Coalescent tree based functional representations for some Feynman-Kac particle models
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Coalescent tree based functional representations for some Feynman-Kac particle models Pierre Del Moral — Frédéric Patras — Sylvain Rubenthaler. N° 5969 Août 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5969--FR+ENG. Thème NUM.
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(86) tvbdfNrf]²1fpqwde5ws¨³3 n1G¹rN¸ w tvbdf;zft{}® 1zt{} ®f5dbqpsr¯{}rN´ tbdf<r¯f egdsf®}rgzfNdzf<r¯fNw;tr(tbdf¾s{}r¯tzv{¥©dxdt{w¦«j1z¯tv{®¥f f¼;®¥¼q{wd{w w ©1r¯;z©1{¥wdegf<s{xde´º¹ { tvb¦;©1r¯tv®¥f<rzfN®tf<!t StfNw;tv{®s«¬x1w1·tv{¥;w1r G ´tv¶q{¥wd3¼®xdfNrK{w [0, 1] ¸ w-tb1{rº w@tf µqt<´tbdfHzt{} ®f X fN¼®¼fNrzs{wdjtv%tu¹º(rf1zvtf egf<vbHwd{}r¯e5rN¸"² {zrutv®¥p;´{¥t e-¼fNr"«¬zve r{ tvf x ∈ E ´tv(wdtbdfz x ∈ E N zs{wdPtf®fe-fNw;tvzp(tzw1r{¥t{w1r ¸ a$bdfNw_´d{¥t{}r©1rzv©SfN]¹ {¥tb¾-dz;©1©1{¥®{¥tp 1 − G (x ) ´Hw1 d®} f<{w¾wxsµs{¥®{zp] fe-ftfzvpruttf M (x ) r@,tdx d b N f z. ¹ {}rf { tºzvfe5{¥w1r{¥w5tvbdfjregfBr{ tvf¸" ª«»¹ºfB®¥ft T ©Sf tvbdfBzvw1s;e ©1r¯;zdt{wtv{egf;´;tbdfNw/{ tº{}rwdtºs{ / xd®¥t X =c t5bdfN¶/tb1t wH η (f ) = E[f (X ) | T ≥ n] ; ¸ γ (f ) = E[f (X ) 1 ] wtv{¥e-f3bd;eg;fw1fxHr
(87) rf tt{wd;r G = G wH M = M ´1¹ºf%b1G¼f 0. q En. 0. n. n. n−1. n. n. xn ∈En. n. n. xn ∈En. n. n. n. n. n. n. n. n. n. n. 0≤k<n. n n≥0. n. n−1. 0 0. n. k. 0 n. n. 0 0. n. 0 n. 0 n. 0 n. 0 n. n. n. n. 0 n. c n. n. n−1. n−1. k. n. n. n. n c n. b. n n≥0. 0. 0 n. n. n. n. 0. n−1. n. n−1. n. n. n. n. n. n. n. c n. T ≥n. n. n. n. n. n. c n. n. γn (1) = P (T ≥ n) ∼ e−λ. n. a$bdfS@r¯{¥tv{¥¼f¾w1r¯tvw;t {r/±e-fN;r¯x1zf «jtvbdf>r¯tzvfwdtvb w1¦tzdd{wd f uEfN tvr5«jtb1f¾©1r¯tv; ®f¸ 4
(88) p¦±®fe-e5±« zsb1w r´ λ {w1 {}sfλ¹ {¥tb[tvbdf±®;z{¥tb1eg{} _p;dxdwd¼ fµsSwdfNw;t λ «tb1ftzw1r{ tv{¥;w ;HfNzvt;z Q(x, dy) = ¸ ¤ bdfNwdf¼fNzB{ tjf µs{r¯tr´dtvbdf(zvzf<r¯Sw1s{wd-f{fNws«¬xdw1 t{w h « Q zvfdzvfNrfw@tvr$tbdf(zvxdwH r¯tvtf«tvbdf G(x) M (x, dy) Sfztvz Q ¸ a$bdf<r¯f-²1fpqwde5ws¨³3(e-ssf®}r%w®}r©Hf-zfN;zvsf<>r%]w1txdz®t{e-f-d{rv zvf tv{V<t{w§«ºw;tv{w@xdxHrjt{egf-1z¯¨ t{} ®f/©1rzvst{w±egdsf®}rrvr¯d {}tf<§¹ {¥tb r¯;e-f/nsbdzv¿ss{wdfNz3SfztvzN¸]a ©Sf/e-zvf-dzvfN{rf´ ®¥ft(x1rrxddS@r¯f-tbHt . 0. Ú Ú Îv5à. .
