performing an integration by parts then integrating between 0 and 𝑡 lead to

𝑃 𝑡 𝑓 (𝑥) =
Thus, if (L, Dom(L)) is the infinitesimal generator of (𝑃 𝑡 ) 𝑡≥0 (this is necessarily a close operator), L = 1 2 △ on 𝒞 2 c (R 𝑑 , R) ⊂ Dom(L). The latter space being dense in the underlying space C z (R 𝑑 , R) with respect to ‖•‖ ∞ and Dom(L) with respect to the graph norm ‖•‖ ∞ + ‖L•‖ ∞ , (37) characterizes (L, Dom(L)) when C z (R 𝑑 , R) is the ambient Banach space.

In other words, we recover that the infinitesimal generator of the Brownian motion is L = 1 2 △ with a suitable domain. This could of course be easily obtained from the Itô's formula. Also, we see the link between the heat equation (B.1), the Brownian motion and the heat semi-group.

B.2 The transport semi-group

Let us consider now the flow (𝒴 𝑡 ). Since 𝒴(𝑥) is solution to 𝒴 𝑡 (𝑥) = 𝑥+ Dividing each side by 𝜖 and passing to the limit implies that 𝜕 𝑡 𝑄 𝑡 𝑓 (𝑥) = 𝑏(𝑥)∇𝑄 𝑡 𝑓 (𝑥). Conversely, it is also possible to start from the transport equation 𝜕 𝑡 𝑢(𝑡, 𝑥) = 𝑏(𝑥)∇𝑢(𝑡, 𝑥) to construct the flow 𝒴 through the so-called method of characteristics, that is to find the paths 𝒵 : R + → R 𝑑 such that 𝑢(𝑡, 𝒵 𝑡 ) is constant over the time 𝑡.

Lemma 7 .Remark 5 .

 75 Both (Z 𝐸 𝑘,𝑛 (𝑥)) 𝑛≥1 and (Z 𝑉 𝑘,𝑛 (𝑥)) 𝑛≥1 are uniformly integrable. Provided that Z 𝐸 𝑛,𝑛 (𝑥) and Z 𝑉 𝑛,𝑛 (𝑥) converge in distribution to some random variable Z 𝑇 , the uniform integrability of these random variables is sufficient to assert that lim 𝑛→∞ E[Z 𝐸 𝑛,𝑛 (𝑥)𝑔(ℬ 𝑇,0 (𝑥))] = lim 𝑛→∞ E[Z 𝑉 𝑛,𝑛 (𝑥)𝑔(ℬ 𝑇,0 (𝑥))] = E[Z 𝑇 (𝑥)𝑔(ℬ 𝑇,0 (𝑥))].

  function 𝑔 which is bounded or in L 2 (R 𝑑 ), the space of square integrable functions.Since the marginal distribution of the Brownian motion 𝐵 at any time 𝑡 is normal one with mean 0 and variance 𝑡,𝑃 𝑡 𝑓 (𝑥) = E[𝑓 (𝑥 + 𝐵 𝑡 )].Using Fourier transform or computing derivatives,𝜕 𝑡 𝑃 𝑡 𝑓 (𝑥) = 1 2△𝑃 𝑡 𝑓 (𝑥), 𝑥 ∈ R 𝑑 , 𝑡 > 0.

𝑃 1 2

 1 𝑡 𝑔(𝑥) -𝑔(𝑥))𝑓 (𝑥) d𝑥 𝑡 𝑓 (𝑥)△𝑔(𝑥) d𝑥. Then, passing to the limit and since 𝑓 is freely chosen, lim𝑡→0 𝑃 𝑡 𝑔(𝑥) -𝑔(𝑥) 𝑡 = △𝑔(𝑥), ∀𝑥 ∈ R 𝑑 , 𝑔 ∈ C 2 c (R 𝑑 , R).

  ∫ 𝑡 0 𝑏(𝒴 𝑠 (𝑥)) d𝑠, the Newton formula for𝑓 ∈ C 1 c (R 𝑑 , R) implies that 𝑓 (𝒴 𝑡 (𝑥)) -𝑓 (𝑥) = ∫ 𝑡 0 𝑏(𝒴 𝑠 (𝑥))∇𝑓 (𝒴 𝑠 (𝑠)) d𝑠.Hence, the infinitesimal generator of (𝑄𝑡 ) 𝑡≥0 is B = 𝑏∇• whose domain Dom(B) is the closure of C 1 c (R 𝑑 , R)to the graph norm (see e.g. [17, § II.3.28, p. 91]). The semi-group (𝑄 𝑡 ) 𝑡>0 is also linked to a PDE, called the transport equation. We have seen in Proposition 2 that 𝑥 ↦ → 𝒴 𝑡 (𝑥) is differentiable. It is actually of class C 1 (R 𝑑 , R). Thus, 𝑄 𝑡 𝑓 (𝑥) = 𝑓 ∘ 𝒴 𝑡 (𝑥) is also differentiable. Applying the Newton formula to 𝑓 ∘ 𝒴 𝑡 = 𝑄 𝑡 𝑓 and using the flow property of 𝒴 𝑡 leads to 𝑓 (𝒴 𝑡+𝜖 (𝑥)) -𝑓 (𝒴 𝑡 (𝑥)) = 𝑄 𝑡 𝑓 (𝒴 𝜖 (𝑥)) -𝑄 𝑡 𝑓 (𝑥) = ∫ 𝜖 0 𝑏(𝒴 𝑠 (𝑥))∇𝑄 𝑡 𝑓 (𝒴 𝑠 (𝑥)) d𝑠.

