nonlinear markov semigroups and interacting lévy type
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HAL Id: inria-00080360https://hal.inria.fr/inria-00080360v2
Submitted on 20 Jun 2006
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Nonlinear Markov semigroups and interacting Lévy typeprocesses
Vassili Kolokoltsov
To cite this version:Vassili Kolokoltsov. Nonlinear Markov semigroups and interacting Lévy type processes. [ResearchReport] RR-5932, INRIA. 2006. inria-00080360v2
appor t de r ech er ch e
ISS
N02
49-6
399
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NIN
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--59
32--
FR
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NG
Thème NUM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Nonlinear Markov semigroups and interacting Lévytype processes
Vassili N. Kolokoltsov
N° 5932
June 2006
Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)
Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30
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/ )J%KY_sZ ik^[cKMOSika]N3luqsqrO*vO*q>f[cKMOxIbÀ8O9acO9^syO1O*[\£ O9O*^3[]KMOkO*^sO*azluqsp8ivZcb[cbrfb[\X3pMa]O9ZcO*a]f¤br^M$O*kivq_M[cbrik^sZhiu½ivacN ¬ f®)lk^sx3[]KMOSYbr^MO[]bryO9df_slu[cbrik^sZ iu k ¤¿lua]Xb^f[cO9a]lvy¡[]b^M$psluac[cbeyqrO9Z ¬ C;k®)beZ [cKMOxMbÀDO*a]O*^sy*O8O[\£4O*O*^y*ik^sxMb[]biv^sluqrqX$p1ifZwb[cbrkOSlu^sx3Zw[ca]ik^MvqX$yik^1xIb[]biv^sluqrqrX$p8ivZcb[]bvOkO9^MO*azlo[]ikazZ ¦Mika[]KMOy*lvZwOiu xIbeZcy*acO*[cO
X[cKMbeZik1ZwO9acolo[]biv^beZ%bqrqr_sZ\[]a]lu[cO;xYXghacivp1ifZwb[cbrik^ C C
_Ma <1a]Zw[Z\[]ac_sy[c_Mazluqa]O9Zc_Mq[yzKsluazlky[cO*a]bDK*O9Zh[cKsOyik^1xIb[]biv^sluqp8ivZcb[cbrfb[\Xlu[ <MnIO9xp1ivb^f[zZ ·Y» · P
X = Rd = /1 P D P #
F1C2
c (X) I 1 9
x = (x1, . . . , xn))' ,% 1 *1
Ω
(
n∑
i=1
ωiδxi
)
g =
n∑
j=1
[
aj(x)g(xj) + (bj(x),∇)g(xj) +1
2(cj(x)∇,∇)g(xj )
+
∫
(g(xj + y) − g(xj) − χ(y)(y,∇)g(xj))νj(x; dy)
] ¬ C u®
g ∈ C2
c (X)*>P / 4
cj IP R P F / R2* (
νj %? = *χ
FIL ¬ C9®+* aj , bj , cj , νj P ∑ni=1 ωiδxi