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(91) ( . zvf1zf<r¯fNw;t$tvbdf3tzwHr¯{¥t{wHr$«5 w@t{w@xd;x1r$tv{¥e-f(k>z¶¼/dzvs f<rr$¹ {¥tb¾{ws½1wd{¥tf<r¯{e5® ;fwdfNz¯¨ t;z ´1sxdzv{¥w1--rbd;z¯t Sfzv{¥d]« tv{¥e-f ∆ rd{wtvbdfr¯fNw1rfjtvb1t. M (x, dy) = M ∆ (x, dy) L. ¤ 1 £ ∆ M (f )(x) − f (x) ∆→0 ∆. L(f )(x) = lim. «¬zr¯x /{¥fNw;t®p zfNxd®}z]«¬xdw1·tv{¥;w1r f w E ¸ _ft x1r ®r¦r¯x1dS;rf>tbHttvbdf±StfNw;tv{®j«¬xdw1·tv{¥;w1r b1G¼f>tb1f§«¬;ze ´q«¬;z re-fPwd;wwdfN;tv{¼fjfNwdfzvp-«¬xdw1 t{w ¸" wtvbd{}r ;r¯f;´q{¥t {}r$zfN;s{®¥p/bdfN¶fN]tbHt$tb1f G(x) = G (x) = e {¥ws½Hwd{ tvfNr{¥e5®_fw1fzt;z
(92) « tvbdf31zt{} ®f3©1rzvst{;w/tzwHr¯{¥t{w1r V −V (x)∆. ∆. Q∆ (x, dy) = e−V (x)∆ M ∆ (x, dy). {r {¼fw ©@p/tvbdf(nsvb1zsd{¥wd;fz$SfNzvt;z. ¤ 1 £ ∆ Q (f )(x) − f (x) ∆→0 ∆. L(f )(x) − V (x)f (x) = lim. iw1sfNz]zvfNrw1©d®¥p ¹
(93) fN¶ rvrxde-st{w1rN´
(94) tbdf>H{z {w1{sf>zvfNrSfN·tv{¥¼fN®¥p¦¹ { tvb tbdf>t; fN{¥;fwq¼®xdf´ w1 tb1f zvxdw1]r¯tvtvfj«_tvbdfnsvbdzvss{wd;fzº;Sfztvz L = L(λ,−h)V ¸a$bdfPfNr¯t{e5tv{¥;w]« tbdf<r¯f%rSfN tz®E|@x1w;tv{ tv{¥f<r
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(96) zf< fNw;t®p zf< f{¼fN g®¥t «"t¯tvfw@t{w {wegdsfzvwwqxde-fzv{N®»dbqpqr{}rN´dwH |;xHw;tvxde&bdfe-{}rutvzp;¸ wHtx1zv®Y´Hw1¾ e-e-w¶fNp{sf<-{w¾e-;r¯t«tvbdf(k¾;w;tf¨
(97) z®5ddzvGµq{e5tv{¥;we-sdf®}r {}r t]r¯{e(xd®}tvf5Sdxs¨ ®t{w «"¹
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(99) ;z 1zt{} ®fNr$e-{e-{¶q{¥wd-tb1f3f¼;®¥xdt{w «Kg1z¯tv{®¥f%{w¾w©1rzv©d{¥w1e-f<s{¥xde¸ºa$bdfNrf%e-sdf®}r w ©Sf({¥w@tfzvdzvf tvfN¾;rB/w1tvxdzv® e-fNw½1fN®¾1z¯tv{} ®f(ddzvGµs{¥e5tv{¥;w «"tvbdfd{r¯tzv{¥©1xst{w n1¹ η ¸ ¤ f(fw1tvbd{}rjrfN·tv{w ¹ { tvb¾gdzvfN{rf3sfNrv zv{st{w « tvbdfNrf3wdwd®{wdfNz$e-sdf®}r,¸ 4
(100) p5tvbdfk>z¶¼51z;Sfztp]w1]tvbdf3e(xd®¥t{d®{Ntv{¥¼f(rutvzx1 txdzvf « ; ¸ý ´d{ t{}r$f<r{¥®p]vb1fN¶fN]tvb1t tbdf nHG¹ (η ) rt{}ru½1f<rºtvbdf%«¬®®G¹ {wdgf<|@x1tv{¥;w ; ¸ η =Φ (η ) a$bdfgtzw1r¯«¬zve5tv{¥wHr Φ : P(E ) → P(E ) zf-sf ½HwdfN§«¬;z3wqp>1{z (η , f ) ∈ (P(E ) × B (E )) ;r «¬®®¥¹r η (Q (f )) ¹ {¥tb Φ (η )(f )= Q (x , dx ) = G (x ) × M (x , dx ) η (Q (1)) htv{fPtvb1t