We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł p-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier-Stokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic Navier-Stokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1

As a Generalization to [37] where the dimension-free Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic p-Laplace equation in Hilbert space.

. This paper analyzes the behavior of a variety of tracking algorithms for single layer threshold networks in the presence of random drift. We use a system identification model to model a target network where weights slowly change and a tracking network. Tracking algorithms are divided into conservative and nonconservative algorithms. For a random drift rate of fl, we find upper bounds for the generalization error of conservative algorithms that are O(fl 2=3 ) and for nonconservative algorithms that are O(fl). Bounds are found for the Perceptron tracker and the least mean square (LMS) tracker. Simulations show the validity of these bounds and also show that the bounds are tight when fl is small and the number of inputs n is large. These results show that the Perceptron tracker and the LMS tracker can work well in slowly changing nonstationary environments. 1 Random Drift Applied to Neural Networks 2 1 Introduction In this paper we study the performance of learning algorithms for s...

Abstract. In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition. 2000 MR subject classification: 60J45