' 0Á½O[ _sZZw[]lkaw[£b[]K.lyivN0NO9^f[iv^m[cKsO iv_Ma]acL9kO []KMO*ivacO9N $§ qriYikik^A[cKMO0pMa]iYiuhikKsbrZ[cKMO9ika]O*N ¬ ZwO9O >DC;A¥ > ~A> >Ï@ A®ZcKMio£Z[cK1lo[([]KMOyzKslka]lvy¡[]O*a]bLK;lo[cbrik^.beZ(lky[c_slkqqrXAvbvO*^ ^Mik[(ik^MqrXivayiv^sxIb[cbrik^slkqqrXp8ivZcb[cbrkOªikp8O*azlo[]ikazZ* M_I[lkqrZci ikayik^1xIb[]biv^sluqrqrX p1ifZwb[cbrkOªqrbr^MO9lka3_M^sy[cbrik^slkqrZ
ivI[]lkb^sO9x0YX <snYbr^M []KMOlkacv_MN0O*^f[]Z PSluN0O*qrXkfb[hbrZ4ZcKMio£^[cKslu[ b[cKMOqrbr^MO9lka)_M^sy[cbrik^slkq (Ag)(x) :C2
c 7→ Rd brZ$ 2%7P R 4lu[ x Yb O b Ag(x) ≥ 0£KMO9^MO*vO*ahl^Miv^I¤¥^MO*flo[cbrkO
gGlk^MbeZwKMO;Z
lu[x[cKsO*^
Ag(x®SKslvZivacN ¬ C;®Sbraca]O9Zcp8O9y¡[]bvO*qrXiu)[]KMO0pMacivp1O9aw[]bO;Ziu Ag(y) b^2iu[]KMO*ap8ikbr^v[zZ y
âã&ä½âÖÞ
C;
Pio£ qrO[ _sZyzKMiYivZcO$lpsluac[cb[cbrik^2iu4_M^Mb[\XmlkZlÐlkN0bqrXmiunZcNiYik[cKA^sik^I¤¥^MO*flo[]bvO _M^sy[cbrik^sZ
χi
i = 1, . . . , nZc_syzKA[cKslu[ ∑n
i=1 χi = 1lu^sx2O9lvyzK
χiO9df_slkqrZik^sO$br^´l^MO9bvKY1ivacKMiYiIxmik
xi¬ lu^1xy*ik^sZcO9df_MO9^f[cqrX2olu^sbrZcKMO9Z$br^lm^MO9bvKY1ivacKMiYiIx iulk^YX2ik[cKMO9a$p8ikbr^f[
xl£b[]K
l 6= i® H4X qbr^MO9lkacb[\X
Ω(∑n
i=1 ωiδxi) =
∑ni=1 Ωi
%£b[]KΩjg = (
∑ni=1 ωiδxi
)(χjg) qrO9lkacqrX O;lkyzK _M^sy¡[]biv^sluq ΩigbeZy*ik^sxIb[cbrik^1luqrqX p1ifZwb[cbrkOAlu[xib^©[]KMOm_sZc_slkqZwO9^sZcOkb O g(xi) = 0
brN0pMqbrO9ZΩig ≥ 0
ikal^sik^A^sO*vlu[cbrkO(_M^sy¡[]biv^
g ∈ C2c (X) MO*^sy*O3lupMpMqrXYb^Ml <MnIO9xAp1ivb^f[ kO9a]Zcbrik^ iuh[cKMO ik_saca]L*vO[]KMO*ivacO9N [ci0O;lkyzK
ΩiIiv^MOikI[zlubr^sZ%acO9pMa]O9ZcO*^f[]lu[cbrik^ ¬ C o®
§ Zhl yivacivqqelua]X£ OZcKslkqqMivI[]lkb^0^Mio£©l yzKslka]lvy¡[cO9acbDK9lu[cbrik^ikD[cKMOZw[ca]ik^M y*ik^sxIb[cbrik^1luqMp8ivZcb[]bYb[\X ·Y» ·
X
D P/ 9 1
" 1 Ak : D 7→ Csym(Xk)
,%< ,%I1R ,%/
Akg(x1, . . . , xk) =
k∑
j=1
[akΠj1(x1, . . . , xk))g(xj)
+ (bkΠj1(x1, . . . , xk)),∇)g(xj) +1
2(ckΠj1(x1, . . . , xk))∇,∇)g(xj)
+
∫
(g(xj + y)g(xi) − χ(y)(y,∇)g(xj))νk(Πj1(x1, . . . , xk); dy)], ¬ C @v®>1 P? 1
Πjl
Rd (1 1 FR JP= j
l* 4
ck P( ,% νj " I 1R 9 P F9/ R2 1( 2%/ %// R=!"1* χ I L ¬ C;® * >P 1 = 1 ak, bk, ck, νk
(% / R R P( 1 P R F 1! #1 R DR
x2, . . . , xk P R 13 4 x1"
" 1 FAk : D 7→ Csym(Xk)
,% 2%9P R I?,%/
Akg(x1, . . . , xk) =k∑
j=1
ak(Πj1(x1, . . . , xk))g(xj) +Bkg+(x1, . . . , xk), ¬ C9~v®
>1ak
1 / ¬ C k®? Bk 7 ,% P R RP P !"
