## 1. Introduction

Let us consider a compact Riemannian manifold M of dimension d endowed with its normalized Riemannian measure $dx$ ($x\in M$).

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${P}_{t}f\ge 0$ if

$f\ge 0$. It is represented by a stochastic differential equation in Stratonovitch sense ([

1])

where

where

$t\to {w}_{t}^{i}$ is a flat Brownian motion on

${\mathbb{R}}^{m}$ Classically, the Stratonovitch diffusion

${x}_{t}\left(x\right)$ can be approximated by its Wong–Zakai approximation.

Let ${w}_{t}^{n,i}$ be the polygonal approximation of the Brownian path $t\to {w}_{t}^{n}$ 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

${L}^{k}={(-1)}^{k}{\sum}_{I}^{m}{X}_{i}^{2k}$. 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

${P}_{t}^{k}$ on

${L}^{2}\left(dx\right)$Let

${L}^{f,k}$ be the generator on

${\mathbb{R}}^{m}$ (

$\left({w}_{i}\right)\in {\mathbb{R}}^{m})$. By [

3], it generates a semi-group

${P}_{t}^{f,k}$ on

$C\left({\mathbb{R}}^{m}\right)$, the space of continuous functions on the flat space endowed with the uniform topology, which is represented by a heat-kernel:

where

$(w=\left({w}_{i}\right))$. In [

4], it is noticed that heuristically

${P}_{t}^{f,k}$ is represented by a formal path space measure

${Q}^{f,k}$ 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 [

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. Then ${\left({Q}_{1/n}^{k}\right)}^{n}\left(f\right)$ converge in ${L}^{2}\left(dx\right)$ to ${P}_{1}^{k}f$ if f is in ${L}^{2}\left(dx\right)$. This means that if we give f in ${L}^{2}\left(dx\right)$ 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}\left(dg\right)$${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

$\lambda $ associated to eigenvectors

${f}_{\lambda}$.

$\lambda \ge 0$ if

$\lambda $ belongs to the spectrum.

**Theorem** **2.** (Wong–Zakai) Let $f=\sum {a}_{\lambda}{f}_{\lambda}$ such that ${\sum}_{\lambda}{\left|{a}_{\lambda}\right|}^{2}{C}^{\lambda}<\infty $ for all $C>0$. Then ${\left({Q}_{1/n}^{k}\right)}^{n}\left(f\right)$ converges in ${L}^{2}\left(dg\right)$ to ${P}_{1}^{k}f$. 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

${\mathbb{R}}^{d}$ to simplify the exposition. Let be

where

$\left(\alpha \right)=({\alpha}_{1},\dots {\alpha}_{d})$ is a multi-index on the flat space with length

$\left|\left(\alpha \right)\right|=\sum {\alpha}_{i}$. We suppose that the function

${a}_{\left(\alpha \right)}\left(x\right)$ are smooth with bounded derivatives at each order. We consider if

$y=({y}_{1},\dots ,{y}_{d})$We consider the symbol associated to the operator

We suppose that we are in an elliptic situation: for all

xWe suppose that the operator is positive bounded below. We can consider in this case the parabolic equation starting from

$f\in {L}^{2}\left(dx\right)$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

${P}_{t}^{x}$ satisfying the parabolic equation

Parametrix method states that

in

${L}^{2}\left(dx\right)$ when

$n\to \infty $ 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 transform

$\widehat{f}$ of a function which belongs to

${L}^{2}\left(dx\right)$. We get

such that

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

$x\to {X}_{i}\left(x\right)$ are smooth vector fields. Moreover, the part of order 1 of the operator define a smooth vector field

${X}_{0}$. In such a case,

${P}_{t}^{a}\left(x\right)$ is represented by an Itô stochastic differential equation starting from

x$t\to \left({w}_{t}^{i}\right)$ is a flat Brownian motion on

${\mathbb{R}}^{m}$. We have

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

$t\to {x}_{t}^{n}\left(x\right)$ for the uniform norm tends to the law of

$t\to {x}_{t}\left(x\right)$. In particular, if

f is a bounded continuous function,

when

$n\to \infty $. However, in such a case,

The Calculus on flat Brownian motion shows

Let

$t\to {h}_{t}=({h}_{t}^{1},\dots ,{h}_{t}^{m})$ a finite energy path in

${\mathbb{R}}^{m}$ starting from 0. We consider the energy norm

At the formal path integral point of view, the law of the flat Brownian motion

$t\to {w}_{t}$ is the Gaussian law

where

$dD\left(h\right)$ 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

${x}_{t}^{h}\left(x\right)$ starting from

xThe Wong–Zakai approximation explains that at a formal point of view, the solution of the Stratonovitch Equation (

