Download An introduction to probability theory and its applications by William Feller PDF

By William Feller

ISBN-10: 1861873743

ISBN-13: 9781861873743

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Example text

Un r´ esultat de convergence. — Pour toute fonction f : x → a + bx, Pt f − Eβ (f ) 2 = e−t |b|, et f 2 = a2 + b2 Ainsi, pour toute fonction f et pour tout t 0, Pt f − Eβ (f ) 2 1 2 . e−t f 2. (1) Remarquons que cette in´egalit´e est satur´ee pour les fonctions f (x) = bx. 3) s’´ecrit d´ ef. 1 2 E β (f ) = |f (−1) − f (1)| = b2 = −Eβ (f Lf ) . 1 du chapitre 1, pour toute fonction f , Varβ (f ) Ainsi, pour la famille d’op´erateurs (Pt )t −Eβ (f Lf ) . 1. — Pour toute fonction f , pour tout t suivantes sont v´erifi´ees : (i) Pt f − Eβ (f ) 2 e−t f 2 , (ii) Varβ (f ) −Eβ (f Lf ).

Notons qu’avec la notation matricielle, cette propri´et´e est ´evidente. Copyright (c) Soci´ et´ e Math´ ematique de France. 2. 3. Propri´ et´ es de sym´ etrie et d’invariance. — Soient f, g deux fonctions de E, et t 0. On a Eβ (f Pt g) = Eβ (gPt f ) . Cette propri´et´e de sym´etrie implique en particulier l’invariance de l’op´erateur Pt sous la mesure β : pour toute fonction f et pour tout t 0, Eβ (Pt f ) = Eβ (f ) . Pour le voir, il suffit d’appliquer la propri´et´e de sym´etrie `a g = 1I, o` u 1I d´esigne la fonction constante ´egale ` a 1, puisque Pt 1I = 1I.

La premi`ere de ces notions sera ´etudi´ee plus en d´etail au chapitre 4. Copyright (c) Soci´ et´ e Math´ ematique de France. 8. 2. — On dira qu’un semi-groupe (Pt )t 0 est ultracontractif de Lp (µ) dans L∞ (µ) si Pt est un op´erateur born´e de Lp (µ) dans L∞ (µ) pour tout t > 0, c’esta-dire s’il existe une constante c(t) telle que ` Pt p→∞ c(t). On dira qu’il est imm´ediatement hypercontractif si pour tout p ∈ [2, ∞[ et tout t > 0, il existe une constante c(p, t) telle que Pt 2→p c(p, t). 3. — Sur Rn , pour une fonction Φ donn´ee telle que exp(−Φ) est int´egrable, on consid`ere la mesure de probabilit´e dµ(x) = Z −1 exp(−Φ(x))dx, de d´ ef.

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