P FICsym
∞ (Xk)
"; 1 P= #% ! 1 = 1 1 / ' A L 8 1 1 = 1
ak, bk, ck, νkF1 PP R,%9
x1, . . . , xk / R= νk 3 P 49 P
1L %(" ' ¬ be®%¦sacivN J%KMO9ika]O*N Cb[ikqrqio£Z [cK1lo[ikalkacsb[]a]lkacX3xIbÀ8O9acO9^f[ x1, . . . , xk
Akg(x1, . . . , xk) =k∑
j=1
[ajk(x1, . . . , xk)g(xj)
+ (bjk(x1, . . . , xk),∇)g(xj) +1
2(cjk(x1, . . . , xk)∇,∇, )g(xj)
+
∫
(g(xj + y) − g(xj) − χ(y)(y,∇)g(xj))νjk(x1, . . . , xk; dy)],
ääÉNPORQRSRT
C9 L
£KsO*a]OO9lvyzKcjkbeZ l p8ivZcb[cbrkOSxMO <s^Mb[cOSN3lo[ca]bnDfO9lvyzK
νjk
beZ l$Á½`9fXN0O9lvZw_Ma]O RªivacO9iokO*a;kxI_sO[ci([cKMOZcXYN0NO*[ca]Xiu_M^1y¡[cbrik^1ZAkgika
g ∈ C2c (X)M[]KMO_M^sy¡[]biv^sZ
ajk, b
jk, c
jk, ν
jk
xMO*p8O*^sxik^x1, . . . , xkZcXYN0NO*[ca]bry9luqrqX3br^[cKMO ZcO*^sZcO[cKslu[9sZ]lGX
ajΠjl = al ikalkqq j, l lk^sx ajΠlm = aj ika m 6= jl 6= j
_MO0[]i[]KMbeZZwXYN0N0O[]acX2b[beZp1ifZcZcbrMqO0[]iacO9£acb[cO[cKsOlu8iokO$ika]N(_MqellkZ ¬ C@k® C0£b[]K ak = a1k
bk = b1k
ck = c1k
νk = ν1
k ¢ [ a]O*N3lubr^sZ[ci^Miu[]bry*Oh[]Kslo[)xI_sO []i[cKMOyiv^f[cbr^f_sb[\XvG[cKMbeZacO9pMa]O9ZcO*^f[]lu[cbrik^a]O*N3lkb^sZ4oluqrbexO9kO9^iva^Miu[^MO;yO;ZcZ]lua]bqrXxIbÀ8O9acO9^f[%p8ikbr^f[]Zx1, . . . , xk ¬ bbÖ®#H4X$[cKMO iv_Ma]acL9kO4[cKMO9ika]O*N lupMpsqbrO9x[]i Xk fl y*ik^sxMb[]biv^sluqrqX$p1ifZwb[cbrkO Bk
br^C∞(Xk)
KslkZivacN
Bkf(x1, . . . , xk) = a(x1, . . . , xk)f(x1, . . . , xk)
+
k∑
j=1
(bj(x1, . . . , xk),∇xj)f(x1, . . . , xk)) +
1
2(c(x1, . . . , xk)∇,∇)f(x1, . . . , xk)
+
∫
(f(x1 + y1, . . . , xk + yk) − f(x1, . . . , xk) −k∑
i=1
(χ(yi)(yi,∇xi)f(x1, . . . , xk))
× ν(x1, . . . , xk; dy1 · · · dyk). ¬ uv®
§ pMpMqrXYb^M[]KMbrZ[ci f = g+ lu^1xy*ikN0pslkacbr^M£b[]K ¬ C o®4XYbrO*qexMZ4[cKsO acO;dv_sba]O9xacO;Zw_sq[ ¬ bbrbe® H4XSiv^MOhlk^sx[cKsO Z]luN0O)pMa]iIyO9xM_MacOv9iv^MO)pMa]iokO;Z <sazZ\[[cKMO%yiv^v[]b^Y_Mb[\XSikaxMbÀDO*a]O*^f[ x1, . . . , xkS[cKsO*^¯ika[\£ i iu$[cKMO9N yivb^sy*brxMb^M1O*[]y ¢ lkqq x1, . . . , xklua]O2xIbÀDO*a]O*^f[9ik^MO y*lk^ _sZcO.[cKMO
