Toward a Wong–Zakai Approximation for Big Order Generators

: We give a new approximation with respect of the traditional parametrix method of the solution of a parabolic equation whose generator is of big order and under the Hoermander form. This generalizes to a higher order generator the traditional approximation of Stratonovitch diffusion which put in relation random ordinary differential equation (the leading process is random and of ﬁnite energy. When a trajectory of it is chosen, the solution of the equation is deﬁned) and stochastic differential equation (the leading process is random and only continuous and we cannot choose a path in the solution which is only almost surely deﬁned). We consider simple operators where the computations can be fully performed. This approximation ﬁts with the semi-group only and not for the full path measure in the case of a stochastic differential equation.


Introduction
Let us consider a compact Riemannian manifold M of dimension d endowed with its normalized Riemannian measure dx (x ∈ M).
Let us consider m smooth vector fields X i (we will suppose later that they are without divergence). Some times vector fields are considered as one order differential operators acting on the space of smooth functions on the manifold M, sometimes they are considered as smooth sections of the tangent bundle of M. We consider the second order differential operator: It generates a Markovian semi-group P t which acts on continuous function f on M P t f ≥ 0 if f ≥ 0. It is represented by a stochastic differential equation in Stratonovitch sense ( [1]) where where t → w i t is a flat Brownian motion on R m Classically, the Stratonovitch diffusion x t (x) can be approximated by its Wong-Zakai approximation.
Let w n,i t be the polygonal approximation of the Brownian path t → w n t for a subdivision of [0, 1] of length n.
We introduce the random ordinary differential equation Wong-Zakai theorem ( [1]) states that if f is continuous We are motivated in this paper by an extension of (6) to higher order generators. Let us consider the generator L k = (−1) k ∑ m I X 2k i . We suppose that the vector fields X i span the tangent space of M in all point of M and that they are divergent free. L k is an elliptic postive essentially self-adjoint operator [2] which generates a contraction semi-group P k t on L 2 (dx) Let L f ,k be the generator on R m ((w i ) ∈ R m ). By [3], it generates a semi-group P f ,k t on C(R m ), the space of continuous functions on the flat space endowed with the uniform topology, which is represented by a heat-kernel: where (w = (w i )). In [4], it is noticed that heuristically P f ,k t is represented by a formal path space If we were able to construct a differential equation in the Stratonovitch sense These are formal considerations because in such a case the path measures are not defined. We will give an approach to (9) by showing that some convenient Wong-Zakai approximation converges to the semi-group. We introduce, according to [5] the Wong-Zakai operator where As a first theorem, we state: Theorem 1. (Wong-Zakai) Let us suppose that the vector fields X i commute.
. This means that if we give f in L 2 (dx) that To give another example, we suppose that M is a compact Lie group G endowed with its normalized Haar measure dg and that the vector fields X i are elements of the Lie algebra of G considered as right invariant vector fields. This means that if we consider the right action on L 2 (dg) R g 0 we have We consider the rightinvariant elliptic differential operator It is an elliptic positive essentially selfadjoint operator. By elliptic theory ( [2]), it has a positive spectrum λ associated to eigenvectors f λ . λ ≥ 0 if λ belongs to the spectrum.
We refer to the reviews [3,6] for the study of stochastic analysis without probability for non-markovian semi-groups.
Let us describe the main difference with the Wong-Zakai approximation of these semi-groups and the the traditional parametrix approximation of these by slicing the time. We work on R d to simplify the exposition. Let be where (α) = (α 1 , ...α d ) is a multi-index on the flat space with length |(α)| = ∑ α i . We suppose that the function a (α) (x) are smooth with bounded derivatives at each order. We consider if y = (y 1 , ..., y d ) We consider the symbol associated to the operator We suppose that we are in an elliptic situation: for all x We suppose that the operator is positive bounded below. We can consider in this case the parabolic equation It has a unique solution. The parametrix method consist to freeze the starting point x by considering the family of operators We consider the family of non-markovian semi-groups P x t satisfying the parabolic equation We introduce the kernel in L 2 (dx) when n → ∞ At the point of view od path integrals, parametrix is related to the formal path integrals of Klauder (see [4] for a rigorous approach). Consider the Fourier transformf of a function which belongs to L 2 (dx). We get( such thatP By using the inverse of the Fourier transform,the lefthandside of (28) gives an approximation of Klauder path integral on the phase space.
Hamiltonian path integrals are not well defined as measures. Let us consider the case of an order 2 differential operator We suppose that where x → X i (x) are smooth vector fields. Moreover, the part of order 1 of the operator define a smooth vector field X 0 . In such a case, P a t (x) is represented by an Itô stochastic differential equation Itô stochastic differential equations can be approximated by the Euler scheme if we consider a subdivision [t k−1 , t k ] of [0, 1] of mesh 1/n. t n is the biggest t k smaller than t. The approximation of the Itô equation is starting from x, by stochastic calculus ([1]), the law of t → x n t (x) for the uniform norm tends to the law of t → x t (x). In particular, if f is a bounded continuous function, when n → ∞. However, in such a case, The Calculus on flat Brownian motion shows Let t → h t = (h 1 t , ..., h m t ) a finite energy path in R m starting from 0. We consider the energy norm At the formal path integral point of view, the law of the flat Brownian motion t → w t is the Gaussian law where dD(h) is the formal Lebesgue measure on the finite energy paths (which does not exist) and z a normalized constant, called the partition function, which is infinite and not well defined.
We introduce the solution of the ordinary differential equation The Wong-Zakai approximation explains that at a formal point of view, the solution of the Stratonovitch Equation (4) x 1 (x) can be seen as x h 1 (x) where h is chosen according the formal Gaussian measure (38). This remark is not suitable for Itô equation.
Bismutian procedure [7] is the use of the implicit function theorem for h → x h 1 (x) to study the heat-kernel associated to the semi-group P t . It was translated in semi-group theory in [5] by introducing some Wong-Zakai kernels associated to the semi-group generated by L. The long term motivation of this paper is to implement Bismut procedure in big order operators of Hoermander's type.
Proof of Theorem 1. L k is an elliptic positive operator. By elliptic theory [2], it has a discete spectrum λ associated to normalized eigenfunctions f λ . Since R m |p f ,k 1 (w)| 2 dw < ∞, Q k 1/n is a bounded operator on L 2 (dx). Moreover Q k The main remark is that we can compute explicitely Q k 1/n f λ . We put t = 1/n. Formaly Namely, by ellipticity and because the vector fields X i commute with L k , we can conclude that the L 2 -norm of X Let us show how to prove this estimate. We have Since X i commutes with L k , we have Therefore X i f λ is still an eigenfunction associate to λ. Therefore, X i is a linear operator on the eigenspace E λ associate to λ which is of finite dimension by elliptic theory. By Garding inequality [2] We use for that (L k + C) 1/2k is an elliptic pseudo-differential operator of order 1 (see the end of this paper). E λ is an eigenspace for (L k + C) 1/2k associated to the eigenvalue (λ + C) 1/2k . Therefore X i is a linear operator on E λ with norm smaller than C(λ 1/2k + 1).
It is enough to compute The main remark (see the end of this paper) is if one of the l i is not a multiple of 2k, we have We ignore some immediated problems of signs. Therefore, B n = 0 if n is not a multiple of 2k and is equal because the vector field commute, ,if n = 2kl to We deduce that and In such a case, the Wong-Zakai approximation is exact. It is analog to the classic result for diffusions of 9]). The Stratonovitch diffusion in this case satisfy The map y → exp[∑ m i X i y i t ](x) is defined as follows. We consider the ordinary differential y = (y 1 , ..., y m ) belongs to R m and we put Proof of Theorem 2. Let E λ be the space of eigenfunctions associated to the eigenvalue λ of L k . Since L k commutes with the right action of G, E λ is a representation for the right action of G ( [10]). Therefore rightinvariant vector fields act on E λ . If Z is a rightinvariant vector field, we can consider the L 2 norm of Z f λ for f λ belonging to E λ . We remark that (L k + C) 1/2k is an elliptic pseudodifferential operator of order 1 (C is strictly positive). By Garding inequality [2], f λ is an eigenfunction associated to (L k + C) 1/2k and the eigenvalue (λ + C) 1/2k . Let us consider a polynomial X It acts on E λ and is norm is bounded by ((λ + C) 1/2k + C) ∑ α i for the L 2 norm.
From that we deduce that if f λ is an eigenfunction associated to λ of L k that the series converges and is equal to f λ (x(t 1/2k w)(x)). By distinguishing if w is big or not and using (46), we see that if l = 2kl where Q l ,t λ has a bound on E λ in C l (λ + C) l . We deduce that Q k t acts on E λ by where the norm on E λ of Q λ t is smaller that C exp [Cλt]. However, because the vector fields are without divergence.
To conclude, we remark that the series tends to 0 when n → ∞. Namely each term is bounded by |a λ | 2 C λ and tends simply to 0. The result arises by the dominated convergence theorem.