  For any 0 ≤ 𝑠 ≤ 𝑡, the maps 𝑥 ↦ → 𝒴 𝑡,𝑠 (𝑥) and 𝑥 ↦ → 𝒳 𝑡,𝑠 (𝑥) are differentiable. Besides, ‖∇𝒴 𝑡,𝑠 (𝑥)‖ ≤ exp((𝑡 -𝑠)𝛽) and ‖∇𝒳 𝑡,𝑠 (𝑥)‖ ≤ exp((𝑡 -𝑠)𝛽), ∀𝑡 ≥ 𝑠 ≥ 0. (11) Proof. For 𝑖 = 1, . . . , 𝑑 and 𝒴 𝑡,𝑠 (𝑥) = (𝒴 1 𝑡,𝑠 (𝑥), . . . , 𝒴 𝑑 𝑡,𝑠 (𝑥)), 𝜕 𝑥 𝑖 𝒴 𝑗 𝑡,𝑠 (𝑥) = 𝛿 𝑖,𝑗 +

	∫ 𝑡	𝑛 ∑
	𝑠
	∫ 𝑡	𝑛 ∑
	𝑠

𝑘=1

𝜕 𝑥 𝑘 𝑏 𝑗 (𝒴 𝑠,𝑟 (𝑥))𝜕 𝑖 𝒴 𝑘 𝑠,𝑟 (𝑥) d𝑟. After having considered formally this equation, it is then possible to show that ∇𝒴 𝑡,𝑠 (𝑥) is really the derivative of 𝒴 𝑡,𝑠 (𝑥). By identifying ∇𝒴 𝑡,𝑠 (𝑥) with a matrix (𝜕 𝑥 𝑖 𝑌 𝑗 𝑡,𝑠 (𝑥)) 𝑖,𝑗=1,...,𝑑 , ‖∇𝒴 𝑡,𝑠 (𝑥)‖ ≤ 1 + ∫ 𝑡 𝑠 𝛽‖∇𝒴 𝑟,𝑠 (𝑥)‖ d𝑟 so that (11) for ∇𝒴 𝑡,𝑠 (𝑥) follows from the Grönwall lemma. Similarly, 𝜕 𝑥 𝑖 𝒳 𝑗 𝑡,𝑠 (𝑥) = 𝛿 𝑖,𝑗 + 𝑘=1 𝜕 𝑥 𝑘 𝑏 𝑗 (𝒳 𝑟,𝑠 (𝑥))𝜕 𝑖 𝒳 𝑘 𝑟,𝑠 (𝑥) d𝑟 so that a control similar to the one on ∇𝒴 𝑡 holds for ∇𝒳 𝑡,𝑠 .

  Proof. Let us work on 𝐸 𝑡,𝑠 , the proof being similar for 𝑉 𝑡,𝑠 .Set 𝐹 (𝑥) = E[𝑔(ℰ 𝑡,𝑠 (𝑥))] so that𝐸 𝑟,𝑠 𝐸 𝑠,𝑡 𝑔(𝑥) = E[𝐸 𝑠,𝑡 𝑔(ℰ 𝑠,𝑟 (𝑥))] = E[𝐹 (ℰ 𝑠,𝑟 (𝑥))] = E[E 𝑟,𝑠 (𝑥)𝐹 (ℬ 𝑠,𝑟 (𝑥))] = E[E 𝑟,𝑠 (𝑥)E 𝑠,𝑡 (ℬ 𝑠,𝑟 (𝑥))𝑔(ℬ 𝑡,𝑠 (ℬ 𝑠,𝑟 (𝑥)))] = E[E 𝑟,𝑠 (𝑥)E 𝑠,𝑡 (ℬ 𝑠,𝑟 (𝑥))𝑔(ℬ 𝑡,𝑟 (𝑥))].

			𝑛,(𝑖+1)𝑇 /𝑛 (ℬ 𝑖𝑇 /𝑛,0 (𝑥))
		𝑘-1
	and Z 𝑉 𝑘,𝑛 (𝑥) =	∏	V 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 (ℬ 𝑖𝑇 /𝑛,0 (𝑥)).
		𝑖=0

  𝑛 ≥ 1 and 𝑇 > 0 fixed, let us denote by Δ 𝑛 𝑖 B = Δ 𝑛 𝑖𝑇 /𝑛,(𝑖+1)𝑇 /𝑛 B, the increments of the Brownian motion. For 𝑖 = 1, . . . , 𝑛, set ℱ 𝑛 𝑖 = 𝜎(Δ 𝑛 0 B, . . . , Δ 𝑛 𝑖-1 B), so that (ℱ 𝑛 𝑖 ) 𝑖≥0 is the filtration generated by the increments of the Brownian motion. We fix a starting point 𝑥 ∈ R 𝑑 .

Lemma 6. For each 𝑛, both (Z 𝐸 𝑘,𝑛 (𝑥)) 𝑘≥0 and (Z 𝑉 𝑘,𝑛 (𝑥)) 𝑛≥0 are discrete martingales with respect to (ℱ 𝑛 𝑖 ) 𝑖=1,...,𝑛 . Besides E[Z 𝐸 𝑘,𝑛 (𝑥)] = E[Z 𝑉 𝑘,𝑛 (𝑥)] = 1. Proof. The proof is immediate using the independence of the increments of the Brownian motion. Remark 4. Although Z 𝐸 𝑛,𝑛 (𝑥) and Z 𝑉 𝑛,𝑛 (𝑥) are close, they are certainly not equal as Z 𝐸 𝑛,𝑛 (𝑥) remains positive while Z 𝑉 𝑛,𝑛 (𝑥) is negative with a positive probability.