4)

${x}_{1}\left(x\right)$ can be seen as

${x}_{1}^{h}\left(x\right)$ 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

$h\to {x}_{1}^{h}\left(x\right)$ 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

$\lambda $ associated to normalized eigenfunctions

${f}_{\lambda}$. Since

${\int}_{{\mathbb{R}}^{m}}{\left|{p}_{1}^{f,k}\left(w\right)\right|}^{2}dw<\infty $,

${Q}_{1/n}^{k}$ is a bounded operator on

${L}^{2}\left(dx\right)$. Moreover

if

The main remark is that we can compute explicitely

${Q}_{1/n}^{k}{f}_{\lambda}$. 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 ${X}_{{i}_{1}}^{{\alpha}_{1}}{X}_{{i}_{2}}^{{\alpha}_{2}}\dots {X}_{{i}_{l}}^{{\alpha}_{l}}{f}_{\lambda}$ has a bound in ${\lambda}^{\sum {\alpha}_{i}/2k}{C}^{\sum {\alpha}_{i}}$ in order to deduce that the series in (42) converges.

Let us show how to prove this estimate. We have

Since

${X}_{i}$ commutes with

${L}^{k}$, we have

Therefore

${X}_{i}{f}_{\lambda}$ is still an eigenfunction associate to

$\lambda $. Therefore,

${X}_{i}$ is a linear operator on the eigenspace

${E}_{\lambda}$ associate to

$\lambda $ 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). ${E}_{\lambda}$ is an eigenspace for ${({L}^{k}+C)}^{1/2k}$ associated to the eigenvalue ${(\lambda +C)}^{1/2k}$. Therefore ${X}_{i}$ is a linear operator on ${E}_{\lambda}$ with norm smaller than $C({\lambda}^{1/2k}+1)$.

The main remark (see the end of this paper) is if one of the

${l}_{i}$ is not a multiple of

$2k$, we have

On the other hand, by using the semi-group properties of

${P}_{t}^{k,f}$, we have

We ignore some immediated problems of signs. Therefore,

${B}_{{n}^{\prime}}=0$ if

${n}^{\prime}$ is not a multiple of

$2k$ and is equal because the vector field commute, if

${n}^{\prime}=2k{l}^{\prime}$ to

We deduce that

and that

such that

if

$f=\sum {a}_{\lambda}{f}_{\lambda}$. □

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

$y\to exp\left[{\sum}_{i}^{m}{X}_{i}{y}_{t}^{i}\right]\left(x\right)$ is defined as follows. We consider the ordinary differential equation issued from

x$y=({y}^{1},\dots ,{y}^{m})$ belongs to

${\mathbb{R}}^{m}$ and we put

**Proof** **of** **Theorem** **2.** Let

${E}_{\lambda}$ be the space of eigenfunctions associated to the eigenvalue

$\lambda $ of

${L}^{k}$. Since

${L}^{k}$ commutes with the right action of

G,

${E}_{\lambda}$ is a representation for the right action of

G ([

10]). Therefore rightinvariant vector fields act on

${E}_{\lambda}$. If

Z is a rightinvariant vector field, we can consider the

${L}^{2}$ norm of

$Z{f}_{\lambda}$ for

${f}_{\lambda}$ belonging to

${E}_{\lambda}$. We remark that

${({L}^{k}+C)}^{1/2k}$ is an elliptic pseudodifferential operator of order 1 (

C is strictly positive). By Garding inequality [

2],

${f}_{\lambda}$ is an eigenfunction associated to

${({L}^{k}+C)}^{1/2k}$ and the eigenvalue

${(\lambda +C)}^{1/2k}$.

