www.mdpi.org/entropy/ Markov property and operads

We define heat kernel measure on punctured spheres. The random field which is got by this procedure is not Gaussian. We define a stochastic line bundle on the loop space, such that the punctured sphere corresponds to a generalized parallel transport on this line bundle. Markov property along the sewing loops corresponds to an operadic structure of the stochastic W.Z.N.W. model.


Introduction
In conformal field theory, people look at a Riemann surface Σ with boundary ∂Σ, and the set of maps from Σ into a Riemannian manifold M .The case which will be of interest for us in this present work is when the genus of the Riemann surface is 0. This corresponds to a punctured sphere.We suppose that there are one input loop and n output loop.The map from Σ into M are chosen at random, with the formal probability law: where dD is the formal Lebesgue measure, I(ψ) the energy of the map and Z a normalizing constant called the partition function destinated to get a probability law.Segal [46] has given a series of axioms which should be satisfied by this theory.In particular, conformal field theory predicts the existence of an Hilbert space Ξ associated to each loop space such that the surface Σ realizes a map from Ξ ⊗n into Ξ, if we consider the case of the (n + 1)-punctured sphere.Hom(Ξ ⊗n , Ξ) is the archetype of an operad.Namely, if we consider n elements of Hom(Ξ ⊗n i , Ξ) and an element of Hom(Ξ ⊗n , Ξ), we deduce by composition an element of Hom(Ξ ⊗ n i , Ξ).This composition operation will correspond to the operation of glueing n 1 + n i punctured spheres in a sphere with (1 + n i ) punctured points.For the literature about this statement, we refer to [22], [24], [21], [49].For material about operads, we refer to the proceedings of Loday, Stasheff and Voronov ( [39]).The problem of the measure dµ is that it is purely hypothetical: in the case when the manifold M is replaced by R, it is a Gaussian measure, which gives random distributions (See [42], [48], [19]).But it is difficult to say what are distributions that live on a manifolds.
Our statement is the following: -)Define a measure over the space of spheres with 1 + n punctured points.
-)Define an Hilbert space Ξ associated to each loop space given the punctured points on the sphere.
-)Define associated to the sphere with 1 + n punctured points an element of Hom(Ξ ⊗n , Ξ), such that the application is compatible with the action of sewing spheres along their boundary.
For that, we use the theory of infinite dimensional process, especially the theory of Brownian motion over a loop group of Airault-Malliavin [1] and Brzezniak-Elworthy [7].Let us recall that the theory of infinite dimensional processes over infinite dimensional manifolds has a lot of aspects.The first who have studied Brownian motion over infinite dimensional manifolds is Kuo [27].The Russian school has its own version [4], [11], [5].The theory of Dirichlet forms allows to study Ornstein-Uhlenbeck processes over some loop spaces [12], [2].Our study is related to the theory of Airault-Malliavin, but in order to produce random cylinders, Airault-Malliavin consider a 1+1 dimensional theory: the first 1 is related to the dimension of the propagation time of the dynamics and the second 1 is involved with the internal time of the theory (The loop space).Our theory is 1+2 dimensional, because the internal time of the theory is 2 dimensional.
1+2 dimensional theories were already studied by Léandre in [31] in order to study the Wess-Zumino-Novikov-Witten model on the torus, in [32] in order to study Brownian cylinders attached to branes and in [35] in order to study one of the concretisation of Segal's axiom by using C k random fields.In [30] and in [31], stochastic line bundles are used.In [29], we give a general construction of 1 + n dimensional theory, and we perform a theory of large deviation, in order to compute the action of the theory.In [33], we study stochastic cohomology of the space of random spheres, related to operads (For the aspect of operads related to n-fold loop space, we refer to the proceeding of Loday-Stasheff-Voronov [39]).The problem in [35] is that there is no Markov property of the random field, such that we cannot realize an operad by sewing punctured spheres.
Our goal is to construct a 1 + 2 dimensional Wess-Zumino-Novikov-Witten model on the punctured sphere, which is Markovian on the boundary on the sphere.This Markov property allow us to realize an operad, by sewing random spheres along their boundary.For the material of sewing surfaces, by using the formal measure of physicist, we refer to the surveys of Gawedzki ([19], [16], [17]).
We thank the warm hospitality of Maphysto, department of Mathematics, of the University of Aarhus, where this work was done.

Punctured random spheres and markov property
In order to construct a sphere with 1 + n punctured points, we define first a sphere with 1 + 2 punctured points (a pant), and we sew the pants along their boundary.
We consider a compact connected Lie group G of dimension d, equipped with its bi-invariant metric.We can imbedd it isometrically in a special orthogonal group.
We consider the Hilbert space H of maps from S 1 × [0, 1] into the real line R endowed with the following Hilbert structure: |∂/∂sh(S)| 2 dS + |∂/∂th(S)| 2 dS+ where S = (s, t) belongs to S 1 × [0, 1] We can consider the free loop space of maps from S 1 into R with the Hilbert structure: We can find an element e(s) of this Hilbert space such that where e(s) = λ exp[−s] + µ exp[s] for 0 ≤ s ≤ 1 such that e(0) = e(1) but e (0) = e (1).We add in (2) the Neumann boundary condition: ∂/∂th(s, 0) = ∂/∂th(s, 1) = 0 (5) Let us recall that the Green kernel over [0, 1] associated to the Hilbert space of functions from [0, 1] into R with Neumann boundary condition, associated to the Hilbert structure:  (7) where µ − t , λ − t , µ + t , λ + t depend smoothly on t.The Green kernel associated to the Hilbert structure (2) are the product of the one dimensional Green kernel e s (s )e t (t ) = E s,t (s , t ).
We would like to consider the same Hilbert space with the constraint h(s 1 , 1) = h(s 2 , 1) = 0 for two given times s 1 < s 2 (We can choose another condition, but we choose the simplest condition for the sake of simplicity).When we add this condition, we get another Hilbert space H 1 which is a finite codimensional subspace of the initial Hilbert space H.
We can find an orthonormal basis of the orthogonal complement of H 1 constituted from two maps h 1 (s, t) and h 2 (s, t) which are smooth in (s, t).Let us consider the Brownian motion with values in H.It is a 3 dimensional Gaussian process B u (s, t) where u denotes the propagation time and (s, t) the internal time.The covariance between B .(s, t) and B .(s , t ) is E s,t (s , t ).The Brownian motion with values in H 1 can be seen as where (α 1 , β 1 , γ 1 ) are deterministic constants and B 1 u and B 2 u are two R-valued independent Brownian motion.In the sequel, we will choose this procedure in order to construct the Brownian motion B 1,u (S) with values in H 1 .
