A maxiset approach of a Gaussian white noise model
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Texte intégral
(2) Universit´ es de Paris 6 & Paris 7 - CNRS (UMR 7599). ´ PREPUBLICATIONS DU LABORATOIRE ´ & MODELES ` ´ DE PROBABILITES ALEATOIRES. 4, place Jussieu - Case 188 - 75 252 Paris cedex 05 http://www.proba.jussieu.fr.
(3) A maxiset approach of a Gaussian white noise model C. CHESNEAU. JUIN 2004. Pr´epublication no 920. Laboratoire de Probabilit´es et Mod`eles Al´eatoires, CNRS-UMR 7599, Universit´e Paris VI & Universit´e Paris VII, 4, place Jussieu, Case 188, F-75252 Paris Cedex 05..
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(104) a gT [w b X I YQ /O NO NO ST [w X W^ X ' {K ~ fZ[VWQYX9}cT±KN[[^WV_}1KUTSVWONjON]JX'KØ[~ÑÙ,kVnTSw V = √1 ² VWj TSw[XVWj[X/pIKN^_V_T è Ù zmé NV_lmXp[akTSw[X n n λ [QYORGXQTgV_X9a è é KNj[Z è Ú é ~ w[X9QYXcORQYXRh3KN^W^P}ORj[Z[V_TSVWORj[aqKNQYXagKUTSVWaY£IXZTSO¢KN[[^_'wIXORQYX9 | ~ r VnTgw E = ξ KNj[ZTgw[XwIKNQYZ TSw[QSXaSw[OR^WZ[VWj['XaYTSVWK TgORQ è rtsRé aSO,TSw[X[QYO/ON$OU$w[X9ORQYX9 Û{~ r V_a}9OR,I^_XTSX9Z~ m I " ÈIµ¿P¹S¸«¿[µ¹#Â:¶ : , ='ÃW»¯'µ &9Ê3ÃÌ ÀNÇWµc¾I¸ sup ur−q ω m ({λ ∈ Λ; |βλ | ≤ u|vj |}) < ∞ u>0. · ÈIµc¹Sµ m = P |β |pkψ kp Å ∀S ∈ Λ Õ¨×>µ«µ ¸1ÈIµ¾/Å¢µ5¸U¹SµgÅv´¯µ¹YÎN¶RÀÄgÈ[ÀR¹«Ã:ÀN¾3Å'ÀN¾Pº λ λ p λ∈S qÃÄgÀN¹Yº ω: (S) @=RÕ ikaSV_jI,Tgw[V_akQYX9KNQYPh5VnTkVWakX9KNaY¬TSOaSXXTSwIKUTORQkKN^W^aSX/p[XjI}X9a v = (v ) KNj[Z z = (z ) j j∈N aSp[}«w¤TgwIKUT 0 < v ≤ z qX¯wIK1lmX¢Tgw[X¯X9MGXZ[ZIV_j[ l h ST Iw UK T ± a w jnV j∈N q X (ξ ) ⊆ l (ξ ) j j (r,p,∞) z (r,p,∞) v ~ Iw _V ¢ a _V a
(105) NK j T ST [w X [M _V R R 9X Ya ¢ T ' K { WV Sa cX ¢ T WV j è r Ú é $h q X ¢ [p Ya ¢ T «} [w /O RO Sa X gT [w X Sa NK _^ W^ X Ya ¢ T 9} /O X 9} _V 9X j ST a mj cX [ UK } ST _^ Iw UK T q X Iw 1K ml b X [Z RO [j X _V j gT [w b X 9} RO [j Ya ST YQ Ip c} ST WV RO j è tr Ru é ~ 9" ( & 0 + ,. © O WV _^ W^ [p Ya ST SQ UK gT X ST [w WV a ' X ST [w /O Z h W^ X T Ip a 9} RO [j Sa _V IZ X Q ST [w X Ip [j c} gT _V RO j g :]0, 1] → [1, +∞[ Z[Xc£5j[XZªM V_TSw 1 > σ > 0 ~o
(106) ^_X1KNQY^nmh g MPX9^WONjIRa
(107) TgO L2([0, 1]) KUjIZqXbwIK1lmXRç g(x) = 1 σ. x2. dj = 2. j 2. s. Z. L 2j. 0. 1−σ. σ 1 L 2 2j 2 . dx = √ σ x 1−σ. ml X9QYV_£IXa è r Ø é M X9KNQY¬Ø[~ r hPVn qXvp[aSVWj[w[XORQYX9 Ø[~ r kVnTSw m = √L 2j TSwIXj j 1−σ qXbORM[TgKNV_jTgw[XbX/p[V_lNKU^WX9j[}X è r é p[j[Z[X9Q
(108) TSwIXb}1KU^WVWM[QSKUTSVWORj ν = σ ~ 0 2 ¡j TSw[X¬aSX9}ORjIZ 5KNQT,ONTSw[VWa,IKNPX9QhEqXKN^_
(109) K9{a¯}9ONjIaYVWZ[X9Q¢TSw[X¬aYT«K TgV_aYTSVW}1KU^
(110) ,O3Z[X^ è rté KNj[Z TSw[X K {V_aSXT]PORVWjT]ONPl3VWXcMIp{TEqXVWjlmX9aTgV_mKUTSXkK±ZIVn >XQYX9jT]aYTSQSKUTSX9N m
(111) XX#{5KUjIZ f ORjK±}9OR'IKN}cTg^n aSp[[GONQYTSX9Z
(112) K9lmX9^WXcTMIKNaSVWa
(113) KNQYPX9Z&MK,}X9QTgKNVWj&/j[Otjp[j[}TSVWORj T KNj[ZqXaYTSp[Z{TSwIXbGXQORQ y KNj[}XONGTSw[XKNaSaSO/}9VWKUTSX9Z'wIKNQYZ,TSw[QYX9aSw[OR^WZ[V_jIbXaYTgV_KUTSORQEp[j[Z[X9QEaSOR,Xd p[}«RX9j[w[ORp[[TE}9ORj[Z[V_TSVWORj[a ORj g ~ 3VWj[}9X. #"$. σ+1 2. σj. 22. ãä Þ¬á¢ã ¬ß %. 2. , 4. 2 0. .. å ,ß Þâ. 8. .
