Seminaire de Probabilites XXXV (Lecture Notes in by J. Azema, M. Emery, M. Ledoux, M. Yor

By J. Azema, M. Emery, M. Ledoux, M. Yor

Researchers and graduate scholars within the idea of stochastic procedures will locate during this thirty fifth quantity a few thirty articles on martingale concept, martingales and finance, analytical inequalities and semigroups, stochastic differential equations, functionals of Brownian movement and of Lévy strategies. Ledoux's article includes a self-contained advent to using semigroups in spectral gaps and logarithmic Sobolev inequalities; the contribution by way of Emery and Schachermayer contains an exposition for probabilists of Vershik's concept of backward discrete filtrations.

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Then conditions a and b below are equivalent and conditions c. d. e. and f are equivalent. (a) f is continuous at x . (b) For every neighborhood V of f(x) in Y. there is a neighborhood U of x in X such that feU) c V • (c) f is continuous. (d) For every closed subset C of Y. f- 1(C) is closed in X . (e) If B is a basis for the topology of Y. then for every V E B • f- 1(V) is open in X • (f) For every open subset V of Y. f- 1 (V) is open in X . Proof: (a) - (b): If f is continuous at x and V is a neighborhood of f(x) in Y.

Thus the only critical point of g is O. Since h is constant in a neighborhood of 0, the critical point must be nondegenerate of index A. 0 EXERCISES 1. 2. Determine the rank of the function f : m2 - m2 defined by f(x,y) = (x 2 + 2xy + y2, xy2 + x2y) at each point of its domain. Is f a diffeomorphism? The transformation (r,e,

4. 43 Some Advanced Calculus Sometimes one can use the inverse function theorem to deduce that a function is a (global) diffeomorphism. Example Let g : (-w/2,w/2) --+ IR be the Cm homeomorphism given by get) = tan t. Since g'(t) = sec 2 t ~ 0 for all t E (-w/2,w/2), by the inverse function theorem, g-l is also Cm • Thus g is a diffeomorphism. COROLLARY 3. Let U be an open neighborhood of 0 in IRm and let f : U - IRn be a Cr function having rank m at 0 (so m ~ n) • Suppose f(O) = O. Then there is a Cr diffeomorphism g of a neighborhood of 0 in IRn onto another such neighborhood such that Case I.

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