Let us consider the time t = 1 where the loop splits in two loops given by s 1 and s 2 .We get after this splitting two circles.We consider the Hilbert space H 2 of maps from S 1 × [0, 1] into R submitted to the boundary conditions h(s, 0) = h(s, 1) = 0 with the Hilbert structure: In fact we should introduce some normalizing constant due to the fact that we do not consider the normalized Lebesgue measure over each circles given by splitting the circle into 2 circles.The Green kernel associated to this problem is the product of the Green kernel associated to (3) and the Green kernel associated to the Hilbert space of functions from [0, 1] into R equal to 0 in t = 0 and t = 1 associated to the Hilbert structure We consider an analogous Hilbert space H 3 with the Hilbert structure (9) and the boundary condition h(s, 0) = 0 (without the boundary condition h(s, 1) = 0).The Green kernel in t are of the type and the global Green kernel satisfy to Over each Hilbert space, we consider the Brownian motion B i,. (., .).Let Σ be a pant (The elementary surface).Its boundary is constituted of circles, and we get tubes near the output boundary S 1 × [0, 1/2] and tube near the input boundary S 1 × [1/2, 1].Near the boundary, we consider the Brownian motion with values in H 3 , by taking care that the starting condition h(s, 0) = 0 is inside Σ for an output boundary and this condition is outside Σ for an input boundary.We choose 3 independent Brownian motion B 3 .(.) over H 3 .We multiply these Brownian motions by a deterministic function g(t) equal to 0 only at 0 and 1 such that g(1/2)B 3 . (., 1/2) corresponds to a normalized circle of length 1. Outside these boundary tubes, we consider over the cylinder with constraint h(s 1 , 1) = h(s 2 , 1) = 0, a Brownian motion with values in H 1 , chosen independently of the others Brownian motions, but which intersect the input boundary tube on the cylinder S 1 × [1 − , 1]: we multiply by a smooth function g(t) > 0 which is 0 only in 1 − .When the loop s → h(s, t) splits in two loops, we get two loops: we add the Brownian motion with values in H 2 over each (Two independent one modulo some normalizing constants), and we get two cylinders which intersect the exit tube S 1 × [0, 1/2] over the tube S 1 × [0, ].We mutiply these Brownian motion by a smooth function g(t) > 0, and which is 0 on .
After performing all these glueing operations, we get an infinite dimensional Gaussian process parametrized by [0, 1] × Σ u → B tot,u (.), Which is an infinite dimensional Brownian motion with values in a suitable Hilbert space of functions on Σ which satisfies to the following properties: -)For all S ∈ Σ, u → B tot,u (S) is a Gaussian martingale.
-)(u, S) → B tot,u (S) is almost surely Hölder, and if <, > denotes the right bracket of the martingale theory, we get for u ≤ 1 over each elementary parts of the pant Σ where the construction is done.Moreover, over the pant Σ, (u, S) → B tot,u (S) is almost surely continuous.c)Over each boundary of the pant, u → B tot,u (S) are independent.
In order to curve these Gaussian processes, we use the theory of Brownian motion over a loop group of Airault-Malliavin [1] and Brzezniak-Elworthy [7].
Let e i be a basis of the Lie algebra of G. Lert B i tot,.(.) be d independent copies of B tot,.. (S).We write d u B tot,u (S) = e i d u B i tot,u (S).We consider the equation in Stratonovitch sense: starting from e, the unit element in the group G..
We get (See [29], [31]) for proof in a closed context.Theorem 2.1: Over each elementary part of the pant where the leading Brownian motion is constructed, the random field S → g 1 (S) is almost surely 1/2 − Hölder.Moreover, the random field on Σ: S → g 1 (S) is almost surely continous, and its restriction on each circle on the boundary are independent.
In order to get a general (1 + n) punctured sphere, we sew successively pants, which are independent, except on the boundary, with a glueing condition.This glueing condition is, when we sew an exit loop of a pant to an input loop of another pant, we choose the same Brownian motion on H 3 .We can do that, because the restriction to S 1 × {1/2} are the same.We get by that a tree or a punctured sphere Σ(1, n).We get: Theorem 2.2: Over each punctured sphere Σ(1, n), the random field S → g 1 (S) got after this sewing procedure is almost surely continuous.
By using this procedure, if we consider a (1 + n) punctured spheres Σ(1, n) and n punctured spheres Σ(1, n i ), we can glue the input loop to each Σ(1, n i ) to the output loop of Σ(1, n) and we get a sphere Σ(1, Σn i ).We suppose that all the data in this sewing procedure are independents, except for the Brownian motion in H 3 when we sew an output boundary in Σ(1, n) to an input boundary in Σ(1, n i ).Let us suppose that the random fields are sewed on the loops (∂Σ) i .
We get some thing like a Markov property along the sewing boundary: Theorem 2.3: The random field S → g 1 (S) over Σ(1, n i ) are conditionally independent over each Σ(1, n i ) and over Σ(1, n) conditionally to each (∂Σ) i .
Proof: We remark that for H 3 < B dB tot,u 1 (S)..dB tot,un (S) (18) we get that, after imbedding the group G in a matrix algebra (dB tot,u 1 (S)..dB tot,un (S) − dB tot,u 1 (S )....dB tot,un (S )) (19) and we write dB tot,u (S ) = dB tot,u (S ) − dB tot,u (S) + dB tot,u (s) and we distribute in (18).Let us choose two points on the same component of the boundary S 1 , S 2 in the boundary, and two points S and S" not on the side of the boundary.We get that g 1 (S ) − g 1 (S 1 ) and g 1 (S") − g 1 (S 2 ) are conditionnally independent when we suppose given the random field g 1 (S) on the boundary.Therefore the result.♦

Line integrals
When we consider the random punctured sphere Σ(1, n), we get vertical loops given by s → g 1 (s, t).Since Σ(1, n) is built from elementary pants Σ(1, 2), it is enough to look each vertical loop s → g 1 (s, t) over each elementary pants.They are of 4 types: -)The loop near the input boundary (Hilbert space -)The loops in the body of the pants (Hilbert space H 1 ).
-)The two loops which are created from a big loop (Hilbert space -)The loops near the exit boundary (Hilbert space H 2 ⊕ H 3 ).
Let us consider a one form ω over G.We would like to define for each type of this loop the stochastic Stratonovitch integral: We extend conveniently the one form ω in a smooth form bounded as well as all its derivatives over the matrix algebra where the matrix group is imbeddded.The technics are very similar to the technics of [31], part III.Let dB u be a Brownian motion with values in the Lie algebra of G.We consider the solution of the stochastic differential equation which gives the Brownian motion from e in the Lie group G: The equation of the differential of the differential of the stochastic flow associated to (21) is given (See [23], [26], [6]) by and the inverse of the differential of the the flow is given by an analoguous equation.It can be identified to g u .Let us consider a finite dimensional family B u (α) of Brownian motion in the Lie algebra of G depending smoothly of a finite dimensional parameter α where α lives in a finite dimensional family of Brownian motion.We consider the stochastic differential equation depending on a parameter: The solution of the equation ( 15) has a smooth version in the finite dimensional parameter α. ∂/∂αg u (α) is for instance the solution of the linear stochastic differential equation with second member: This equation can be solved by the method of variation of the constant.We get: We will write s → B tot,. (s, t) = B .(s), and in order to define stochastic line integrals, we will follow the method of [30] and [31], but in this case, it is much more simpler, because there is no conditioning.By using the properties of the Hilbert structure given H
. We don't write the analoguous expression for g 2 .(s).There is a double integral in dw .(2) where the simple derivative of β(s, ∆) in √ ∆s appear and a simple integral where the second derivative in √ ∆s of α(s, ∆s) and β(s, ∆s) appear.(.) is the time of the differential equation (15).Moreover, in law: Let f and h be 2 smooth functions over the matrix space.We suppose they are bounded as well as their derivatives of all orders.We have the estimate which follows from the properties listed after ( 27), (28) ( 29): where we used either Itô integral or Stratonovitch integral.We convert it in Skorokhod integral (whose expectation is 0) and we find a counterterm in O(∆s) (We can suppose that ∆s = ∆t as we will do in the sequel).For that we used the following result: let f a smooth functional with bounded derivatives of all orders in a finite number of g u (s) or in g u (t).Let F the associated Wiener cylindrical functional.