xMO9yivN0p1ifZwb[cbrik^ a]ikN°[]KMO4pMaciYikMiu1J%KMO9ika]O*N Cha]O9xM_sybr^M[]KMO%pMa]ikMqrO*Nµ[cipMa]ioYb^M[cKMOyiv^f[cbr^f_sb[\Xiva&[cKMOivp1O9a]lu[cikaAk(χjg)(x1, . . . , xk) KsififZwbr^M g [ci1Olyiv^sZw[]lu^f[;oik^MO%pMa]iokO;Z½[]KMOyiv^v[]b^Y_Mb[\Xik½[cKsOy*iYO GybrO*^f[]Zakiva []KMbeZ ivp1O9a]lu[civa J%KMO*^yzKMiYivZcb^s g [ci1O lyiv^sZ\[zlu^f[ br^l$^MO9bvKf8ika]KMiYiIxikhlyO9aw[zlubr^ªp1ivb^f[
xik^sO(pMa]iokO;Z%[cKMOyiv^v[]b^Y_Mb[\Xiu
νk J%KMO*^AyzKsififZwbr^M[ci8O(qrbr^MO9lkaSlua]ik_s^sxxiv^MOkO*[]Z4[cKsOyik^f[]b^Y_Mb[\Xiu
bkMlu^sxlu[%qelkZw[[cKsOy*ik^f[cbr^Y_Mb[\Xiu
ckikqrqio£Z J%KY_sZ ivp1O9a]lu[civa]Z
Akb^ ¬ Y® lua]OZ\[]aciv^MkqrX(y*ik^sxIb[cbrik^1luqrqX$p1ifZwb[cbrkOb½lu^sxik^sqX$bD[cKsO*XKslGkO4ika]N
Akg = Bkg+/k!
£b[]KmZwivNOy*ik^sxMb[]biv^sluqrqXp1ifZwb[cbrkOBkbr^C∞(X
® ¬ _Mpª[]iZcikN0O N(_Mq[cbrpMqrbry9lo[]biv^ivp1O9a]lu[civa]Zz®lk^sx([cKY_sZ bD[cKMO9X$yivaca]O9Zcp8ik^sx[ci lYb^MO*[cbey%O9df_slu[cbrik^$ika)ZcikN0ORmlua]kio N0iIxIO*qsiuk¤¥lkacX
br^f[cO9a]lvy¡[cbr^Mpsluac[cbeyqrO9Z ·Y» · P P
X = Rd D
FPC2
c (X) ¬ C;k®
,%P R P,% I ( ! 1I 1 Ak
1 ? R ¬ C@v®$ 1 / % %. ¬ C @k® ck
νk 3( R R2%
P R = P,% R/( 1 19/ R2K∑
k=1
k
m∑
i1=1
· · ·m∑
ik−1=1
ωi1 · · ·ωik−1ck(x, xi1 , . . . , xik−1
) ¬ =C;®
P R = # 1 = K∑
k=1
k
m∑
i1=1
· · ·m∑
ik−1=1
ωi1 · · ·ωik−1νk(x, xi1 , . . . , xik−1
) ¬ kk®
âã&ä½âÖÞ
CG
P R > 7%m* /% L( 17! P R 7P=
ω1, . . . , ωm P
x, x1, . . . , xm νk
(% ¬ k® * ! = !" R P =L(* AkP3 R ¬ C9~f®= Bk
5%¬ uk®9 4 R,% P R c ν ' · ; MO*a]O4£4OZwK1luqrqIpMaciovOh[]Kslo[hlkqq Ak
KslGkO ika]N ¬ C u®&£b[cK n = kuM_I[)p8ivZ]ZwbrMqrX
£b[cKsik_I[4p1ifZwb[cbrYb[\X0iucklu^sx
νk ¦Miva4lkacMb[cazlua]XI<snYO;x x1, . . . , xm[cKMOS_M^sy[cbrik^slkq1ik
gvbvO*^3YX
K∑
k=1
m∑
i1=1
· · ·m∑
ik=1
ωi1 · · ·ωikAkg(xi1 , . . . , xik
) ¬ uv®
beZx = (x1, . . . , xm)
