We consider the general question of estimating decay of correlations for non-uniformly expanding maps, for classes of observables which are much larger than the usual class of Holder continuous functions. Our results give new estimates for many non-uniformly expanding systems, including Manneville-Pomeau maps, many one-dimensional systems with critical points, and Viana maps. In many situations, we also obtain a Central Limit Theorem for a much larger class of observables than usual. Our main tool is an extension of the coupling method introduced by L.-S. Young for estimating rates of mixing on certain non-uniformly expanding tower maps.
Finally, we mention a result which applies directly to certain non-uniformly expanding systems, rather than to a symbolic space or tower. Pollicott and Yuri ( @cite_12 ) consider a class of maps of arbitrary dimension with a single indifferent periodic orbit and a given Markov structure, including in particular the Manneville-Pomeau interval maps. The class of observables considered is dynamically defined; each observable is required to be Lipschitz with respect to a Markov partition corresponding to some induced map, chosen to have good distortion properties. This class includes all functions which are Lipschitz with respect to the manifold, and while some estimates are weaker than comparable results for H "older observables, bounds are obtained for some observables which cannot be dealt with at all by our methods, such as certain unbounded functions.
@cite_12
2289207600
Many methods for planning under uncertainty operate in the belief space, i.e., the set of probability distributions over states. Although the problem is computationally hard, recent advances have shown that belief-space planning is becoming practical for many medium size problems. Some of the most successful methods utilize sampling and often rely on distances between beliefs to partially guide the search process. This paper deals with the question of what is a suitable distance function for belief space planning, which despite its importance remains unanswered. This work indicates that the rarely used Wasserstein distance (also known as Earth Mover’s Distance ( ( EMD ))) is a more suitable metric than the commonly used ( L1 ) and Kullback–Leibler ( ( KL )) for belief-space planning. Simulation results on Non-Observable Markov Decision Problems, i.e., the simplest class of belief-space planning, indicate that as the problem becomes more complex, the differences on the effectiveness of different distance functions become quite prominent. In fact, in state spaces with more than 4 dimensions, by just replacing ( L1 ) or ( KL ) distance with ( EMD ), the problems become from virtually unsolvable to solvable within a reasonable time frame. Furthermore, preliminary results on Partially Observable Markov Decision Processes indicate that point-based solvers with ( EMD ) use a smaller number of samples to generate policies with similar qualities, compared to those with ( L1 ) and ( KL ). This paper also shows that ( EMD ) carries the Lipschitz continuity of the state’s cost function to Lipschitz continuity of the expected cost of the beliefs. Such a continuity property is often critical for convergence to optimal solutions.