It is a smooth functional in the sense of Malliavin Calculus in w .(2), w .(4) and its derivatives D k F (t 1 , .., t k ) have an estimate in O( √ ∆s) k We consider a smooth 1-form ω v in the spaces of matrices with bounded derivatives of all orders which depends smoothly from a finite dimensional parameter v.We suppose that the derivatives in the parameter v are bounded.
We consider 2 N , N being a big integer, and the dyadic subdivision of [0,1] associated to 2 N .We call it s i with s s → g N 1 (s) is piecewise differentiable.We consider the random variable: Let us give the following decomposition of A N v : The Itô term is A N v (δ) and the Stratonovitch counterterm is A N v (<, >).The Itô term can be divided into two pieces: the first one is when in (30) we take the term in g 1 .(s) and the second one is when we take in (31) the term in g 2 .(s).We get the decomposition, of the Itô term in ).But we can use (32), and show that when N → ∞, where C(s, t) is continuous.Moreover, the second part in the Itô term checks clearly: Since the counterterm which is due to the Stratonovitch correction is a "a priori" less diverging, we can see in an analoguous way that: These remarks justify but not prove the following proposition: Proof: Let us forget for the moment the parameter v. Let us write: where B N i is the Bracket term and C N i is the Itô term: We write where and First step: convergence of E N i .In g 2 1 (s i ) whose writing is derived from ( 24) by taking another derivative, there is a linear integral which comes from the second derivative of α(s i + ∆s i ), from a second derivative in β(s, ∆s) in √ ∆s and a double integral which comes from taking only one derivative in β(s, ∆s).The term in the linear integral can be treated in the following way: we get In order to compute g1 (s i ) is the term in the simple integral where we take the second derivatives in √ ∆s of α(s, ∆s) and β(s, ∆s).The terms which are integrated depend continuously from s. Therefore the contribution where we take two derivatives of α(s, ∆s) vanish.It remains to consider the contribution where we take two derivatives of β(s, ∆s).We can replace the terms considered by where we have replaced the term in two derivatives by ∆s j B .(s j + ∆s j ) − B .(s j ).We write behaves as O(1/N )∆s j because ω(g 1 (s)) depends continuously of s and after using the desintegration argument used after (32).
The problem arises when we take the double integral.In order to study the behaviour of its sum, we can replace w .(2) in ( 27) by B .(s i + ∆s i ) − B .(s i ) and take the double stochastic integral which is associated by taking the derivative of the flow φ u (s i ) associted to the equation dg u (s i ) = g u (s i )dB u (s i ).Namely, we consider a double integral of the type For the convergence of E N i , we can assimilate (s i+1 − s i )g 2 u (s i ) with the double integral α u (s i ) after performing these replacements.Let N > N and s j be the dyadic subdivision which is associated.We sum over [s j , s j+1 ] ⊆ [s i , s i+1 ].We get: The sum of the first term tends to 0 in L 2 .The difficult term is to estimate the term in N i .In the double integral which compose α t (s i ), we write We distribute the integrands.Over each dB .(s i + ∆s i ) − dB .(s i ), there is in the double integral a term which B .(s i ) measurable, which is adapted and which depends on a continuous way of s i .Since it depends on a continuous way from s i , we can replace it when we distibute by the corresponding term in s j in α t (s i ).After distributing in α t (s i ) − α t (s j ), the diagonal term are substracting, and it remains to study the process We decompose the semi martingale δ N t into a finite variational part which converges by using (26) to 0 and a martingale part M N t .Namely, we can convert the double Stratonovitch integral which appears in (54) in an Itô integral.The boring term arises when we replace the double Stratonovitch integral by an Itô integral in (54).We would like to show that this martingale tends to 0. For that, we compute its quadratic variation.We get a sum over all quadruple [s j 1 , s j 1 +1 ], [s j 2 , s j 2 +1 ], [s j 3 , s j 3 +1 ] and [s j 4 , s j 4 +1 ].
-First case: let us suppose that all the elements of the quadruple are different.The contribution of each quadruple is in 2 −4N by the properties listed after ( 27), (28), (29) which express that the covariance of B .(s j + ∆s j ) − B .(s j ) and of B .(s j +1 ) − B .(s j ) in term of ∆s j ∆s j and the covariance of (B .(s j + ∆s j ) − B .(s j ) and of B .(t) in ∆s j if t does not belong to [s j , s j+1 ].Namely, if the intervals [s j 1 , s j 1 +1 ], [s j 2 , s j 2 +1 ] do not intersect and if s j 3 and s j 4 do not belong to these intervals, we have only to show by using the Itô formula that because the right Bracket between ∆ s j 3 B(s j 3 and ∆ s j 4 B(s j 4 ) is in O(∆s j 3 ∆s j 4 ) We take the conditional expectation of r v (s j 3 ) and r v (s j 4 ) along the Gaussian space spanned by B .(s j 1 ), B .(s j 2 ), ∆ s j 1 B(s j 1 ) and ∆ s j 2 B .(s j 2 ).We can suppose that r v (s j 3 ) and r v (s j 3 ) are measurable over this Gaussian space.But r v is solution of the stochastic differential equation giving the flow of the Brownian motion over the Lie group, and is therefore a stochastic integral.We use the following rules for calculating different conditional expectation for the solution of this flow.We consider the solution of the stochastic differential equation starting from the identity: where B t and Bt are two independent Brownian motions.We can write A t = W t V t where dV t = V t dB t and dW t = W t V t d Bt V −1 t .after using this remark in order to calculate the conditional expectation, we desintegrate along ∆ s j 1 B .(s j 1 ) and ∆ s j 2 B .(s j 2 ) as in (32), and we conclude by using the consideration following ( 27), ( 28), (29).
They are at most 2 2N 2 4(N −N ) such possibilities.The total contribution is 2 −2N which tends to 0 when N → ∞.
-)Second case: there are 3 different intervals [s j , s j+1 ].This can come from a concatenation of two times d v for u < v in the stochastic integral (54) after converting it in a double Itô integral or a concatenation of the same term d u in the stochastic integral (54).The contribution of each term is 2 −3N by doing as in the first case..They are at most 2 3N 2 −2N such possibilities.The total contribution behaves in 2 −2N which tends to 0 when N → ∞.
-)Third case: there are 2 different intervals [s j , s j+1 ].The contribution of each element which appears is in 2 −2N by doing as in the first case.There are at most 2 N 2 2(N −N ) such terms.The total contribution is in 2 −N which converges to 0 when N → ∞.