¤%yiv^sxIb[cbrik^slkqqrXp8ivZcb[]bvOlk^sxyiv^sZwO;df_MO*^f[cqrXKslvZ%ika]N ¬ C o® ¬ £b[]K m br^sZ\[]O9lvxikn®slkZikqrqio£Z4a]ikN J%KsO*ika]O*N C QSZwbr^M εωj
b^1Z\[]O9lkxiuωj8xIbrfbexIbr^M3YX
εlk^sx[cKMO9^ªqO*[w[]b^M
ε → 0iv^MO3ikM[]lubr^sZ[]Kslo[ ∑m
i=1A1g(xi)KslvZ[cKsOZclkNO0ivacN § Z m beZlkacMb[cazlua]XkD[]KMbeZbrNpsqbrO9Ziv^A[]KMO0ik^MO0Kslk^sxA[cKslu[
A1g(x)KslkZ[]KMO3ZclkNOivacN ivalua]Mb[cazlua]X
x ¬ [cKY_sZ vbYbr^M[]KMO3a]O9df_MbracO;xZw[ca]_sy[c_MazluqDa]O9Zc_Mq[ivaA1®%lk^sxiv^[cKMOiu[]KMO*aK1lu^sx[cK1lo[
K∑
k=2
εk−2m∑
i1=1
· · ·m∑
ik=1
ωi1 · · ·ωikAkg(xi1 , . . . , xik
)
K1lkZ []KMOa]O9df_Mbra]O9xika]N § vlkb^3qrO[c[cbr^M ε→ 0XYbO9qrxsZ)[]KMOSZ]luN0Oa]O*pMa]O9ZcO*^f[zlo[cbrik^ivah[]KMO_M^1y¡[cbrik^1luq
m∑
i1=1
m∑
i2=1
ωi1ωi2A2g(xi1 , xi2).
§ Zlu8iokO[cKsbrZ%brN0pMqbrO9Zik^[cKMO iv^MO Kslu^1x[]Kslo[ω2
1A2g(x1, x1) + 2ω1ω2A2g(x1, x2) + ω22A2g(x2, x2)
K1lkZh[cKMOacO;df_Mba]O9xivacN lk^sx3KMO9^syOluqeZciA2g(x1, x1)
KslkZh[cKMbeZ ika]N ika%lua]Mb[]a]lkacXx1¬ ps_I[ ω2 = 0br^[cKMOªpsacO9fbrik_1Z$O*nIpMacO;ZcZcbrik^1® lu^1x KsO*^sy*OmluqeZwi
A2g(x1, x2)KslvZ[cKMOmZ]luN0Oika]N ivalua]Mb[]a]lkacX
x1, x2¬ [cKY_sZvbYbr^M([]KMO a]O9df_MbracO;xZw[ca]_sy¡[]_MazluqDacO;Zw_sq[iva A2
®slu^1xiv^[]KMO iu[]KMO*aK1lu^sx[cKslu[K∑
k=3
εk−3m∑
i1=1
· · ·m∑
ik=1
ωi1 · · ·ωikAkg(xi1 , . . . , xik
)
K1lkZ4[cKsO acO;dv_sba]O9xika]N ¦sikqrqio£br^M$[cKMbeZpMa]iIyO;xI_Ma]OSbr^sxI_sy[cbrkO9qXXfbrO*qexMZ4[]KMOyqelubrN ikVf[]O*p,C · ¦Ma]ikN []KMO ikI[zlubr^MO9xa]O*pMa]O9ZcO*^f[]lu[cbrik^ika Ak
b[ikqrqio£Z []Kslo[K∑
k=1
m∑
i1=1
· · ·m∑
ik=1
ωi1 · · ·ωikAkg(xi1 , . . . , xik
) =
K∑
k=1
k
K∑
l=1
m∑
i1=1
· · ·m∑
ik−1=1
ωlωi1 · · ·ωik−1
[
ak(xl, xi1 , . . . , xik−1)g(xl) + (bk(xl, xi1 , . . . , xik−1
),∇)g(xl) +1
2(ck(xl, xi1 , . . . , xik−1
)∇,∇)g(xl)
+
∫
(g(xj + y) − g(xj) − χ(y)(y,∇)g(xj))νk(xl, xi1 , . . . , xik−1; dy)
]
.
