Nonlinear Markov semigroups and interacting Lévy type processes
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Nonlinear Markov semigroups and interacting Lévy type processes Vassili N. Kolokoltsov. N° 5932 June 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5932--FR+ENG. Thème NUM.
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(133) ω δ)J%)gKY_sZ ik^[cKMOSika]N3luqs qrO*vO*q>f[cKMOxIbÀ8O9acO9^syO1O*[\£ O9 O*^3[]KMOkO*^sO*azluqsp8ivZcb[cbrfb[\X3pMa]O9ZcO*a]f¤ brxM^MbÀD$O*O*a]kO*iv^sqy*_MO[cbr8ikO^s[\Zh£4iuO*½O*^ivy*acNik^sxM¬ fb[]®)bivlk^s^slux3qrq[]X$KMp1OSifYZwbrb^M[cbrOk[]OSbryluO9^sdfx3_sZwlu[c[ca]brikik^s^MvZ qiuX$ ykik¤¿^1luxIa]Xb[]bbiv^f^s[cO9lua]qrqrlvX$y¡[]p8b^Miv$Zcb[]psbvluO ac[ckbeyO9qr^MO9O*Z az¬loC;[]kik®)azZ beZ ¦M[cKMikOa []KMOy*lv_MZwa O<1iua] Zw[xIbeZ\Zc[]y*acac_sO*y[c[cO _MXazluq[ca]KMO9beZcZ _Mikq[1ZwyzO9Ksacluoazlolk[]ybiv[cO*^a]bDbeK*Z%O9bZhqrqr[c_sKsZ\O[]a]yluik[cO;^1x xIbY[]bXiv^sghluacqivp8p1ivifZcZwbb[c[cbrbrfikb^ [\X C luC [ <MnIO9xp1ivb^f[zZ .
(134) ·Y» · P X = R
(135)
(136) =
(137) /
(138) 1 P
(139) D P #
(140) . F
(141) 1 I
(142) 1
(143) 9
(144) 1 1 2%< . C sym (X k ). . 2%. 9 . . ,%. k. . . .
(145) . . . $. ". . . . . . . . *. . . . "$. . . . ∞. . ( . . /. k. ∞. (%. . ". . *. (-. 3 . ". . ). . . ,%. . . ,%. k. ∞. . :. ,%. k * . 2%. '. . !$". / (. +. (. . . . k. ,%. . +. . '. 1. 1. m. n i=1. . 1 x1. m xm. m. 0. d dt. t. t=0. i xi. . 0.
(146) . . Cc2 (X). n X. Ω. d . . . x = (x1 , . . . , xn ). !. . . )' . . . . ,%. *. . . n X. 1 aj (x)g(xj ) + (bj (x), ∇)g(xj ) + (cj (x)∇, ∇)g(xj ) 2 j=1 Z j + (g(xj + y) − g(xj ) − χ(y)(y, ∇)g(xj ))ν (x; dy). ωi δ x i g =. i=1. . ¬ C u®. P /
(147) j
(148) I P R . P F /
(149) R
(150) j
(151) ?
(152) = χ 2
(153) g ∈ C
(154) cF(X) IL ¬ C9®
(155) c
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(157) lk ac7v_M PN0 RO* ^f [] Z 4 luPS[ luN0YbO*qrO Xkfbb[hbrZ4ZcKMio£ ^[cKs£ luKM[ O9b^MO*[cvKMO*Oahqrbrl^M^MO9lkiva)^I¤¥_M^M^sO*fylo[cbr[cikbrk^sO lkq (Ag)(x) : G k l M ^ e b w Z M K ; O Z 2 d C → 7 R x Ag(x) ≥ 0 g c lu[ x [cKsO*^ Ag(x®SKslvZivacN ¬ ;C ®Sbraca]O9Zcp8O9y¡[]bvO*qrXiu)[]KMO0pMacivp1O9aw[]bO;Ziu Ag(y) b^2iu[]KMO*ap8ikbr^v[zZ y * >. . 4. +*. . . . . 2 *. . (. %. *. . i. '. $ . . 2%. âã&ä½âÖÞ.