This shows us that and we would like to show that α N i → 0 in L 2 .They are two terms to study: -)The contribution of E[α N i α N i ] for i = i .By (32), . By using the consideration of the first step, we can write modulo a term which vanish that To study its convergence, we write: B .(s j + ∆s j ) = α(s i , s j , ∆s j )w .(1) + β(s i , s j ,∆s j )w .(4) + γ(s i , s j , ∆s j )w .(5) (60) We have γ(s, t, ∆t) = C(s, t) (2) >= O( ∆s j ) and < w .(5), w .(1) >= O( ∆s j ).In a similar way, we have < w .(3), w .(1) >= O( ∆s j ), < w .(3), w .(4) >= O(∆s j ) (We used the fact that ∆s j = ∆s j ).With this decomposition, we write the analoguous of ( 30) and (3 1) for g .(s j + ∆s j ) by doing the conditional expectation along the Gaussian processes w .(5), w .(4), w .(2), w .(3) and for g .(s j + ∆s j ).We and The more singular singular tem in There is in the previous contribution a quadratic expression in g 1 1 (s i ).These expressions can be treated exactly as in the first step of the convergence of E N i , by writing < g 1 1 (s i ), g 1 1 (s i ) > as a double integral and relpacing (s i+1 − s i ) < g 1 1 (s i ), g 1 1 (s i ) > by a double stochastic integral where we have removed √ ∆s i w .(1) by ∆ s i B .(s i ).The sum of the others terms tends clearly to 0. In order to show that S 1 < ω v (g 1 (s)), d s g 1 (s) > has a smooth version, we show that the system of derivatives of A N v in v converges in L 2 .We conclude by using the embedding Sobolev theorem as in [23].

♦
We consider a more intrinsic approximation of the line integral.We use if g 1 (s i , t), g 1 (s i+1 , t) are close, conveniently extended over the set of all matrices.We put: We consider ÃN v as in (35) with this new approximation.If we look the asymptotic expansion of F N , we see that the more singular term in d s gN 1 (s, t) and d s g N 1 (s, t) coincides.This justify the following theorem: Theorem 3.2: ÃN v tends in L 2 for the C k topology over each compact of the parameter set to the Stratonovitch integral S 1 < ω v (g(s, t)), d s g(s, t) > which has a smooth version in v.
Remark: We don't know if the Stratonovitch integrals of Theorem III.2 and of Proposition III.1 coincide.In the sequel, we will use the version of Theorem III.1, because it is a geometrical version.
Remark: Instead of integrating over a circle, we can integrate over a segment.

Integral of a two form
We decompose the pant Σ(1, 2) in elementary cylinders (s, t) be the Brownian motion parametrized by these elementary cylinders.Each correlators check all the properties listed in the part IV of [31] such that each correlator is smooth outside the diagonals and its derivative has half limits on the diagonals, such that we can apply the technics of the part IV of [31].where C(t, t ) is continuous, the same being true for the symmetric case.
We imbedd G into a matrix algebra isometrically.Let g(s, t) be the random field parametrized by the torus with values in G. Let 2 N be an integer, and s i be the associated dyadic subdivision of S 1 and t j be the associated dyadic subdivision of a copy of [0, 1].We consider the polygonal approximation of g(s, t), if (s, t) Let us consider a two form ω over G, conveniently extended in a two form ω over the matrix algebra bounded with bounded derivatives of all orders.We suppose that the two form depends on a finite dimensional parameter v.We consider Let us denote by ∆ t j g(s i , t j ) the quantity g(s i , t j+1 )−g(s i , t j ), by ∆ s i g(s i , t j ) the quantity g(s i+1 , t j )− g(s i , t j ) where we have imbedded the group G in a linear space.If i = i , j = j , we will see later that E[∆ s i g(s i , t j )∆ t j g(s i , t j )∆ s i g(s i , t j )∆ t j g(s i , t j )] = O(∆s i ∆t j ∆s i ∆t j ) where we take a quadratic expression homogeneous in each term in each increment.The most diverging term in the quantity When the length of the subdivision tends to zero, the L 2 -norm of this expression tends to D×D C(s, t, s , t )dsds dtdt + This justifies without to prove the following proposition: Proposition 4.1: When N → ∞, the traditional integral A N v tends for the C k topology over each compact of the parameter space in L 2 to the stochastic integral in Stratonovich sense: < ω(g(s, t)), d s g(s, t), d t g(s, t) > (75 where the stochastic integral D g * ω v has a smooth version in v.
Proof: We suppose first that there is no auxiliary parameter.We can write: STEP I: convergence of A N 1 .We repeat the considerations of the part III for s → B .(s, t j ) and t → B .(s i , t).If we fix t j , we get by (30) an asymptotic expansion in order 3.We get expressions in the asymptotic expansion in g 1;.
. (s i ; t j ) and g 3;.(s i , t j ).If we fix s i , we go in (30) to an asymptotic expansion at order 3.We get derivatives in law g .;1 . (s i , t j ), g .;2(s i , t j ) and g .;3 .(s i , t j ).We get: < ω(g(s i , t j )), g(s i+1 , t j ) − g(s i , t j ),g(s i , t j+1 ) − g(s i , t j ) > B N 1 is the Itô term, which is apparently the most diverging when N → ∞.B N 2 is the Stratonovitch counterterm.
Step I.1: convergence of the Itô term B N 1 .We write as in (30) and we write as in (30) g(s i , t j+1 ) − g(s i , t j ) = t j+1 − t j g .;1(s i , t j ) This will lead to stochastic integrals in √ s i+1 − s i g 1;.(s i , t j ) and in √ t j+1 − t j g .;1(s i , t j ) which apparently do not converge and to integrals in (s i+1 − s i )g 2;.(s i , t j ) as in (t j+1 − t j )g .;2(s i , t j ) which will lead to classical integrals.We deduce the following decomposition of the Itô term B N 1 : -)C N 1 is the double stochastic integral in the time direction s and in the time direction t: ∆s i ∆t j < ω(g(s i , t j )), g 1;.(s i , t j ), g .;1(s i , t j ) > (81) -)C N 2 is a stochastic integral in the direction s and a classical integral in the direction t: ∆s i ∆t j < ω(g(s i , t j )), g 1.. (s i ; t j ), g .;2(s i ; t j ) > (82) -)C N 3 is a vanishing term: < ω(g(s i , t j )), g 1;.(s i , t j ), g .;3(s i , t j ) > + i,j -)C N 4 is a classical integral in the time direction s and a stochastic integral in the time direction t: C N 4 = i,j ∆s i ∆t j < ω(g(s i , t j )), g 2;.(s i , t j ), g .;1(s i , t j ) > (84) -)C N 5 is a classical integral in the time direction s and in the time direction t.
C N 5 = i,j ∆s i ∆t j < ω(g(s i , t j )), g 2;.(s i , t j ), g .;2(s i , t j ) > (85) C N 1 is the more "a priori" divergent term when N tends to ∞ and C N 5 will lead to a double classical integral on the torus.
Step I.1.1:For integers N , N such that N > N , we consider C N 1 = i,j C N i,j,1 .We consider a bigger integer N than N and we consider Let us consider first the case where 0 ≤ s+∆s ≤ s ≤ s +∆s ≤ 1 and 0 ≤ t+∆t ≤ t ≤ t +∆t ≤ 1.
We get if f and g are smooth functions with bounded derivatives of all orders: the same asymptotic results being true when we reverse the role of s, t.
The main result is the following: if u does not belong to ]s, s + ∆s[, the same equality being true if we reverse the role of s and t.