tb1f3f¼®xstv{¥;w rfe-{zvxd (Q ) « tvbdf3xdwdwd;ze5®¥{ Vf<s{}rutvz{©dxstv{¥;w n1¹[{}r {¼fw ©@p ¹ { tvb Q = Q Q ; ¸ ∀0 ≤ p < n γ =γ Q w1!tvbdf xdwdwd;ze5®{ VfN°²dfNpqwde-ws¨³3/e-fNrxdzvfNr rvtv{ «¬p ¸ |@xd{¼®fw@t®p´Ktvbdfp Nw°©Sf f µsdzvfNrvr¯fN]{w tfzve5r$«tbdf n1¹ (η ) ¹ {¥tb tvbdγf%«¬zve(xd®}γf (f ) = γ (Q (f )) ¹ {¥tb γ (1) = Y η (G ) ; 0¸ < γ (f ) = η (f ) × γ (1) «¬zBw@p f ∈ B (E ) ¸$ ftxdzvwd{wd-tv5tbdf(db@psr{N®_{w;tvfzvdzftvt{w_´dtbd{}r1zsdx1·t«¬zve(xd®5rbd¹r$tb1ttvbdf _p;dxdwd¼ f µsSwdfNw;t λ {r(zvf®}tvfN±tv¾tvbdf rp@e-stvtv{5e-f<w ®¥;;zv{¥tbde+¼®¥x1f/« tbdfStvfw;tv{® G = G ¹ { tvb zvfNrSfN·t(t>tb1f egf<rxdzvfNr η 1 X ; ¸= log η (G) ' −λ n k¾x1b5e-zvf {}rKtvzvxdf¸ w5tbdfjNrf ¹ bdfNzf tb1fb1e-;fwdf;x1r"tzw1r{ tv{¥;w1r M = M zvf zfN¼fzr¯{©d®f ¹ { tvb5zvfNrSfN tKtv(re-f egf<rxdzvf ´;tb1fB;z;xdw15r¯tvtfBzvfdzvfNrfw;trKtvbdfj®¥{e-{ tv{¥w1½dµsf<5S{w;t
(101) e-fNrxdzvf«»tvbdfPs{r¯tzv{©dxst{w n1G¹ η rq{¥w/tbdfPrfw1rf tb1t$«¬;zµwqp/©SxdwHsfN egf<rxdz©d®fj«¬xdw1·tv{¥;w f ´d¹ºf3b1<¼;fjtvb1t 1 X ; ¸ η (Gf )/η (G) ' µ(hf )/µ(h) V. n. n n≥0. n+1. n+1. n+1. n. n. n. n+1. n. n+1. n+1. n. n+1. n+1. n+1. n+1. n. n+1. n. n+1. n. n. n+1. n. b. n. n+1. n+1. p,n 0≤p≤n n. p. p,n. p,n. n. n. n. p,n−1. n−1. n. n. n. p 0≤p≤n. n. n. n. n. n. p. n. p. 0≤p<n. n. b. n. n. n. p. 0≤p≤n. n. n. n. p. p. 0≤p≤n. ¬ÙÚ ãØ ). ().
(102) .
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(105) ! " $#% & (') * +,- .',/- 0 Z\[. <. \_( .e k]_ba %c2\i a!k]_ % 'GcNa!ki. ! dfe . ". . wHtx1zv®ºe-f<w ½1fN®°1zt{} ®fe-ssf® rvrs {}tf< ¹ {¥tb°tvbdf wdwd®{wdfNzg²1fpqe5ws¨³3)n1¹ ; ¸ý {}r(tvbdf k>zv¶G¼/b1{w ξ = (ξ ) ¹ {¥tbfN®¥fNegfNw;tzvp-tvzvw1r{ tv{¥;w1r$df ½1wdf<]«¬;zwqp F ∈ B (E ) ©qp ³ ´ ³ ³ ´´ ¹ {¥tb m ³ξ ´ = 1 X δ E F (ξ ) | ξ =Φ m ξ (F ) (N ) n. i,N 1≤i≤N n. N n. N n. (N ) n−1. (N ) n. ⊗N. (N ) n−1. n. b. N n. N. (N ) n−1. N n. N. i,N ξn−1. EnN. ¨9¼®xdfN ¸ . ; ;. wtvbdfz tfNze5rN´d{¼fw tbdf( ;ws½1x1zvt{w ξ tzwd¶ (n − 1) ´dtvbdf31z¯tv{} ®f3rpqr¯tfNe ξ tzvwd¶ n ´H ;w1r¯{}r¯tvr$« N {¥w1dfSfw1dfw;tKwH3{sfNw;tv{N®®¥p3s{}rutvz{©dxstvfNzw1s;eD¼zv{©d®¥f<r»¹ {¥tbe-e-w(s{r¯tzv{©dxst{w Φ ³m ³ξ ´´ ¸a$bdfº{wd{ tv{® wd½1xdztv{¥;w ξ ;w1r¯{}r¯tvrº« N {w1sfSfNw1sfw@t w1/{sfNw;tv{N®®¥p/s{}rutvz{©dxstvfN]zvw1se ¼z{}©1®¥f<r¹ {¥tb s{r¯tvz{©dxstv{¥;w η ¸ ®¥tbd;xdbgtb1fsfNSfw1sfNw1 p«Stbdfe-ssf®d;w N {}rrutvz;wd1´sxdf t%tb1f e-fNws¨Y½Hf®}w1txdzvf$«Htvbdf e-ssfN®Y´;¹ºf©d©dzvf¼q{}tvf tv (ξ ) ¹ bdfNw N {}r ½dµsfN»´qf µd fNstf<5¹ b1fw]¹ºfj¹
(106) w@tºtfe-db1;r¯{ Vfjf µsd®{{ tvf®pgtbdf%sfSfNw1sfw1p-;w (ξ ) « v t d b 3 f e s s f®9´sf;¸ H¸Kr${w tbdf3fwH« tbdf3dzvfNrfw@tr¯f<·tv{¥;w_¸ N hBt{} f%tb1t (N ) n−1. i=1. (N ) n. n. (N ) n−1. (N ) 0. i,N 1≤i≤N n. 0. i n 1≤i≤N. ³. Φn m. ³. (N ) ξn−1. ´´. 1. (dxn ) = PN. j Gn−1 (ξn−1 ). Ã. N X. i Gn−1 (ξn−1 ). i Mn (ξn−1 , dxn ). !. r¯jtvb1tKtbdf 1zt{} ®f
(107) e-ssfN®sf¼;®¥¼f<r;r"Pfw1f t{}ºtpqSf$e-ssfN®s¹ {¥tbgdzvS;z¯tv{¥;w1®qrf®fN·tv{¥;w1rN´w1(e(xdtvtv{wtzw1r{¥t{w1r s{ tvtfN>©@p tb1f(H{zB«StfNw;tv{® ¨9tzw1r{¥t{w (G , M ) ¸B w§w@xde-fzv{}® dbqpsr¯{}rN´Htbdfgrf®fN·tv{¥;wtvzvw1r{ tv{¥;w¾{rj«ãtvfw ®®fN-zvfNws½Hxdzt{w-«»tvbdfBS;dxd®}tv{¥;w5«_¹
(108) ®¥¶fNzvrN¸ w5tvbdfjw@tf µqtº«»Htbr¯1; fBk§zv¶G¼gvbH{w5e-sdf® ; ¸ 1 ´@{¥t
(109) {r {¥e-Sztvw;t$tv-wdt{} fPtvb1t tvbdf31zt{} ®f%{¥w@tfNzdzvf ttv{¥;we-ssf®_{}r ;{w g1tvb 1z¯tv{®¥f%e-ssfN®Y¸ tjxdzv{¥wd-tbdfrf®fN t{w rutf;´¹ºfrf®fN·t %1tvbg1z¯tv{} ®f N zs{wdjt3{ tr"½dtvwdfNrvr´@w1tbdfe(xstvt{w5r¯tvf wHr¯{}rutr"{¥w-fµqtfwHs{¥w1Ptvbdfrf®fN tf< 1tvb¾¹ {¥tb§w¾f®fe-fw@tvzp]tvzvw1r{ tv{¥;w¾s{}·ttvfN¾©@ptvbdfk>z¶¼/tzwHr¯{¥t{w1r«Ktbdf(fN®¥fNe-fw;tzvpdzv{¥e-fvb1{¥w ¸ wtvbd{}r w@tfµ@t<´qtvbdf31ztv{®¥f%e-ssf® w ©Hf3{w;tvfz1zftfNr gfNwdfN®¥;{N®S1z¯tv{®¥f%fN¼®xst{w egdsf®9¸ a$bdf ddzv<µs{e-t{wPe-fN;r¯xdzvfNr ;rrs{tvfNj¹ {¥tb%tvbdfH{z_«d²dfNp@w1e-ws¨ª³e-fN;r¯xdzvfNr (η , γ ) zvfsf ½1wdf< ©@p/tb1f%fNe-d{¥zv{}®»d xdHt{w e-fN(ηrxdzvfN,r γ ) ³ ´ j=1. n−1. N n. i=1. n. N n. n. w1]tvbdf3xdwdwd;ze5®¥{ Vf<]1zt{} ®f3s{}rutvz{©dxstv{¥;w1r. n. ηnN = m ξn(N ). zvf%sf½1wdf<]«¬;zwqp f ∈ B (E ) ©@p ¹ {¥tvb γ (1) = Y η (G ) ; ¸ ;`; γ (f ) = η (f ) × γ (1) ²d;z(ztvbdfz(e-d®f tf];r¯pqe-stvt{}5w1®psr¯{}r%«
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