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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.
Let us consider a compact Riemannian manifold M of dimension d endowed with its normalized Riemannian measure ().
Let us consider m smooth vector fields (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 which acts on continuous function f on M
if . It is represented by a stochastic differential equation in Stratonovitch sense ()
where is a flat Brownian motion on Classically, the Stratonovitch diffusion can be approximated by its Wong–Zakai approximation.
Let be the polygonal approximation of the Brownian path for a subdivision of of length n.
We introduce the random ordinary differential equation
Wong–Zakai theorem () 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 . We suppose that the vector fields span the tangent space of M in all point of M and that they are divergent free. is an elliptic postive essentially self-adjoint operator  which generates a contraction semi-group on
Let be the generator on (. By , it generates a semi-group on , the space of continuous functions on the flat space endowed with the uniform topology, which is represented by a heat-kernel:
where . In , it is noticed that heuristically is represented by a formal path space measure such that
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  the Wong–Zakai operator
As a first theorem, we state:
(Wong–Zakai) Let us suppose that the vector fields commute. Then converge in to if f is in . This means that if we give f in that
To give another example, we suppose that M is a compact Lie group G endowed with its normalized Haar measure and that the vector fields are elements of the Lie algebra of G considered as right invariant vector fields. This means that if we consider the right action on
We consider the rightinvariant elliptic differential operator
It is an elliptic positive essentially selfadjoint operator. By elliptic theory (), it has a positive spectrum associated to eigenvectors . if belongs to the spectrum.
(Wong–Zakai) Let such that for all . Then converges in to . This means that
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 to simplify the exposition. Let be
where is a multi-index on the flat space with length . We suppose that the function are smooth with bounded derivatives at each order. We consider if
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 starting from
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 satisfying the parabolic equation
We introduce the kernel
Parametrix method states that
in when At the point of view od path integrals, parametrix is related to the formal path integrals of Klauder (see  for a rigorous approach). Consider the Fourier transform of a function which belongs to . We get
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 are smooth vector fields. Moreover, the part of order 1 of the operator define a smooth vector field . In such a case, is represented by an Itô stochastic differential equation starting from x
is a flat Brownian motion on . We have
Itô stochastic differential equations can be approximated by the Euler scheme if we consider a subdivision of of mesh . is the biggest smaller than t. The approximation of the Itô equation is
starting from x, by stochastic calculus (), the law of for the uniform norm tends to the law of . In particular, if f is a bounded continuous function,
when . However, in such a case,
The Calculus on flat Brownian motion shows
Let a finite energy path in starting from 0. We consider the energy norm
At the formal path integral point of view, the law of the flat Brownian motion is the Gaussian law
where 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 starting from x
The Wong–Zakai approximation explains that at a formal point of view, the solution of the Stratonovitch Equation (4) can be seen as where h is chosen according the formal Gaussian measure (38). This remark is not suitable for Itô equation.
Bismutian procedure  is the use of the implicit function theorem for to study the heat-kernel associated to the semi-group . It was translated in semi-group theory in  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.
is an elliptic positive operator. By elliptic theory , it has a discete spectrum associated to normalized eigenfunctions . Since , is a bounded operator on . Moreover
The main remark is that we can compute explicitely . We put . Formaly
Namely, by ellipticity and because the vector fields commute with , we can conclude that the -norm of has a bound in in order to deduce that the series in (42) converges.
Let us show how to prove this estimate. We have
Since commutes with , we have
Therefore is still an eigenfunction associate to . Therefore, is a linear operator on the eigenspace associate to which is of finite dimension by elliptic theory. By Garding inequality 
We use for that is an elliptic pseudo-differential operator of order 1 (see the end of this paper). is an eigenspace for associated to the eigenvalue . Therefore is a linear operator on with norm smaller than .
It is enough to compute
The main remark (see the end of this paper) is if one of the is not a multiple of , we have
On the other hand, by using the semi-group properties of , we have
We ignore some immediated problems of signs. Therefore, if is not a multiple of and is equal because the vector field commute, if to
We deduce that
if . □
In such a case, the Wong–Zakai approximation is exact. It is analog to the classic result for diffusions of Doss–Sussmann ([8,9]). The Stratonovitch diffusion in this case satisfy
The map is defined as follows. We consider the ordinary differential equation issued from x
belongs to and we put
Let be the space of eigenfunctions associated to the eigenvalue of . Since commutes with the right action of G, is a representation for the right action of G (). Therefore rightinvariant vector fields act on . If Z is a rightinvariant vector field, we can consider the norm of for belonging to . We remark that is an elliptic pseudodifferential operator of order 1 (C is strictly positive). By Garding inequality ,
is an eigenfunction associated to and the eigenvalue .
Let us consider a polynomial . It acts on and is norm is bounded by for the norm.
From that we deduce that if is an eigenfunction associated to of that the series
converges and is equal to . By distinguishing if w is big or not and using (46), we see that if
Moreover, by (47) and (48)
where is the number of of equal to j. By using (47) and (48), we recognize in (59)
For , we recognize . Let us compute the norm of the previous element. It is bounded by
For , we recognize .
We recognize in the previous sum
We deduce a bound of the operation given by (59) in .
By the same argument, we have a bound of on in .
In order to conclude, we see that on
where has a bound on in . We deduce that acts on by
where the norm on of is smaller that .
because the vector fields are without divergence.
To conclude, we remark that the series
tends to 0 when . Namely each term is bounded by and tends simply to 0. The result arises by the dominated convergence theorem. □
2. Some Results on Linear Operators
We work on functions f with values in , but it is possible to work in . We refer to  for details.
Let us begin to work on . We consider a smooth function on called a symbol such that
We define the operator associated to the symbol a by
acting on smooth function with bounded derivatives at each order. We suppose
We sat the operator is a pseudodifferential operator of order p. This notion is invariant if we do a diffeomorphism of 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 . 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 . We say that is symmetric if
It is called positive if
and strictly positive if there exists such that for all f
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 .
We can consider the eigenvalue problem: for what , there exists a not equal to 0 such that
In the compact case, this problem is solved for a positive symmetric elliptic pseudodifferential operator. belongs to a discrete subset of called the spectrum of . The solutions of (76) constitute a linear subset of finite dimension which constitutes an orthonormal decomposition of . Each element of is smooth.
If is a strictly positive elliptic pseudodifferential operator of order p, we can define is power for any positive real (). It is still a strictly positive pseudodifferential of order . The eigenspaces are the same, but associated to the eigenvalue . Therefore is a pseudodifferential operator of order 1.
If is a strictly positive pseudo differential operator of order 1, it satisfy to the Garding inequality
Let us look to the case of non-compact set by looking the driving semi-group . First of all, by classic results (see  for instance)
Let us recall that the main difference in this work with respect of the case of diffusion is that the flat heat-kernel can change of sign. Moreover, if f is a polynomial, the series
converges and only in fact a finite numbers of terms are different to zero. This shows
This last formula explains (47) and (48) modulo some minor problems of signs.
We continue in this paper our previous works (see [6,11] for reviews) which implement stochastic analysis in non-Markovian semi-groups (they do not preserve positivity). The traditional Wong–Zakai approximation of Stratonovitch diffusion is interpreted in this framework, for the case of higher order elliptic operators under Hoermander’s form. Computations are done by using the global property of the generator. This gives a new approximation than the parametrix one, which was done by freezing the starting point in the generator.
The author has read and agreed to the published version of the manuscript.
This research received no external funding.
Conflicts of Interest
The author declares no conflict of interest.
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