We use the fact that the Green kernel associated to the two dimensional problem is the product of the Green kernels associated to the one dimensional problem by the remark following (6).Moreover In order to simplify the exposure, we writte ∆t = ∆t = ∆s = ∆s .We conditionate B .(s, t) and B .(s , t ) by w .(3),, w .(4), w .(5), w .(6).We use the formula (56) in order to compute this conditionating for g(s, t) and g(s , t ), and after the Clark-Ocone formula (See [43]) in order to compute the conditional of h(g(s, t)) as an Itô integral in w .(3), w .(4), w .(5) and w .(5) with term bounded by √ ∆s by (92).We get to take the expectation of the product of four Itô integral or 5 or 6.We can estimate its expectation by using the Itô formula and ( 93), (94) by applying iteratively the Itô formula and the Clark-Ocone formula.We reduce iteratively the length of the iterated integral we have to compute.The same result holds by the same arguments for: if we suppose that ∆s = ∆s = ∆t = ∆t .We deduce from the previous considerations that: C(s, t, s , t )dsdtds dt = 0 (97) Let us now study the behaviour of when N → ∞.
By the previous considerations, the contributions of the T k,l strictly interior to T i,j and of the T k ,l strictly interior to T i,j vanish.Therefore, it is enough to study the contribution of C 1,N i,j,1 = ∆s i ∆t j , < ω(g(s i , t j ),g 1;.(s i , t j ), g .;1(s i ; t j ) > − i ∆s i ∆t j < ω(g(s i , t j )), g 1;.(s i , t j ), g .;1(s i , t j ) > (99) ] tends to 0 when N → ∞.We will see later (See Step I.1.2, Step I.1.3and Step I.1.4)that we can replace √ ∆s i g 1;.(s i , t j ) by ∆ s i g(s i , t j ) and ∆t j g(s i , t j ) by ∆ t j g(s i , t j ). it is enough therefore to consider the behaviour of and to show that By using the technics of the next steps, we can replace ∆ s i g(s i , t j ) by √ ∆s i g 1;.(s i , t j ) and ∆ t j g(s i , t j ) by ∆t j g .;1(s i , t j ) and ∆ t j g(s i , t j ) by ∆t j g .;1(s i , t j ) and ∆ t j g(s i , t j ) by ∆t j g .;1(s i , t j ).We get two quantities C 5,N i,j,1 and C 6,N i,j,1 We compute i,j =j E[(C 5,N i,j,1 C 5,N i,j ,1 )].There are two contributions.The first one is when we consider twice the same s i .There are 4 types of increments which appear (s i , t j ), (s i , t j ), (s i , t j and (s i , t j j ).We take the conditional expectation along ∆ s i B .(s i ), t j ), ∆ t j B .(s i , t j ), ∆ s i B(s i , t j ) and ∆ t j B .(s i , t j ) or more precisely along the Brownian motion which arise from the diagonalisation (89) of the Brownian motions B .(s i , t j ), B .(s i , t j ), B .(s i , t j ) and B .(s i , t j ).The Stratonovitch integrals g 1;.(s, t) and g .;1(s, t) are in fact Itô integrals.Moreover we can compute the conditional law of g(s i , t j ), g(s i , t j ), g(s i , t j ) g(s i , t j ) by using ( 56) and the Clark -Ocone formula to express the quantities which appear in this way as stochastic integral which are martingales and whose bracket with the others tems can be estimated by (89).There is a product of Martingale Itô integrals, whose expectation can be estimated by using succesivly the Itô formula and the Clark Ocone formula.We conclude by using (4.27), (4.28) and (4.30).We get that the contribution when there is one coincidence leads to a term in O(1/N )∆s i ∆t j ∆t j .When there is no coincidence, we condition by ∆ s i B .(s i , t j ), ∆ t j B .(s i , t j ), ∆ s i" B .(s i" , t j ) and ∆ t j B .(s i , t j ), or more precisely by the Brownian motions arising from the diagonalisation (89).We proceed as before, and we get a contribution in o(1/N )∆ s i ∆ s i" ∆ t j ∆ t j Therefore E[ i,j =j C 5,N i,j,1 C 5,N i,j ,1 ] → 0. By the same type of trick and performing the conditional expectation along the increment ∆ s B .(s, t) and ∆ t B .(s, t) or more precisley by conditioning along the Brownian motions which appears in the diagonalisation (89) in C 6,N i,j,1 C 6+,N i,j ,1 and after using the Clark-Ocone formula, we see that the quantity i,j =j E[C 6,N i,j,1 C 6,N i,j ,1 ] → 0. The same holds for . By the considerations which will follow in the next step, it is enough to study the behaviour of < ω(g(s i , t j )), ∆ s i g(s i , t j ), ∆ t j g(s i , t j ) >} where we do the summation over [s i , s i +1 ] ⊆ [s i , s i+1 ] and [t j , t j +1 ] ⊆ [t j , t j+1 ].In GN i,j,1 , we write: and we deduce a decomposition of GN , we can replace ∆ s i g( i , t j ), ∆ t j g(s i , t j ) by √ ∆s i g 1;.(s i , t j ).We can replace ∆t j g .;1g(s i , t j ) and ∆ s i g(s i , t j ) by √ ∆s i g 1;.(s i , t j ) and ∆ t j g(s i , t j ) by ∆t j g .;1(s i , t j ) by ∆t j g .;1(s i , t j ).We get G 3,N i,j,1 and G 4,N i,j,1 .
We have 6 terms to estimate: . We can do the multiplication term by term in each product which appear.In each term, we distribute another time.There are 4 terms where two expressions in g 1;. and g .;1appear.We condition by the set of increments in the leading Brownian motion which appears in these expressions, or more precisely of the terms which appear after the diagonalisation (89) in ∆ s B(s, t) and ∆ t B(s, t).We use (57) and the Clark-Ocone formula (See [43]).We use (89), and (93).When we develop, there is the possibility that we get exactly 4 times s i , s i" , t j and t j" , which lead to a contribution in O(1/N ) i =i",j =j" ∆s i ∆s i" ∆t j ∆t j" .There is a contribution when there are 3 different s i , t j , t j" or s i , s i" , t j which lead to a contribution in i,j =j" O(1/N )∆s i ∆t j ∆t j" or i =i",j O(1/N )∆s i ∆s i" ∆t j and a contribution where we get only two times s i and t j which leads to a contribution in i,j O(1/N )∆s i ∆t j .Therefore, i,j G 3,N i,j,1 tends to 0 in L 2 .
By the same argument, i,j G 1,N i,j,1 and 2,N i,j tend to 0 in L 2 .By using this type of argument, we can get the requested limits.
Step I.1.2Study of the convergence of the terms C N 2 and C N 4 where we mix stochastic integral and classical integral.
This term is simpler to treat than the double stochastic integral, which is most diverging, which appears.But it leads to some complications, because in g .;2(s, t), there are some double stochastic integral in the dynamical time u which appears.We write We consider a bigger integer N and we write: We have the following behaviour: . (s, t)g .;2(s, t)g (s , t ) or more precisely by the terms which arise from the diagonalisation in (89).We write the double Stratonovitch integral which appears in g .;2(s, t) or g .;2(s , t ) as double Itô integral and a simple integral.After using the Clark-Ocone formula, the expectation of the product of at most 8 term and at least 2 Itô integrals hasto be computed.We use Itô formula successivly and Clark-Ocone formula successivly in order to get our estimate.