Let us consider a polynomial ${X}_{{i}_{1}}^{{\alpha}_{1}}\dots {X}_{{i}_{l}}^{{\alpha}_{l}}={Z}_{l}$. It acts on ${E}_{\lambda}$ and is norm is bounded by ${({(\lambda \phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}C)}^{1/2k}+C)}^{\sum {\alpha}_{i}}$ for the ${L}^{2}$ norm.

From that we deduce that if

${f}_{\lambda}$ is an eigenfunction associated to

$\lambda $ of

${L}^{k}$ that the series

converges and is equal to

${f}_{\lambda}\left(x\left({t}^{1/2k}w\right)\left(x\right)\right)$. By distinguishing if

w is big or not and using (46), we see that if

$l\ne 2k{l}^{\prime}$Moreover, by (47) and (48)

where

$2k{l}_{j}^{\prime}$ is the number of of

${\alpha}_{i}$ equal to

j. By using (47) and (48), we recognize in (59)

For

${l}^{\prime}=1$, we recognize

$tL$. Let us compute the

${L}^{2}$ norm of the previous element. It is bounded by

For ${l}^{\prime}=1$, we recognize $tL$.

We recognize in the previous sum

We deduce a bound of the operation given by (59) in $\frac{{t}^{{l}^{\prime}}{C}^{2k{l}^{\prime}}}{\left({l}^{\prime}\right)!}{(\lambda +C)}^{{l}^{\prime}}$.

By the same argument, we have a bound of $\frac{{t}^{{l}^{\prime}}}{{l}^{\prime}!}{\left({L}^{k}\right)}^{{l}^{\prime}}$ on ${E}_{\lambda}$ in $\frac{{t}^{{l}^{\prime}}}{{l}^{\prime}!}{C}^{{l}^{\prime}}{(\lambda +C)}^{{l}^{\prime}}$.

In order to conclude, we see that on

${E}_{\lambda}$
where

${Q}_{\lambda}^{{l}^{\prime},t}$ has a bound on

${E}_{\lambda}$ in

${C}^{{l}^{\prime}}{(\lambda +C)}^{{l}^{\prime}}$. We deduce that

${Q}_{t}^{k}$ acts on

${E}_{\lambda}$ by

where the norm on

${E}_{\lambda}$ of

${Q}_{t}^{\lambda}$ is smaller that

$Cexp\left[C\lambda t\right]$.

However, if

$f=\sum {a}_{\lambda}{f}_{\lambda}$However,

because the vector fields are without divergence.

To conclude, we remark that the series

tends to 0 when

$n\to \infty $. Namely each term is bounded by

$|{a}_{\lambda}{|}^{2}{C}^{\lambda}$ 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

$\mathbb{R}$, but it is possible to work in

$\mathbb{C}$. We refer to [

2] for details.

Let us begin to work on

${\mathbb{R}}^{d}$. We consider a smooth function on

${\mathbb{R}}^{d}\times {\mathbb{R}}^{d}$ 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

${\mathbb{R}}^{d}$ 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

$x\in M$. 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

It is called positive if

and strictly positive if there exists

$C>0$ such that for all

fIf 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

$\lambda $, there exists a

${f}_{\lambda}\in {L}^{2}\left(dx\right)$ not equal to 0 such that

In the compact case, this problem is solved for a positive symmetric elliptic pseudodifferential operator. $\lambda $ belongs to a discrete subset of ${\mathbb{R}}^{+}$ called the spectrum of ${L}^{a}$. The solutions of (76) constitute a linear subset of finite dimension ${E}_{\lambda}$ which constitutes an orthonormal decomposition of ${L}^{2}\left(dx\right)$. Each element of ${E}_{\lambda}$ is smooth.

If

${L}^{a}$ is a strictly positive elliptic pseudodifferential operator of order

p, we can define is power

${\left({L}^{a}\right)}^{\alpha}$ for any positive real

$\alpha $ ([

11]). It is still a strictly positive pseudodifferential of order

$p\alpha $. The eigenspaces are the same, but associated to the eigenvalue

${\lambda}^{\alpha}$. Therefore

${\left({L}^{a}\right)}^{1/p}$ 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

We refer to [

2] for this material.

Let us look to the case of non-compact set by looking the driving semi-group

${P}_{t}^{k,f}$. First of all, by classic results (see [

12] 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

${p}_{1}^{f,k}\left(w\right)$ 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.