  Let 𝜑 be a positive function of class C 2 b (R 𝑑 , R 𝑑 ). Consider the semi-group (𝑃 𝑡 ) 𝑡≥0 generated by P = L + V. Then the infinitesimal generator P 𝜑 of (𝑃 𝜑 𝑡 ) 𝑡≥0 is given by

	Proposition 10.
	𝑃 𝜑 𝑡+𝑠 𝑓 for any 𝑠, 𝑡 ≥ 0
	so that (𝑃 𝜑 𝑡 ) 𝑡≥0 is still a semi-group.

  of the Euler scheme follows from the recursive inequality|𝜉 𝑖+1 (𝑥) -𝒳 𝑡 𝑖+1 ,0 (𝑥)| ≤ |𝜉 𝑖 (𝑥) -𝒳 𝑡 𝑖 ,0 (𝑥)| + 𝜌 𝑖 + 𝑇 𝑛 |𝑏(𝜉 𝑖 (𝑥)) -𝑏(𝒳 𝑡 𝑖 ,0 (𝑥))| ≤ exp (𝑇 ‖∇𝑏‖ ∞ ) |𝑏(𝒳 𝑟,0 (𝑥)) -𝑏(𝒳 𝑡 𝑖 ,0 (𝑥))| d𝑟 ≤ 𝑇 𝑛 ‖∇𝑏‖ ∞ sup 𝑟∈[𝑡 𝑖 ,𝑡 𝑖+1 ]

		𝑖
		∑	𝜌 𝑗 (36)
		𝑗=1
	with
	𝜌 𝑖 =	∫ 𝑡 𝑖+1 𝑡 𝑖

In the domain of SDE, among others, the Ninomiya-Victoir scheme[START_REF] Ninomiya | Weak approximation of stochastic differential equations and application to derivative pricing[END_REF] relies on an astute way to compose the operators.

In[START_REF] Leandre | Applications of the Malliavin calculus of Bismut type without probability[END_REF], R. Léandre gives an interpretation of the Girsanov formula and Malliavin calculus in terms of manipulation on semi-groups.

This is one of the central ideas of Malliavin calculus to express the expectation involving the derivative of a function as the expectation involving the function multiplied by a weight.

This is, X 𝑡 (𝜔) = 𝜔(𝑡) when the probability space Ω is C(R + , R).

These semi-groups are actually time-homogeneous. We however found it more convenient to keep the time dependence for our purpose.

These semi-groups satisfies the far more finer properties of being Feller, as for (𝑋 𝑡 ) 𝑡≥0 , but we do not use it here.

This condition is stronger than the one given in the original article of M. Kac on that subject.

Acknowledgement. I wish to thank K. Coulibaly-Pasquier, V. Bally and A. Kohatsu-Higa for some motivating and interesting discussions on this approach. This article is a follow-up of a talk given at the "groupe de travail" of the Probability and Statistics teams of Institut Élie Cartan de Lorraine (Nancy) on the link between the Trotter-Kato-Lie and the Feynman-Kac formula, and I am grateful to the audience for his/her patience.


Small firms in the sustainable transformation of food industry: Entangling entrepreneurship and activism in grassroots innovation processes

Keywords: grassroots innovazione, PMI, cooperazione, industria alimentare grassroots innovation, SME, food industry, cooperation

azione imprenditoriale ed economica: attivisti ed imprenditori sociali non sono attori distinti con logiche separate ma usano tattiche e piani d'azione simili. Questo articolo si interessa alla combinazione dell'attivismo civico e politico e l'imprenditorialità e vuole dimostrare come tali interazioni possano esistere e funzionare in modo innovativo in piccole imprese al fine di contribuire alla trasformazione di un settore nell'ottica di uno sviluppo sostenibile. Viene studiato in particolar modo il settore dell'industria alimentare e dei cosiddetti sistemi alimentari alternativi. In un contesto di crescente interesse e sensibilità verso modelli alimentari più sostenibili, come riescono le piccole imprese a combinare l'attivismo e l'imprenditorialità per inquadrare i processi di grassroots innovazione e tradurre questo quadro in modello organizzativo?

