Stochastic evolution of inviscid Burgers fluid

Summary

We study a stochastic Burgers equation using the geometric point of view initiated by Arnold for the incompressible Euler flow evolution. The geometry is developed as a Cartan-type geometry, using a frame bundle approach (stochastic, in this case) with respect to the infinite-dimensional Lie group where the evolution takes place. The existence of the stochastic Burgers flow is a consequence of the control in the mean of the energy transfer from low modes to high modes during the evolution, together with the use of a Girsanov transformation.

Introduction

Many distinguished authors have made notable contributions to the stochastic Burgers equation, of which a small sample appears in our very short bibliography. It is not our purpose to review those contributions; it is perhaps appropriate that we underline here that which seems to us the novelty of our approach.

We start from the viewpoint of geometrization of inertial evolution initiated in [Arnold 1966] and systematically developed in [Ebin and Marsden 1970; Brenier 2003; Constantin and Kolev 2002], based on infinite-dimensional Riemannian geometry; the classical approach of [Ebin and Marsden 1970] is to use Banach-modeled manifold theory; inherent difficulties appear in the construction of exponential charts and in the introduction of appropriate function spaces. We circumvent these difficulties by using the viewpoint [Malliavin 2007] of Itô charts, Itô atlas; in short Itô calculus makes it possible to compute any derivative of a smooth function f on the path p of a diffusion from the unique knowledge of its restriction f|_p. Then no more function spaces are a priori introduced: the path of the diffusion constructs dynamically its canonical tangent space, built from the evolution of the system.