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Constructive quantization: approximation by empirical measures

Steffen Dereich, Michael Scheutzow, Reik Schottstedt (2013)

Annales de l'I.H.P. Probabilités et statistiques

In this article, we study the approximation of a probability measure μ on ℝ d by its empirical measure μ ^ N interpreted as a random quantization. As error criterion we consider an averaged p th moment Wasserstein metric. In the case where 2 p l t ; d , we establish fine upper and lower bounds for the error, ahigh resolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions....

Controllability of three-dimensional Navier–Stokes equations and applications

Armen Shirikyan (2005/2006)

Séminaire Équations aux dérivées partielles

We formulate two results on controllability properties of the 3D Navier–Stokes (NS) system. They concern the approximate controllability and exact controllability in finite-dimensional projections of the problem in question. As a consequence, we obtain the existence of a strong solution of the Cauchy problem for the 3D NS system with an arbitrary initial function and a large class of right-hand sides. We also discuss some qualitative properties of admissible weak solutions for randomly forced NS...

Convergence of sequences of iterates of random-valued vector functions

Rafał Kapica (2003)

Colloquium Mathematicae

Given a probability space (Ω,, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates f ⁿ : X × Ω ℕ → X defined by f¹(x,ω) = f(x,ω₁), f n + 1 ( x , ω ) = f ( f ⁿ ( x , ω ) , ω n + 1 ) , and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).

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