Introduction

The sustainable transformation of an industry is a complex and collective process, which implies networks of state, civil society, market-based actors and institutions [START_REF] Smith | The governance of sustainable socio-technical transitions[END_REF]. As a process of discovering opportunities in market failures that detract from sustainability [START_REF] Dean | Toward a theory of sustainable entrepreneurship: reducing environmental degradation through entrepreneurial action[END_REF], or as a process of transforming institutions by altering or creating norms, property rights and government legislation [START_REF] Schaltegger | A framework for ecopreneurship. Leading bioneers and environmental managers to ecopreneurship[END_REF][START_REF] Pacheco | Escaping the green prison: entrepreneurship and the creation of opportunities for sustainable development[END_REF], entrepreneurship can help create a sustainable future [START_REF] Gibbs | Sustainability entrepreneurs, ecopreneurs and the development of a sustainable economy[END_REF]. The literature in Social Movement Theory, Organization Studies and Entrepreneurship emphasizes on the linkages between social movement action and economic organization. Indeed, social movement and social entrepreneurship studies tend to be more and more linked [START_REF] Van Den Broek | Dotcauses for sustainability: combining activism and entrepreneurship[END_REF]: activists and social entrepreneurs do not represent separate and distinct actors with different logics of action [START_REF] Vasi | New heroes, old theories? Toward a sociological perspective on social entrepreneurship[END_REF], but tend to transfer their tactics, such as framing, mobilization, protest and negation [START_REF] Mair | Entrepreneurship in and around institutional voids: A case study from Bangladesh[END_REF]. If linking of social entrepreneurs with social movements makes sense at a regime level, the combination of their practices is still not evident at the organizational level. This paper explores how activism and entrepreneurship can be combined in an innovative way by small firms in order to contribute to an industry's transformation towards sustainable development. The industry that is being investigated is the food industry where a wide literature is available to describe how social activists struggle with the market to create new rules of consumption and production [START_REF] Dubuisson-Quellier | Citizenship and Consumption: Mobilisation in Alternative Food Systems in France[END_REF]. However, the role of small business, acting in this field, is less developed. Indeed, the development of AFN -Alternative Food Networks [START_REF] Goodman | Alternative Food Networks. Knowledge, Practice and Politics[END_REF], even based on the participation and the cooperation with consumers, is deeply based on firms that activate new exchange and organizational mechanisms for providing food products. In a context of growing interest and sensitivity towards more sustainable food models, how do these small firms combine activism and entrepreneurship to frame grassroots innovation processes and translate this frame into organizational model? The paper is organized as follows. In the next section, we present previous research that focused on the combination of activism and entrepreneurship in small firms. Then, we explain the research method used, based on a case study conducted within a French cluster of small firms operating in organic, local food and/or short food supply chains. Finally, we present and discuss the case study findings, which allow us to highlight the challenges and the ways of combining the best of these two worlds.

Entangling entrepreneurship and activism in grassroots innovation processes: a conceptual framework

We propose to combine the literature on AFN, grassroots innovation processes, social entrepreneurship, social movement and institutional work to explore innovation practices and processes through which small firms can combine activism and entrepreneurship to change the food industry.

Alternative Food Networks as grassroots innovation processes

The sustainable food supply chain is a mosaic-like field of Alternative Food Networks (AFN). There are many different expressions of the search for sustainable practices, from WEAK experiences, enacting partial change, to STRONG experiences, committed to radically change practices [START_REF] Watts | Making re-connections in agro-food geography: alternative systems of food provision[END_REF]. "Strong AFNs" seem better suited to create social and political change because they challenge the foundations of the conventional food system (Follett, 2009). Increasingly, they represent spaces where producers and consumers go beyond the practices and relationships related to food provisioning and become engaged together in new, more significant forms of food citizenship [START_REF] Renting | Building Food Democracy: Exploring Civic Food Networks and Newly Emerging Forms of Food Citizenship[END_REF]. These spaces of interaction and development of new food practices can be seen as expression of a broader movement aimed at reshaping and repositioning economy within society, that of Solidarity Economy [START_REF] Laville | The Solidarity Economy: An International Movement[END_REF]. Strong Alternative Food Networks (SAFN) prioritize social usefulness, democratic governance, fair trade and support of peasant and organic agriculture of economic projects. They are both political and economic in nature: they want to criticize the actual food provision chain and act to develop another. Although not relevant in size, these firms built around new approaches to food matters contribute to democratize the mainstream agro-food system, both culturally and practically. At its basis, there are significant processes of growth of knowledge and empowerment involving producers and consumers. SAFN are seen as market agencements, sociotechnical arrangements capable of market action (Le Velly R. & Dufeu I., 2016). They are not passive intermediaries between pre-existing supply and demand. They participate in educating and equipping consumers, but they also act on supply by establishing a reference frame of practices that growing numbers of people are assumed to expect. Markets' effects must be seen as the results of their various arrangements. As they contribute to create new borders between civic and economic action, to change market rules and finally to advocate alternative pathways of innovation, SAFN contribute to produce grassroots innovation [START_REF] Horwitch | The interlinking of entrepreneurs, grassroots movements, public policy and hubs of innovation: The rise of Cleantech in New York City[END_REF][START_REF] Smith | Grassroots innovation movements: challenges and contribution[END_REF][START_REF] Fressoli | When grassroots innovation movements encounter mainstream institutions: implications for models of inclusive innovation[END_REF]. Grassroots innovation are innovation coming from the 'grassroots', meaning that it is generally a result of a bottom-up process emanating from communities and users. Based on prior work on social movement framings of grassroots innovation, [START_REF] Fressoli | When grassroots innovation movements encounter mainstream institutions: implications for models of inclusive innovation[END_REF] propose to consider frames and models of inclusive innovation as two different -and recursively connected -aspects of the process of building alternative pathways of innovation. The frame allows to bring the actors together. It is a particular way of problematizing the world, constructed ad hoc to facilitate the emergence of collective action. But frame do not necessarily constitute a blueprint for mobilization and socio-technical experimentation. The frame needs to be translated into model, and this model in turn has to be legible and meaningful to framings associated with mainstream institutions. The implementation of a model can also test ideas drawn from different frames and allow processes of learning that would eventually create feedback and transform the framings as well. [START_REF] Fressoli | When grassroots innovation movements encounter mainstream institutions: implications for models of inclusive innovation[END_REF] argue that it is important to look at models as devices for opening spaces and processes of experimentation, empowerment and alternative ways of knowledge production. While studying grassroots innovations in the sustainable food sector, [START_REF] Rossi | Beyond Food Provisioning: The Transformative Potential of Grassroots Innovation around Food[END_REF] explores their internal dynamics and how they manage their growth and the diffusion of their innovation out of the niche. She shows that the translation of the frame into alternative practices is a complex process, which requires proper changes in multiple domains, and the active engagement of all stakeholders. In a perspective of growth, this translation of the frame into model has to face organizational efficiency requirements and greater complexity. Notably, an iterative collective learning process is crucial to align attitudes and behaviour between new and old network members. Previous research in social entrepreneurship and social movement highlight the combination of activism and entrepreneurship in sustainability-driven small firms.