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(116) XVWRwTSa è Ù rté p < q p q wIKNaMPX9XjVWjTSQSOZIp[}X9ZMd pI}«RX9j[w[ORp[{TVWj r Ú s Ù é KNj[ZBVWZ[X^_¤p[aSX9ZKU TgXQ
(117) KNQSZ[abVWjBTSwIX,}9ONj{y TSXc3TONo
(118) KN^WZ[XQNj{yP3R¢p[j[Z Tgw[XORQm~w[X}^KNè aSaSXaONd p[}#RXj[wIORp[{TkqX9V_RwTga}«wIKNQSKN}TSX9QYV
(119) 9XvTSw[X MPORp[jIZ[XZIj[XaSaqONLaSOR,X®VWjTSX9NQgKN^PORPX9QSKUTSORQYaqORj L p aS5KU}9X9aÍ^WVWRXkTSwIX KNQYZ[yJVnTSTS^_XqO/O/ZK {VW'KN^ ORPX9QSKUTSORQORQ
(120) TSwIX kVW^WMPX9QTTSQSKNj[aYORQY&~]xIORQ}9ON'[w^WXTSX±Tgw[XORQmhIaYX9X±Tgw[XbMGO/OROU*/TgXVWj è r ÚRÚ |mé ~ (*) 9 [+ )-, . xIQYOR K,/j[Otjp[j[}TSV_ORj KNj[Z K¯}9OR'IKN}cTg^n¬aSp[[GONQYTSX9Z¬K9lmX9^_XT T : [0, 1] 7→ [0, 1] MIKNaSVWa ξ hRqXZ[X9j[ONTSX ξ T = {ψ (T (.)); λ ∈ Λ} TSw[X}9OR,5KU}TS^_¢aSp[[GORQTSX9Z¯
(121) KNQYGXZ,
(122) K1lRX9^WXcT]MIKNaYVWa λ KNZIKN{TSX9ZTSO [0, 1] ~. #" !. &. . 0,. &. * , *. kX9QYX±qX±VWjlmX9aTgV_mKUTSX±TSw[XbRXOR'XTSQYVW}9KN^[QYORGXQTgV_X9aON@TSw[XbMIKNaSV_a ξ T w[VW}«w¬TSQSp[^n¬Z[XGXj[Z&ORj¬TSw[X
(123) KNQY[VWj[,:KN}cTSORQ T ~ ÀN¾Pº B, D ·͸þIÂÓµ¹#Ì ÀNÇ »'¸ R Õ µ»9ÀN¶ , + + ) , ) - @ ) t@% A@µc Hp 1<p<∞ ÂÈ[ÀN À¢ÁµgÀt»Ê3¹YÀmÔÇWµ#Ê3¾GÄcÂøU¾ T : B 7→ D ÌUµ¹#à .͵«»ÂÈIµ]¿P¹S¸«¿Iµc¹#¶ H à ®ÀN¾Pº'¸U¾IÇ ¶Ã k ÈIµ¹SµbµYÆmÃ_»Â»±À ÁµgÀt»cÊ3¹YÀmÔ9ÇWµ #Ê3¾GÄcÂøU¾ S : D 7→ B ÀN¾PºÀÁµSÀt»Ê3¹YÀmÔÇWµ
(124) ¿[¸1»ÃÂ:ÃÌUµ #Ê3¾Gp ÄcÂøU¾ w ∈ A (D) »cÊIÄgÈ& ÈIÀN ¼ p À m Õ µ ¢ À N ¾ P º À m Õ µ 9 Õ • S(T (x)) = x T (S(x)) = x L¸U¹ÀNÇÇÁµgÀt»cÊ3¹YÀÔÇWµ¿[¸1»ÃÂ:ÃÌUµ3#Ê3¾GÄcÂøU¾ z Å R z(T (x))dx = R z(x)w(x)dx • B D [w ± X RO _^ W^ tO _V Ij ¯ ST [w 9X RO YQ 9X ê Sa [w tO
(125) a ST Iw b X _^ WV [j PM X T q X 9X j d [p #} N 9X [j [w RO [p [
(126) T q 9X _V R w gT
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(128) NK YQ P 9X Z
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