Stochastic flows of diffeomorphisms

Abstract

Publisher Summary This chapter discusses the stochastic flows of diffeomorphisms. The solutions of an ordinary differential equation (O.D.E.) on a manifold define a one-parameter subgroup of the group of diffeomorphisms called a dynamical system; solutions of a stochastic differential equation (S.D.E.) define a continuous random motion of diffeomorphisms called a stochastic dynamical system or a stochastic flow of diffeomorphisms. It is called a Brownian motion on the group of diffeomorphisms as it is a continuous motion on the group with independent increments. The chapter discusses similar problems in more general framework of continuous random motions on the group of diffeomorphisms than Brownian motions. A semi-martingale on the space of vector fields called the velocity field can be associated. A semi-martingale on the space of vector fields generates a semi-martingale on the group of diffeomorphisms through S.D.E. so that it is the velocity field of the latter. The chapter establishes a one-to-one correspondence among a class of semi-martingales on the group of diffeomorphisms and a class of semi-martingales on the space of vector fields. A semi-martingale on the group of diffeomorphisms is a Brownian motion if the corresponding semi-martingale on the space of vector fields is a Wiener process.