(158) C;.
(159)
(160) . Pio£ qrO[ _sZyzKMZciY_sivyzZcKAO$[cl Kslups[ luacP[cb[cbrik^2iu4_M^Mb[\luXm^slkx2ZO9llvyzÐK lkN0bqrXmO9dfiu_s lknqrZZcikN^siYO$ikbr[c^´KAl^sik^M^IO9¤¥b^MvKYO*f1loiv[]acbKMviYO iIxm_M^siky [cbrik^sZ luχ^1x i = 1, . . . , n y*ik^sPZcO9df_MO9^f[cqrX2olu^sbrZcPKMO9Z$br^lm%^M£ O9χbb[]vK KY=1iv1acKMiYiIx Piulk^YχX2ik[cKMO9a$p8ikbr^f[ x qrO9£ lkbac[]qrK X lO;lk6=yzK i® _M^sH4y¡X[]bivq^sbrx^MluO9q lk¬ acb[\X Ω( g) Ωg b^seZik^Ay*ik^s^sO*xIvb[cluωbr[cikbrδ^1kO(luqr)qX_M=^sp1y¡[]ifbZwivb[c^ brkOAΩlu[ x b^©Ω[]KMMgOmO*^s=_sy*ZcO3_s(lkluqpMpMZwO9qrXY^sbZcω^MOk δ bl O )(χ <MnIg(x r b 0 N M p q r b 9 O Z k i avl O ) = 0 Ωg ≥ 0 9 O A x 1 p v i b f ^. [ k 9 O ] a c Z r b k i ^ . u i h c [ M K O k i s _ c a ] a * L []KMO*ivacZhO9N l y[civi0acivO;lkqqeyzluK a]XΩ£ IOivZc^MKsgOlkq∈ikqMIivC[zIlu[]brlk(X) ^sbZ%^0ac^MO9 iopM£©a]O9Zcl O*yz^fKs[]lulk[ca]brlviky¡^ [cO9¬ acC Db K9olu® [c brik^ikD[cKMOZw[ca]ik^M y*ik^sxIb[cbrik^1luqMp8ivZcb[]bYb[\X § .
(161) ·Y» ·
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(164) 9 1 i. n i=1. n i=1. n i=1. i xi. i. i. i. 2 c. l. n i=1. j. i. i xi. . j. i. i. i. i. . i. . .
(165).
(166) 1
(167) . ". . X. D. . . . Ak : D 7→ C sym (X k ). Ak g(x1 , . . . , xk ) =. k X. .
(168) I 1R
(169) / ,%< . . ,%. . ,%. [ak Πj1 (x1 , . . . , xk ))g(xj ). j=1. 1 + (bk Πj1 (x1 , . . . , xk )), ∇)g(xj ) + (ck Πj1 (x1 , . . . , xk ))∇, ∇)g(xj ) 2 Z + (g(xj + y)g(xi ) − χ(y)(y, ∇)g(xj ))νk (Πj1 (x1 , . . . , xk ); dy)],. ¬ C @v®. 1 P ? 1
(170) Π Rd 1
(171) 1
(172) F R JP = j
(173) l
(174) P ν j Ijl
(175) 1R . P F 9/
(176) R 1 /
(177) //
(178) R = χ I
(179) ck
(180) L ¬ C;®
(181) P 1 = 1 a , b , c , ν
(182) / R R P 1 P R F
(183) 1 # 1
(184) R
(185) D R x , . . . , x k k
(186) k P kR
(187) 13
(188) x 2 k 1
(189) 1
(190) F A : D 7→ C sym (X k )
(191) 9 P R I
(192) ? / >. . (. ( . . ,%. ". . 9 . *. . . !. >. . . . . 2 . . . . . . 2%. . %. (%. . . ,%. . . *. 4. !"1*. . ". . . ( . 2%. . ( . 4 . ". . ,%. k. Ak g(x1 , . . . , xk ) =. k X. ¬ C9~v®. ak (Πj1 (x1 , . . . , xk ))g(xj ) + Bk g + (x1 , . . . , xk ),. 1 1
(193) /
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