Entangling entrepreneurship and activism in practice

In the literature, social entrepreneurship and social movement studies tend to be more and more linked [START_REF] Mair | Entrepreneurship in and around institutional voids: A case study from Bangladesh[END_REF][START_REF] Spear | Social Entrepreneurship and Social Movements[END_REF][START_REF] Van Den Broek | Dotcauses for sustainability: combining activism and entrepreneurship[END_REF][START_REF] Vasi | New heroes, old theories? Toward a sociological perspective on social entrepreneurship[END_REF]. Social enterprises share the pursuit of revenue generation with organizations in the private sector as well as the achievement of social (and environmental) goals of non-profit organizations [START_REF] Di Domenico | Social bricolage: Theorizing social value creation in social enterprises[END_REF]. Lots of conventional entrepreneurship practices (like new value creation via a process of bricolage, the creative destruction of organisational models, and catalytic exogenous change) can be applied equally well to social entrepreneurship [START_REF] Nicholls | Fair Trade: Towards an Economics of Virtue[END_REF]. But social entrepreneurship is a mosaic-like field from market-driven (who are in a win-win logic and do not necessarily strive for radical social change) to sustainability-driven. Sustainability-driven entrepreneurs have principles of organisation design, which diverge in important ways from the conventional principles of entrepreneurship. [START_REF] Parrish | Sustainability-driven entrepreneurship: principles of organization design[END_REF] has highlighted five design principles of a 'perpetual reasoning'. It includes resource perpetuation (i.e. produce benefit streams), benefit stacking (i.e. stack as many benefits as possible onto each operational activity), strategic satisficing (i.e. strategically identify satisfactory outcomes of multiples objectives), qualitative management (i.e. use expected quality of outcomes and processes as decision criteria) and worthy contribution (i.e. structure benefit streams to privilege worthy recipients). While most small firms operate under conditions of resource scarcity [START_REF] Torres | Specificity and Denaturing of Small Business[END_REF], sustainability-driven small firms face a specific set of challenges because they purposely locate their activities in areas where markets function poorly. To face these challenges, they use social bricolage [START_REF] Di Domenico | Social bricolage: Theorizing social value creation in social enterprises[END_REF], which is a process that involves making do, the refusal to be constrained by limitations and improvisation (like conventional entrepreneurship bricolage) but also social value creation, stakeholder participation and persuasion (which seems to be specific to social entrepreneurship). As they need to establish support and to access resources from market-based institutions, sustainability-driven small firms may be more susceptible to isomorphic pressures. Indeed, they are exposed to the same risk of institutionalization than social movements [START_REF] Spear | Social Entrepreneurship and Social Movements[END_REF]. Tactics used by social movements, such as framing, mobilization, protest and negation can be useful for social entrepreneurs [START_REF] Mair | Entrepreneurship in and around institutional voids: A case study from Bangladesh[END_REF]. So, social movement theory is relevant to the study of sustainability-driven small firms. Issued from a dynamic social process, which includes conflictual forces (Tourraine, 1985), social movements are alternative spaces for developing new meanings and new systems of praxis, which can be continually reproduced. To achieve their desired social change, activists -members of social movements -often draw on entrepreneurial strategies that help them to mobilize and exploit resources -such as donations, media attention and moral support -in a more efficient and creative way (Van den Broeck et al., 2012). Activists and social entrepreneurs do not represent separate and distinctly actors with different logic of action [START_REF] Vasi | New heroes, old theories? Toward a sociological perspective on social entrepreneurship[END_REF]. Van den Brock et al. (2012) propose a new framework, which positions transformational social entrepreneurs (term used by [START_REF] Nicholls | Fair Trade: Towards an Economics of Virtue[END_REF] as a specific type of social entrepreneurship that combines (radical) social change and the fulfillment of social needs. This type is similar to the social engineer of Zahra et al. 2010) or the sustainability-driven entrepreneur of [START_REF] Parrish | Sustainability-driven entrepreneurship: principles of organization design[END_REF].

Fulfilling social needs High Low

Desire for social change High Transformational social entrepreneurs [START_REF] Zahra | A typology of social entrepreneurs: Motives, search processes and ethical challenges[END_REF][START_REF] Nicholls | Fair Trade: Towards an Economics of Virtue[END_REF] Activists Low Social entrepreneurs Entrepreneurs Transformational social entrepreneurs -entrepreneurs who combine activism and entrepreneurshipintroduce rules and organize their dissemination through institutional work (Lawrence and Suddaby, 2006). These authors divide the process of institutional creation into three categories. The first includes advocacy, defining, vesting, and reflecting the "political work in which actors reconstruct rules, property rights and boundaries that define access to material resources" (Lawrence and Suddaby, 2006: 15). The second set of practices, constructing identities, changing norms and constructing networks, emphasize actions in which actors' belief systems are reconfigured. The final category includes activities that involve associating new practices with the old, theorizing and elaborating chains of cause and effect, and educating, i.e. providing actors with the necessary skills and knowledge to support the new institution.

