Young Measures on Topological Spaces: With Applications in Control Theory and Probability Theory

ISBN-13: 9781402019630 / Angielski / Twarda / 2004 / 320 str.

Charles Castaing; Paul Raynaud de Fitte; Michel Valadier

Opis

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 u on ? xR, called Young measure. In Functional Analysis formulation, this is the narrow convergence to u of the image of the Lebesgue measure on ? by ? ? (?, u (?)). In the disintegrated form (u ), the parametrized measure u 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(XA) 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? ].

Recenzje

From the reviews:

"This book presents a wealth of results on Young measures on topological spaces in a very general framework. It is very likely that it will become the reference and starting point for any further developments in the field." (Georg K. Dolzmann, Mathematical Reviews, 2005k)

Spis treści

Preface. 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. Application. Applications in Control Theory. Semicontinuity of Integral Functionals using Young Measures. Stable Convergence in Limit Theorems of Probability Theory.