We have analoguous formulas we don't write.Therefore: Let us study now the behaviour of By the considerations which will follow, it is enough to study C N i,j,2 = ∆t j < ω(g(s i , t j )), ∆ s i g(s i , t j ), g .;2(s i , t j ) > − i ,j ∆t j < ω(g(s i , t j ), ∆ s i g(s i , t j ), g .,2(s i , t j ) > But we can write: such that: In C 1,N i,j,2 and C 2,N i,j,2 , we can replace, by the considerations which will follow, ∆ s i (g(s i , t j ) by the quantity √ ∆s i g 1,.(s i , t j ) and ∆ s i (g(s i , t j ) by √ ∆s i g 1,.(s i , t j ).We get expressions C 3,N i,j,2 and C 4,N i,j,2 .We distribute the term which appear in (C 4,N i,j,2 C 4,N i,j ,2 , there are 4 terms with increments √ ∆s i g 1;.(s i , t j ) √ ∆s i" g 1;.(s i" , t j" ) and ∆t j g .;2(s i , t j ) and ∆t j" g .;2(s i" , t j" ) which appear.We condition by the Brownian motions which are got after diagonalising the increments of the leadings Brownian motions which appear in these formulas and we get as before a norm in L 2 which tends to 0.
We have to study 3 terms: ] and the last one ] is the most complicated to treat.We write: ∆s i ∆t j < ω(g(s i , t j )), g 1;.(s i , t j ), g .,2(s i , t j ) >} +{ i ,j ∆s i ∆t j < ω(g(s i , t j )), g 1;.(s i , t j ),g .;2(s i , t j ) − g .;2(s i , t j ) >} +{ i ,j By the previous considerations, we have only to estimate ] and E[ i,j =j C 7,N i,j,2 C 7,N i,j ,2 ] as well as the sum where there exist other coincidences of indices i, i , j, j .We have to estimate the analoguous quantities where we mix C 5,N i,j,2 and C 6,N i,j ,2 , the term where we mix C 5,N i,j,2 and C 6,N i,j ,2 and C 7,N i,j ,2 and the term where we mix C 6,N i,j,2 and C 7,N i,j ,2 .We will omit to write the details of the convergence of these mixed term to 0. Clearly, Namely, if we do the multiplication of each term in the sum, there are 6 increments which appear . Their mutual covariances satisfy to (92), ( 93) and (95) because j 1 = j 2 and because we don't have to consider when we do the multiplication term by term to consider the interaction between ∆ s i 1 (s i 1 , t j 1 ) and ∆ s i 1 B(s i 1 , t j 1 ) and the interaction between ∆ s i 2 B(s i 2 , t j 2 ) and ∆ s i 2 B(s i 2 , t j 2 ).We conclude after conditioning along these increments, or more precisely the Brownian motions which appear when we use the diagonalization (89).This allows us to show (114). Moreover, Namely, when we do the product term by term in (115), there are 6 increments which appear , and the terms ∆ t j 2 B(s i 2 , t j 2 ) ∆ t j 2 B(s i 2 , t j 2 ).We can apply (92), ( 93) and (95) to these increments because we don't have to take the covariance between ∆ t j 1 B(s i 1 , t j 1 ) and ∆ t j 1 B(s i 1 , t j 1 ) and the covariance between ∆ t j 2 B(s i 2 , t j 2 ) and ∆ t j 2 (s i 2 , t j 2 ).
Let us consider the most complicated term C 7,N i,j,2 because in g .;2(s i , t j ) and in g .;2(s i , t j ) in (114), it is not the same subdivision in t j .But since we consider there are 6 increments to see.They are B(s i 2 , t j 2 ) and we don't have to consider the correlation between ∆ t j 1 B(s i 1 , t j 1 ) and ∆ t j 1 B(s i 1 , t j 1 ) and the correlation ∆ t j 2 B(s i 2 , t j 2 ) and ∆ t j 2 (s i 2 , t j 2 ).We can apply (92), ( 93), (95) for the correlations we consider, and we can conclude as previously.
By the same reason The same arguments arise when we consider: It remains to treat the case where there are two coincidences, that is to treat the case of ] and E[(C 7,N i,j,2 ) 2 ], after doing the same restriction about the mixed terms.But as a matter of fact, we can show simply that We have namely the correlators between the following increments to consider: (s i 2 , t j ) and ∆ s i 2 B(s i 2 , t j 1 ).But we have t j 1 ≥ t j and t j 2 ≥ t j .Therefore: because t j 1 ≥ t j and because e has half derivatives in 0. This remark allows us to repeat the previous considerations as well as to use (92), ( 93) and (95).Moreover E[(C 6,N i,j,2 ) 2 ] → 0 (122) We have no difficulty to show that because we don't have to consider the covariance of a g 1;0 (s i , t j ) and a g 1;0 (s i , t j ) and because < g 1;.(s i , t j ), g 1;.(s i" , t j ) >= CO( ∆s i ∆ s i" ).The difficult part is to show that E[(C 7,N i,j,2 ) 2 ] → 0, because two different subdivision [t j , t j +1 ] and [t j , t j+1 ] appear and because t j ∈ [t j , t j+1 ].We write the details of this limit, because it the most complicated, the others limits are simpler.We write: i,j,2 = δs i ∆t j < ω(g(s i , t j )), g 1;.(s i , t j ),g .;2(s i , t j ) − g .;2(s i , t j ) > + ∆s i ∆t j < ω(g(s i , t j )), g 1;.(s i , t j ) − g 1;.(s i , t j ), g .;2(s i , t j ) > − < ∆s i ∆t j < ω(g(s i , t j )), g 1;.(s i , t j ) − g 1;.(s i , t j ), g .;2(s i , t j ) By the previous considerations, the terms E[ (C can be treated by Cauchy-Schwartz inequality.We proceed for that as it was done in the previous part.We remark, by the same considerations as in the first part, that it is enough to replace ∆t j g .;2(s i , t j ) by a double stochastic iterated integral 0<u<v<1 α u (s i , t j )(dB u (s i , t j+1 ) −dB u (s i , t j ))α v (s i )(dB v (s i , t j+1 ) − dB v (s i , t j )) where α u and α v are B(s i , t j ) measurable.By the same argument, we replace ∆t j g .;2(s i , t j ) by a double stochastic integral 0<u<v<1 α u (s i , t j )(dB u (s i , t j +1 ) − dB u (s i , t j )) α v (s i , t j )(dB v (s i , t j +1 ) − dB v (s i , t j ) where α u (s i , t j ) and α v (s i , t j ) are B .(s i , t j ) measurable.To study the behaviour when N → ∞, we can replace without difficulty in this last expression α u (s i , t j ) by α u (s i , t j ).We write: and we distribute in the first term of (124).The diagonal terms cancel, and we have to estimate when N → ∞ the behaviour of C 11,N i,j,2 = ∆s i < ω(g(s i , t j )), g 1,.(s i , t j ), where we sum over [t k , t k+1 ] ⊆ [t j , t j+1 ] and [t k , t k +1 ] ⊆ [t j , t j+1 ] for the sharper dyadic subdivision associated to 2 N .Instead of taking the following expression in time 1, let us take it in time r.We get a process C 11,N i,j,2,r (We replace g(s i , t j ) by g r (s i , t j ), g 1;.