Ergodic Theory of Infinite Dimensional Systems with Applications to Dissipative Parabolic PDEs

This paper concerns the ergodic theory of a class of nonlinear dissipative PDEs of parabolic type. Leaving precise statements for later, we first give an indication of the nature of our results. We view the equation in question as a semi-group or dynamical system St on a suitable function space H , and assume the existence of a compact attracting set (as in Temam [15], Chapter 1). To this deterministic system, we add a random force in the form of a “kick” at periodic time intervals, defining a Markov chain X with state space H . We assume that the combined effect of the semi-group and our kicks sends balls to compact sets. Under these conditions, the existence of invariant measures for X is straightforward. The goal of this paper is a better understanding of the set of invariant measures and their ergodic properties. In a state space as large as ours, particularly when the noise is bounded and degenerate, the set of invariant measures can, in principle, be very large. In this paper, we discuss two different types of conditions that reduce the complexity of the situation. The first uses the fact that for the type of equations in question, high modes tend to be contracted. By actively driving as many of the low modes as needed, we show that the dynamics resemble those of Markov chains on R with smooth transition probabilities. In particular, the set of ergodic invariant measures is finite, and every aperiodic ergodic measure is exponentially mixing. The second type of conditions we consider is when all of the Lyapunov exponents of X are negative. As in finite dimensions, we show under these conditions that nearby orbits cluster together in a phenomenon known as “random sinks”. The conditions in the last paragraph give a general understanding of the structure of invariant measures; they alone do not guarantee uniqueness. (Indeed, it is not the case that for the equations in question, invariant measures are always unique;