Some Results on Linear Operators
We work on functions f with values in R, but it is possible to work in C. We refer to [2] for details. Let us begin to work on R d . We consider a smooth function on R d × R d called a symbol a(., .) such that We define the operator L a associated to the symbol a by acting on smooth function with bounded derivatives at each order. We suppose We sat the operator L a is a pseudodifferential operator of order p. This notion is invariant if we do a diffeomorphism of R d with bounded derivatives at each order. This explain that we can define an elliptic operator on a smooth compact manifold M. On each space of the tangent bundle, we introduce a metric strictly positive which depends smoothly on x ∈ M. We say that the manifold is equipped of a Riemannian structure. In such a case, we can introduce the analog of the normalized Lebesgue measure which is called the Riemannian measure dx. We say that L a is symmetric if If we consider vector fields X on M as differential operators, we can consider their adjoint: such that the operators considered in this work are symmetric. Moreover, there are alliptic of order 2k. We can consider the eigenvalue problem: for what λ, there exists a f λ ∈ L 2 (dx) not equal to 0 such that L a f λ = λ f λ In the compact case, this problem is solved for a positive symmetric elliptic pseudodifferential operator. λ belongs to a discrete subset of R + called the spectrum of L a . The solutions of (76) constitute a linear subset of finite dimension E λ which constitutes an orthonormal decomposition of L 2 (dx). Each element of E λ is smooth.
If L a is a strictly positive elliptic pseudodifferential operator of order p, we can define is power (L a ) α for any positive real α ( [11]). It is still a strictly positive pseudodifferential of order pα. The eigenspaces are the same, but associated to the eigenvalue λ α . Therefore (L a ) 1/p is a pseudodifferential operator of order 1.
If L a is a strictly positive pseudo differential operator of order 1, it satisfy to the Garding inequality the generator. This gives a new approximation than the parametrix one, which was done by freezing the starting point in the generator.