(s i , t j ) by g 1;.r (s i , t j ) and the double integral between 0 and 1 by a double integral between 0 and r.Let us consider the finite variational part V N r = V N i,j,2,r and the martingale part M N r = M N i,j,2,r associated to this process.Let us begin to study the finite variational part of this process V N r .This can come from a contraction between ω(g(s i , t j )) and g 1;.(s i , t j ) which leads to a term in ∆s i , which is multiplied by a term in ∆s i .But the L 2 norm of the sum t k =t k can be estimated.We decompose first t k =t k in a martingale term and a finite variational term.There is first a contraction between α v and dB v (s i , t k +1 ) − dB v (s i , t k ) which leads to a term in t k +1 − t k The stochastic integral in u can be estimated.We see the martingale term.By Itô formula Therefore the L 2 norm of this term behaves in √ t j+1 − t j .But since there is (t k +1 − t k ) in time u, we have a behaviour of this contribution in ∆s i (t j+1 − t j ) 3/2 whose sum vanish when N → ∞.The second term comes from a contraction between dB u (s i , t k+1 ) − dB u (s i , t k ) and dB v (s i , t k +1 ) − dB v (s i , t k ) which leads to a term in (t k+1 − t k )(t k +1 − t k ) and therefore to a contribution in (t j+1 − t j ) 2 .Therefore the total contribution is in ∆s i (t j+1 − t j ) 2 , whose sum vanish when N → ∞, because < g 1;.(s i , t j ), g 1;.(s i" , t j ) >= O √ ∆s i ∆s i" There is a contraction between ω(g(s i , t j )) and dB v (s i , t k +1 )−dB v (s i , t k ) which is in (t k +1 −t k ).This term cancel, because when we take the square of the L 2 norm of the sum, it behaves in i ,i" ∆ s i ∆ si" I i ,i" , where I i ,i" where I i ,i" is a sum of quadruple t k , t k" , t k 3 , t k 4 which behaves in O(t j+1 − t j ) 3 and a sum i ∆s i I i where I i has a bound in (t j+1 − t j ) 3/2 .The sum of these terms vanish, when N → ∞ (See part III for analoguous considerations).
Let us estimate the martingale term M N i,j,2,r .Let us estimate the L 2 norm of M N r .We use Itô formula.It behaves as i ,i" ∆s i" ∆s i I i,i + i ∆s i I i where I i ,i" has a bound in (t j+1 − t j ) 3/2 and I i the same.Therefore the L 2 norm of M N r vanish when N → ∞.
Step I.1.3:study of the behaviour of the double classical integral C N 5 .We writte C N i,j,5 = ∆s i ∆t j < ω(g(s i , t j )), g 2;.(s i , t j ), g .;2(s i , t j ) > We consider N > N and study: We write with C 2,N i,j,5 = T i ,j ⊆T i,j ∆s i ∆t j < ω(g(s i , t j )) − ω(g(s i , t j ), g 2;.(s i , t j )., g .;2(s i , t j ) > (130) and ∆s i ∆t j {< ω(g(s i , t j ), g 2;.(s i , t j ), g .;2(s i , t j ) > − < ω(g(s i , t j ), g 2;.(s i , t j ), g .;2(s i , t j ) > It is clear that C 2,N i,j,5 → 0 in L 2 because g 2;2 (s i , t j ) is bounded in L 2 .In order to estimate C 3,N i,j,5 , we can replace ω(g(s i , t j ) by ω(g(s i , t j )).We can replace ∆s i g 2;.(s i , t j ) by a double stochastic integral in the dynamical time u I 2;.(s i , t j ) as it was done in (126) and do the same transformation for the other g 2;. and g .;2which appear in C 3,N i,j,5 such that we have only to show that 4,N i,j,5 → 0 in L 2 where C 4,N i,j,5 =< ω(g(s i , t j ), I 2;.(s i ,t j ), I .;2(s i , t j ) > − T i ,j ⊆T i,j < ω(g(s i , t j )), I and we distribute in I 2;.(s i , t j ) and I .;2(s i , t j ).We get that the expression I 2;.(s i , t j ) is equal to the expression . (s 1 i , s 2 i , t j ) and that I .;2= t 1 j ,t 2 j ∈[t j ,t j+1 ] I .;2(s i , t 1 j , t 2 j ) after distributing in these stochastic integral.Only the contribution where s 1 i = s 2 i and t 1 j = t 2 j do not vanish when N → ∞, by the same considerations than in (54).These terms are nothing else, modulo some small error terms than I 2;.(s i , t j ) and I .;2(s i , t j ).We have only to show that i,j C 5,N i,j,5 → 0 in L 2 where C 5,N i,j,5 = T i ,j ⊆T i,j (< ω(g(s i , t j )), I 2;.(s i , t j ), I .;2(s i , t j ) > − < ω(g(s i , t j )), , I 2;.(s i ,t j ), I .;2(s i , t j ) > But we can show that the L 2 norm of I 2;.(s i , t j ) − I Step I.1.4:study of the vanishing term C N 3 .We write C N 3 = N i,j,3 where the L 2 norm of C N i,j,3 is in O(∆s i ∆t 3/2 j ).But we have if s i = s i , by using the previous technics E[< ω(g(s i , t j )), g 1;.(s i , t j ), g .;3(s i , t j ) >< ω(g(s i , t j )), g 1,.(s i , t j ), g .,3(s i , t We write in probability: The residual term converges to 0 by the previous arguments.It remains to treat the main term.We recall: The integral of the first term of (138) leads to the convergence of the sum of random quantities of a type analoguous to already considered quantities, which contains some "brackets" of the type < ∇ω(g(s i , t j )).∆ s i g(s i , t j ), ∆ s i g(s i , t j ), ∆ t j g(s i , t j ) > which converges by the methods used before.We can treat by the same method the convergence of < ∇ω(g(s i , t j ))(g(s i , t j+1 ) − g(s i , t j )), ∆ s i g(s i , t j ), ∆ t j g(s j , t j ) > which converge by the same methods as before.The term in (t−t j )(s−s i ) (∆t j ∆s i lead to analoguous terms.If we consider the term where the square of g N (s, t) − g(s i , t j ) appear, there is a term where the quantity < ∇ 2 ω(g(s i , t j )); ∆ s i g(s i , t j ) 2 , ∆ s i g(s i , t j ), ∆ t j g(s i , t j ) > appears whose sum vanishes in L 2 by the same considerations as in Step I.1.4.The only problem comes when we take sum corresponding more and less to the double bracket of (s, t) → g 1 (s, t) of the type i,j < ∇ 2 ω(g(s i , t j )).∆ s i g(s i , t j ).∆ t i g(s i , t j ), ∆ s i g(s i , t j ), ∆ t i g(s i , t j ) > whose treatment is similar to step I.1.3by expanding a product of integrals into iterated integrals of length 2.
Step II: convergence of A N 2 and A N 3 .The treatment for A N 2 and A N 3 are similar.So we will treat only the case of A N 2 .We write: Step II.1: convergence of B N 1 .