Small firms in the sustainable transformation of food industry: the GRAP case

Method and presentation of the case

We collected data on a cluster of small firms operating in organic, local food and/or short food supply chains in the Lyon area in France. The city of Lyon is known for its gastronomy culture based on the proximity of the periurban rural areas that bring a diversity of fresh products. However, this gastronomic tradition has long been concerned mainly with high-income people or tourists who could afford a great reputed restaurant. That is why, since 10 years, new initiatives emerge that aim to reactive a diffused culture of food, based on the quality of the products, reasonable prices and strong relationships between producers and consumers.

The empirical research adopted is a mix qualitative methodology. First, we proceeded with 8 semistructured interviews of the two founders, employees and members of the GRAP organization. Second, we adopt an ethnographic research; we became actors of the process we study [START_REF] Greenwood | Participatory action research as a process and as a goal[END_REF] through our direct participation in some activities of GRAP, notably the construction of an organizational cluster, which the objective was to manage the cooperation between sustainable food sector firms of the solidarity economy. All the observations have been done between 2013 and 2016; the founders have been interviewed several times, once a year in order to understand the transformation and the strategy of the organization. Finally, our methods were constructivist in the sense that we considered both the manifestation of action but also the reflexivity of actors in the way they design their own action.

GRAP (Groupement Régional Alimentaire de Proximité) was founded in Lyon in January 2013 and has the status of a SCIC (Société Coopérative d'Intérêt Collectif) 1 under the French law. GRAP currently hosts 21 businesses (64 workers) operating in organic, locally produced food and/or short food supply chains (grocers, import business, restaurant, supermarket, restaurant) for a provisional turnover of 5 millions €. Issues addressed by the project are the change of scale of short food supply chains, professionalization of the actors in the sector, improvement of working conditions, minimizing entrepreneurial risks.

GRAP is a cooperative framework for undertaking projects where entrepreneurs can emerge from isolation and cooperate with other entrepreneurs, exchange products, services and support. It offers several services to small firms, including legal hosting, stock management, accounting, financial engineering, training, entrepreneurship support and business development, etc. It also aims to create synergies between the different firms (by pooling orders, transport and storage facilities, etc.) and promotes cooperative governance and innovative way of doing business and working in the food sector.

The small firms are selected for their ethical approach, the ecological and social impact of their activity, the economic viability of their project and the potential for cooperation and synergies with the other small firms already present in GRAP. The majority became part of GRAP during the start-up of the business, with the exception of 3 petits pois and Prairial.

Name Activity

petits pois Grocery

Petits The study of GRAP allowed us to explore how activism and entrepreneurship issues are combined in the framing of the organization. In this case, two levels were identified in the translation process of this frame into a model (2.2). First an innovative organizational model is constructed to resolve the growth/singularity dilemma. Second, work participation and work balance as a considered as crucial issues to permit grassroots innovations. A discussion also examines the underlying challenges facing GRAP, in order to further document the implications of these results.

Translation of frame into an innovative model

The translation from frame to model was mainly done in two directions. First, the challenge was to build an organizational model that enables the cooperation between small firms and mutualize resources to get broader market spaces. Second, the translation process was targeted to work participation and work balance issues.

An innovative organizational model to solve the growth/singularity dilemma

The first challenge was to develop an innovative organizational model based on cooperation and pooled resources in order to facilitate the development of small firms implicated and the creation of new firms. In other words, in order to defend their niche and diffuse it, these small firms have to invent entrepreneurial forms based on stable and innovative business models. Their stated objective was as follows: "to devise, study, experiment with and promote new forms of cooperation and economic innovation in the social, solidarity and environmental economy" (General Meeting). Moreover, the project is resolutely offensive and aims to take "market shares to the big suppliers and distributors, not to people who develop the same type of project." (Founder 2) This meant therefore inventing technical solutions and new tools with which to consolidate and develop sustainability-driven small firms, as well as demonstrating the innovative nature of these solutions.

The definition of the organizational and business form of GRAP has been the crucial point of this innovation process. Indeed, in order to facilitate its legitimacy and diffusion, it was necessary to demonstrate that the proposal was both original and innovative. One of the founders explains their intention: "Our first idea was to facilitate the emergence of new projects and precisely to organize the interactions and synergies between projects. That is why we have been inspired both by the Mondragone experience in Spain and by the industrial districts in Italy. Actually, our first name was « the district of solidarity ». Then, we saw that entrepreneurs work better together when they belong to the same organization: we looked for a kind of organization that protects people and projects through a formal entity but also that guarantees autonomy for them." (Founder 1)

The both experiences of Mondragone Cooperative2 in Spain and industrial district are mobilized as a general frame. It helps to design a direction to their project and to give a proof that such project can be realized. However, this two general experiences do not refer to the same objective: the Mondragone experience is directly linked to an alternative and cooperative experience while industrial district refers more to a territorial productive organization for SME. The third reference of Coopératives d'Activités et d'Emplois (CAE) is more relevant in the sense that it brings the operational form of the project GRAP. A CAE is a Business and Employment Cooperative (BEC) that serves as the legal entity and surrogate for emerging projects. It thereby allows the entrepreneurs involved to benefit from employee status, pooled support services (accounting, support, etc.) and any potential operational synergies with other entrepreneurs. Collective projects may emerge from the proximity of entrepreneurs. Born in Lyon in 1995, the CAE have gradually evolved into "shared enterprises": establishment of a worker's representation, formation of important indivisible reserves and taking of collective risks (Veyer and Sangiogio, 2006). However the collective and cooperation action is only based, in the case of CAE, on the share of risks and the cooperative principles. It means that single entrepreneurial activities are not linked ex ante and the complementarities among activities are not fully exploited. The CAE reference brings to the founders of GRAP the boundary and the form of the cooperation model, but they translate it into a single-sector perspective and precisely in the food sector, in order to generate external effects of cooperation but also collective strategies.

