Toward a Wong–Zakai Approximation for Big Order Generators

Rémi Léandre

doi:10.3390/sym12111893

Abstract

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.

Highlights

Let us consider m smooth vector fields Xi (we will suppose later that they are without divergence)
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 Xi.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 introduce the solution of the ordinary differential equation xth ( x ) starting from x dxth ( x ) =

Summary

Introduction

Let us consider m smooth vector fields Xi (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: m (Wong–Zakai) Let us suppose that the vector fields Xi commute.

Results

Conclusion