Preprint C a Class of Velocity Elds with Known Lagrangian Law

We introduce a large class of random velocity elds on the periodic lattice and in discrete time having a certain hidden Markov structure. The generalized Lagrangian velocity (the velocity eld as viewed from the location of a single moving particle) has similar hidden Markov structure, and its law is found exactly. As a result, the law of the trajectory of an individual particle is known in principle. The rate of convergence to equilibrium of the generalized Lagrangian velocity is studied in small numerical examples and in rigorous results giving absolute and relative bounds on the size of the spectral gap. The eeect of molecular diiusion on the rate of convergence is also investigated. 1 Background and introduction Let U = fU(x; t); x 2 R d ; t 0g be a stochastic process taking values in R d. We think of U as a random velocity eld and consider the motion of a particle whose position follows the velocity eld according to dX t dt = U(X t ; t); t > 0; (1.1) assuming suucient smoothness of U. In some cases we also make the particle subject to molecular diiusion; then the trajectory equation is dX t = U(X t ; t)dt + (X t ; t)dW t ; t > 0; (1.2) where > 0 and W is a Wiener process independent of U. One major goal of statistical uid mechanics is to determine the law of the particle's trajectory X t ; t 0; from knowledge of the law of the velocity eld U. A slightly simpler goal is to determine the law of the Lagrangian velocity process U(X t ; t); t 0, which is the particle's velocity under (1.1) or its drift under (1.2). (By contrast, U is called the Eulerian velocity eld.) These goals are rarely stated so plainly, perhaps because they have proven so far beyond our ability for so long.