Young Measures on Topological Spaces

With Applications in Control Theory and Probability Theory
2004. Auflage 2004
ISBN: 978-1-4020-1963-0
Verlag: Springer

Buch, Englisch, Band 571, 320 Seiten, Format (B × H): 164 mm x 244 mm, Gewicht: 1430 g

Reihe: Mathematics and Its Applications

Classicalexamples of moreand more oscillatingreal–valued functions on a domain N ?of R are the functions u (x)=sin(nx)with x=(x ,.,x ) or the so-called n 1 1 n n+1 Rademacherfunctionson]0,1[,u (x)=r (x) = sgn(sin(2 ?x))(seelater3.1.4). n n They may appear as the gradients?v of minimizing sequences (v ) in some n n n?N variationalproblems. Intheseexamples,thefunctionu convergesinsomesenseto n ameasure µ on ? ×R, called Young measure. In Functional Analysis formulation, this is the narrow convergence to µ of the image of the Lebesgue measure on ? by ? ? (?,u (?)). In the disintegrated form (µ ) ,the parametrized measure µ n ? ??? ? captures the possible scattering of the u around ?. n Curiously if (X ) is a sequence of random variables deriving from indep- n n?N dent ones, the n-th one may appear more and more far from the k ?rst ones as 2 if it was oscillating (think of orthonormal vectors in L which converge weakly to 0). More precisely when the laws L(X ) narrowly converge to some probability n measure, it often happens that for any k and any A in the algebra generated by X ,.,X, the conditional law L(X A) still converges to (see Chapter 9) 1 k n which means 1 ??? C (R) ?(X (?))dP(?)?? ?d b n P(A) A R or equivalently, ? denoting the image of P by ? ? (?,X (?)), n X n (1l ??)d? ?? (1l ??)d[P? ].

Zielgruppe

Research

Autoren/Hrsg.

Castaing, Charles

Raynaud de Fitte, Paul

Valadier, Michel

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Operations Research Spieltheorie
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis

Inhaltsverzeichnis

Generalities, preliminary results.- Young measures, the four stable topologies: S, M, N, W.- Convergence in probability of Young measures (with some applications to stable convergence).- Compactness.- Strong tightness.- Young measures on Banach spaces. Applications.- Applications in Control Theory.- Semicontinuity of integral functionals using Young measures.- Stable convergence in limit theorems of probability theory.