The organizational model created by GRAP has the same advantage in solving the growth/singularity dilemma than the hypogroup [START_REF] Marchesnay | De l'hypofirme à l'hypogroupe : naissance, connaissance, reconnaissance[END_REF] which is an organizational configuration adapted to sustain small businesses development. It permits to articulate the professionalization of management and more formalized modes of control on tone hand, and the conservation of features specific to the small size of his company on the other hand (Debray, 20002). But GRAP's model is also innovative because small firms involved in it have the objective to support and secure the creation of new businesses in order to extend their niche and to promote the sustainable transformation of the food sector. To do this, they constitute indivisible reserves and allocate times, skills and employees to the support of others entrepreneurs.

Work participation and work balance as a crucial issue

Innovation practices are also related to work issues. Indeed the attention of founders has been focused on the conditions in which activities would be done. Work participation is defined by [START_REF] Neumann | Why people don't participate in organizational change[END_REF] as the structures and processes for organizing individual autonomy in the context of group responsibility and linked to system-wide influence. It can take a number of diverse forms as formal or informal, direct or not… It treats about the degree of involvement in decision and more broadly to the capacity of people to get a voice to the way the firm is organized and worked [START_REF] Glew | Participation in Organizations: A Preview of the Issues and Proposed Framework for Future Analysis[END_REF]. Here, the previous activist experience of founders and employees is crucial since the question of participation appears as a recurrent question. Precisely, a lot of them explain that the GRAP experience was a way "to go on with activism but to achieve more concrete objectives" (employee 1). They already get the tools and ideas of collective participation in decision-making and it deals with translating it into a business context. The objectives was so to create a non-vertical and mobile organization where people are involved in decision-making whatever its job position and where job position can change and evolve, according individual preference and work balance.

First, the organization creates spaces of collective decision where the whole diversity of stakeholders and employees are involved. The statute of employment and business cooperative brings the formal rules for the decision-making. The work governance is organized according the position of people in the organization: job employees, associated and integrated entrepreneurs and others shareholders are part of the board where the main decision and the strategy is designed once a year. Moreover, GRAP creates workgroups with specific purpose, as the integration of new incumbents. The composition of such workgroups is not only based on the skill criteria but also on the diversity of people represented. Second, the capacity of people to get a voice is regulated to guarantee the real participation of people and to prevent the conscious and unconscious effects of domination. The members of the organization have therefore created a kind of toolbox, coming from the field of popular education and the new fields of conceptualization of direct democracy (sociocracy) Most of these arrangements concern group facilitation, designation of specific responsibilities and decision-making. Meetings are animated using techniques of brainstorming, positioning on cursors, Forum Theater, etc. Some devices require participants to ask themselves crucial questions to break down inhibitions and taboos: the dishonest questioning system consists in asking the participants taboo questions directly related to an issue of the organization and pushing them to develop arguments. The principle of election without a candidate is used systematically to assign a function, a role or authority to a person, without it being a candidate. Another area of innovation practices related to the work issue concerns the balance between private and professional life. All the actors involved in GRAP are very careful to the negative effects of work activities and employment, such tiredness, exhaustion but also boredom. The imperative was therefore to foresee changes in the allocation of functions, in order to make commitments visible and to guarantee the rotation of experiments. The principle of job rotation is enforced and since the beginning of the organization, people move from a position to other. Moreover, sabbatical leaves are planned, especially for those most involved: it is a matter of relieving these people but also encouraging the rotation of responsibilities and pushing the structure to empower each small business owner. Further deliberation is encouraged with the possibility of new members being admitted to the board or to be spokesperson.

Finally GRAP has to make visible issues and problems and to tackle individual difficulties and interindividual conflicts that the work may generate. The case of the functions of mediator and guardian angel illustrates well this idea. Indeed, like any organization, inter-individual tensions may occur within the organization. It was then decided to take charge of these tensions with the creation of a function of mediator. The mediator has a vocation to intervene before and during the friction, in order to defuse it. The function of mediator in fact proved quite ineffective, because it intervenes too late in the relationship. After reflections and discussions, the function of mediation has been reformulated in principle of "the guardian angel". Indeed, the idea is that individuals, all involved in stressful situations (creation of activity, economic uncertainty, etc.), need to be listened to. It is then a question of dedicating to each, a person available to listen to the concerns. This solution has the advantage of intervening before of a potential crisis and of introducing a reciprocal relationship since all the members of the organization are concerned.

Discussion and conclusion

Our case study highlights the challenges and the ways of combining the best of activism and entrepreneurship to allow sustainability-driven small firms to contribute to an industry's transformation towards sustainable development, despite their limited resources. We specially focused our analysis on a single case study in the Lyon area and we considered how actors translate their objectives (to set up an alternative food networks able to compete with the conventional firms) into practices and strategy. Two levels were identified. First, an innovative organizational model is constructed to resolve the growth/singularity dilemma. Second, work participation and work balance are considered as crucial issues to permit grassroots innovations. First, the GRAP's innovative organizational model facilitates the development of small firms (pooling of services and resources) but also the growth of the niche (creation of new sustainability-driven small firms, political activities to overcome market barrier). It directly contributes to the structuration of an organizational field in the Lyon area and allows the emergence of a broader alternative food system that implies the participation of public authorities and civil society.