T i,j < ω(g(s i , t j )), The integral over T i,j is constant.We write: The term in γ 2 i,j can be treated as in step I.1.The term in γ 1 i,j can be treated as in step I.1, because the increments between ∆ s i B(s i , t j ) and ∆ s i B(s i , t j+1 ) > satisfy to (121), and we can do as in the treatment of (121) Step II.2: convergence of B N 2 .We use (137) and we conclude as in step I.2.
Step III: convergence of A N 4 .We write: Step III.1: convergence of B N 1 .We write with the notations of (142): The integral over T i,j is constant.In order to treat the sum, we write the second γ 1 i,j + γ 2 i,j as δ 1 i,j + δ 2 i,j where and δ 2 i,j = −g(s i , t j+1 ) + g(s i , t j ) (146) and we perform the limit as in the previous considerations.
Step III.2: convergence of B N 2 .We write and we use (137) for α N (s, t) a suitable function of (s, t).
When the form depends on a finite dimensional parameter, we show that the approximation of the stochastic integrals converge for all the derivatives of ω and we conclude by using the Sobolev imbedding theorem as in [23].That is we consider the integrals which converge in L 2 for all multiindices α. ♦ We would like to get the same theorem with a more intrinsic approximation gN (s, t) of the random field g(s, t).As in the part III, the finite dimensional approximations of the integral T 2 gN, * ω will converge in L 2 , but we don't know if they will converge to the same limit integral of T 2 g N, * ω.
For that if g(s, t j ) and g(s, t j+1 ) are close, we use the functions: F N (t, g(s, t j ), g(s, t j+1 ) = exp[ t − t j t j+1 − t j log(g(s, t j+1 )g −1 (s, t j )]g(s, t j ) (149) conveniently extended to the whole sets of matrices.We approximate g(s, t j+1 ), g(s, t j ) as follows: F N (s, g(s i , t j+1 ), g(s i+1 , t j+1 )) = exp[ s − s i s i+1 − s i log(g(s i+1 , t j+1 )g −1 (s i , t j+1 ))]g(s i , t j+1 ) (150) conveniently extended over the whole matrix algebras as well as its inverse.Moreover, F N (s, g(s i , t j ), g(s i+1 , t j )] = exp[ s − s i s i+1 − s i log(g(s i+1 , t j )g −1 (s i , t j ))]g(s i , t j ) (151) conveniently extended as well as its inverse to the set of all matrices.We take as approximation: gN (s, t) = exp[ t − t j t j+1 − t j log(F N (s, g(s i , t j+1 , g(s i+1 , t j+1 )) (F N ) −1 (s, g(s i , t j ), g(s i+1 , t j )))]F N (s,g(s i , t j ), g(s i+1 , t j )) (152) We have the asymptotic expansion: F N (t, g(s, t j ), g(s, t j+1 ) = g(s, t j ) + t − t j t j+1 − t j (g(s, t j+1 ) − g(s, t j ))+O(( t − t j t j+1 − t j ) 2 (g(s, t j+1 − g(s, t j )) 2 ) (153) We imbed in this expression the approximation of g(s, t j+1 ) and of g(s, t j ).This shows that, in the expansion of gN (s, t), the more singular term is the same in (70), modulo some more regular terms which converge.The main Itô integral is the same, but we don't know if the correcting terms are the same.We get the main result of this part: Theorem 4.2: when N → ∞, the traditional integral ÃN v = T 2 (g N ) * ω v converges in L 2 to the stochastic Stratonovitch integral: < ω(g(s, t)), d s g(s, t), d t g(s, t) > (154) Moreover, D g * ω v has a smooth version in v.
Remark: we ignore if the stochastic integral of Theorem IV.2 is equal to the stochastic integral of Proposition IV.1.In the rest of this paper, we will use the version of Theorem IV.2.
Remark: we can consider in the previous theorem a 2-tensor which is not necessarily a 2-form.
5 Stochastic W.Z.N.W. model on the punctured sphere Let us consider the 3-form closed Z-valued ω over G which is supposed simple simply connected, which at the level of the Lie algebra of G is equal to We extend ω in a 3-form over the whole matrix algebra bounded with bounded derivatives of all orders.We can suppose that ω is Z-valued on G. Let Σ(1, n) be a (1 + n) punctured sphere.We deduce a family of loops s → g(s, t).Let s → g(s, t) such a loop.We repeat the considerations of [28] and [31] in order to define over such loop group L i t (G) the stochastic 2-form: < ω(g(s, t)), d s g(s, t), .> We can define for that the following poor stochastic diffeology (see [10], [46] for the introduction of this notion in the deterministic case).Let Ω be the probability space where the random (1 + n) punctured sphere is defined: Definition 5.1: A stochastic plot of dimension m of L(G) is given by a countable family (O, φ i , Ω i ) where O is an open subset of R m such that: i)The Ω i constitute a measurable partition of Ω.
ii) φ i (u)(.)= {s → F i (u, s, g(s, t))} where F i is a smooth function over O × S 1 × R N with bounded derivatives of all orders (R N is the matrix algebra where we have imbedded G).
iii)Over Ω i , for all u ∈ U φ i (u)(.)belongs to the loop group L(G).
We identify two stochastic plots (O, φ 1 i , Ω We can look at the apparatus of [28], [30], [31] to define a stochastic line bundle ξ i t over L i t (G), with curvature 2π √ −1kτ st (ω) for k an integer.Let us recall how to do (See [28], p 463-464): let g i be a countable system of finite energy loops in the group such that the ball of radius δ and center g i for the uniform norm O i determine an open cover of L(G).We can suppose that δ is small.The loop g i constitutes a distinguished point in O i .We construct if g belongs to O i a distinguished curve joining g to g i , called l(g i , g): since δ is small, g i (s) and g(s) are joined by a unique geodesic for the group structure.l u (g i , g) is the loop s → exp g i (s) [u(g(s) − g i (s))] where g(s) − g i (s) is the vector over the unique geodesic joining g i (s) to g(s) and exp the exponential of the Lie group associated to the canonical Riemannian structure over the Lie group.This allows to define over O i a distinguished path joining g(.) to g i (.).We choose a deterministic path joining the unit loop e(.) to g i (.) l i (e(.), g i (.)), and by concatenation of the two paths, we get a distinguished path joining g(.) to e(.) l i (g(.), g i (.)) over O i .
The second step is to specify a distinguished surface bounded by l i (e(.), g(.)) and l j (e(.), g(.)),where g(.) belongs to O i ∩ O j .Since δ is small, there is a path u → exp g i (.) [u(g j (.) − g i (.))] joining g i (.) to g j (.).Because L(G) is simply connected, because G is two-connected, the loop constituted of the path joining e(.) to g i (.), the path joining g i (.) to g j (.) and the path joining g j (.) to e(.) can be filled by a deterministic surface in the smooth loop group.We can moreover fill the small stochastic triangle constituted of l .(g i (.), g(.)), l .(g j (.), g(.)) and the the exponential curve joining g i (.) to g j (.) by a small stochastic surface (See [28] for analoguous statements).We get a surface B t i,j (g(.)) which satisfies to our request and which is a stochastic plot.By pulling back (See [28], [30], [31]), we can consider the stochastic Z-valued form τ st (ω) and integrate it over the surface B t i,j (g(.)).We put ρ t i,j (g(.)) = exp[− √ −12πk B t i,j (g(.)) (See [30]).