According to [START_REF] Hockerts | Greening Goliaths versus emerging Davids. Theorizing about the role of incumbents and new entrants in sustainable entrepreneurship[END_REF], a true industry transformation towards sustainable development can be achieved only when some sustainability-driven small firms become more business-like and seek to grow proactively. But growth creates a paradoxical tension [START_REF] Smith | Managing social-business tensions: a review and research agenda for social enterprise[END_REF][START_REF] Weinberg | Distinguishing among green businesses: growth, green and anomie[END_REF] for sustainability-driven small firms: although it allows a firm to improve its social and environmental impacts, sustainability values might be only partially maintained [START_REF] Parrish | Sustainability-driven entrepreneurship: principles of organization design[END_REF] when growing, and growth might in turn threaten these expected impacts [START_REF] Smith | Managing social-business tensions: a review and research agenda for social enterprise[END_REF]. Thus, the case of GRAP enables to consider the plurality of forms of AFN. The GRAP case is particularly relevant because it is an integrated organization that implies the main actors of the food sector (producers, entrepreneurs, employees, intermediary actors and even public authorities) but excluding consumers. The project is precisely to ensure firms and to create new opportunities niches for firms. In others words, while others alternative food networks are based on the active participation of consumers [START_REF] Brunori | On the New Social Relations around and beyond Food. Analysing Consumers' Role and Action in Gruppi di Acquisto Solidale (Solidarity Purchasing Groups)[END_REF], the case of GRAP highlights another path of development of AFN.

Second, the case of GRAP underlines the fabric of innovation. Innovation is viewed as the art of attracting a growing number of allies who are making you stronger and stronger (Alkrich et al., 1988). At the same time as their innovation was being conceptualized, this information had to be disseminated. GRAP's actors undertook actions designed to promote the new model and build strategic alliances with other actors in order to overcome market barrier and to gain political access and influence [START_REF] Pinkse | Sustainable Entrepreneurship and Corporate Political Activity: Overcoming Market Barriers in the Clean Energy Sector[END_REF]. Their objective therefore was "to secure recognition for their initiative as a genuine social innovation that can be supported in the same way as technological innovation". Their strategic intentions clearly extended beyond the context of food alone, and their ambition was to secure legitimacy within the social and solidarity economy and entrepreneurship.

To achieve this, GRAP sought the involvement of researchers and intervened in training sessions that addressed the theme of the SSE. Academic output and research articles are perceived as tools of dissemination and persuasion that are just as effective as, for example, press committees in developing the legitimacy of an enterprise. It is also felt that cooperation with researchers reinforces the conceptualization process, as researchers are supposed to provide more objective analysis of their practices and strategies. Third, the paper has shown that all the actors involved in GRAP are very careful to the negative effects of power on decision-making and of stressful work on life. The organization of GRAP is not an ideal-type; it is based on a strong commitment of people involved that implies numerous working hours and a flow of stressful situations. But, all the actors are conscious of this peculiar situation, that defines the mix between activism and entrepreneurship, and justify the continual flow of innovation practices related to work issues. GRAP can be thus described as a reflexive and innovative organization: the process in which these actors are involved accurately corresponds to the "multi-actor whirlwind", a term used by [START_REF] Akrich | A quoi tient le succès des innovations? Premier épisode: L'art de l'intéressement[END_REF] in reference to innovation. This is clearly a collective process of conceptualization built up over time and made up of learning curves, ongoing back-andforth exchanges and negotiations whose importance cannot be determined in advance. It is a process that has not been completed, and the new rules, which this organization intends to produce, have not yet been formalized and are not taken for granted. Finally, the paper brings a contribution to better understand how entrepreneurship and activism can be combined. Our analysis has been focused on operations of translation of frame into strategic action and organization forms. Based on a single case, the results cannot be generalized. However, the analysis enables to underline some assumptions that could be extended to others alternative food sectors in a comparative perspective. Also, it could be interesting to investigate others fields in order to identify new tactics and strategy of entrepreneurial actions that could generate a social change.

Figure 1 -

 1 Figure 1 -From frame to model in grassroots innovations processes

Table 1 -Transformational social entrepreneurs, desire for social change and fulfill social needs (Van den Broeck et al. 2012)

 1

Table 2 -Presentation of the small firms which are part of GRAP in 2017

 2

	zestes	Short supply chain (Mediterranean products)
	La fabrique des producteurs Restaurant
	La Fourmillière	Bakery
	Elodie D. chocolatière	Chocolate maker
	La Super Halle	Grocery + farm shop + restaurant
	Des Vins et des Hommes	Short supply chain (Wine)
	Coolivri	Delivery service
	Coeur d'Artichaut	Grocery
	Prairial	Grocery
	Croc Bauge	Grocery
	L'épicerie du coin	Grocery
	Le Grain dans le Grenier	Grocery
	Les bon plans de Tonton	Short supply chain
	Mélim'élo	Food craft
	Le pain de Garance	Bakery
	Chocolalala	Chocolate maker
	Philosophie végétale	Caterer
	Gout'chou	Caterer
	Mamie Marie	Grocery
	Vrac en Vert	Grocery

A SCIC (Société Coopérative d'Intérêt Collectif) is a Co-operative Company of Collective Interest. SCIC are collective interest cooperatives bringing together employees, users, volunteers, and local authorities to implement local development projects.

The Mondragon Corporation is a corporation and federation of worker cooperatives based in the Basque region of Spain. It was founded in the town of Mondragon in 1956 by graduates of a local technical college