ääÉNPORQRSRT
C; L
§ Z)[]KMbeZh_s^sy¡[]biv^sluq8KslkZh[ci$1OZw[ca]ik^MvqX0yiv^sxIb[cbrik^slkqqrX0p1ifZwb[cbrkOvk[]KMOSa]O9df_Mbra]O9x0p8ivZcb[]bYb[\Xpsacivp1O9aw[\Xikclu^1x
νikqrqio£Z4a]ikN J%KMO*ivacO9N C
» » ÐI ¸ P9 K = 2
19 P9' 11R % / P R #%3 ¬ C;k® ' )¢ ^ªy*lvZwO K = 2
[cKsO p1ifZwb[cbrYb[\Xik ¬ 1C9®4acO;lkxMZlvZ [cKMO p8ivZcb[]bYb[\Xik[cKsO N0lu[ca]bn
c1(x) + 2
m∑
i=1
ωic2(x, xi)¬ of®
iva luqrq ^slo[]_Ma]lkqm½p8ivZcb[cbrkO0^Y_MN(8O*azZ
ωjlk^sx2p1ivb^f[]Z
x, xjj = 1, . . . ,m MO*^sy*O c1 brZ lkq£%lGXIZp8ivZcb[cbrkO ¬ pM_I[ ωj = 0
ika%luqrqj® JipMa]iokOZw[ca]ik^M$yik^1xIb[]biv^sluq8p8ivZcb[]bYb[\X3ik^MOSKslkZh[cipMa]iokO[]Kslo[
c(x, y)beZ%p8ivZcb[]bvOxMO <s^Mb[cOivaluqrq
x, y H4_I[b[]KMO*a]OOnIbrZw[ x, y Zw_1yzK[]Kslo[ c(x, y) beZ%^Miu[p8ivZcb[]bvOxMO <s^Mb[cOvI[cKsO*^fXyzKMiYivZcb^Mqelua]kOO9^Mik_skK^f_sN(8O*a%iup1ivb^f[zZx1, . . . , xm
^MO9lkayIik^sO£4ik_sqrxkO[
l0N3lu[ca]bniu&ika]N ¬ ov®[cKslu[beZ^Mik[p1ifZwb[cbrkO(xMO <s^Mb[cO ¬ O*vO*^ivaluqrq ωj = 1® J%KMbeZyiv^f[cazlkxIbey¡[]biv^y*ikN0pMqrO[]O9Z4[cKsO pMaciYik J&luYbr^M$br^f[cilky*y*ik_M^f[%OnIpMqrbry*b[]qX0[]KMOZwXYN0N0O[ca]X3iukO*^sO*azlo[civa]Zhb^ ¬ f® ¬ ^Miu[]OS[]Kslo[%br^ ¬ f® ika ¬ @v®[]KMO0_sZcOik ^Miv^sZcXfN0N0O[]acbey$kO*^sO*azlo[civa]ZSika[]KMO*braZwXYN0N0O[]acbDK9lu[cbrik^sZZcp1O;ybL<sO9Z[]KMO3ZclkNOO9df_slu[cbrik^1®
lkqqrio£Z&[ci(kO[hl_sZwO*_MqsO;df_MbolkqO9^v[)a]O*psacO;ZwO9^v[zlo[]biv^(iva ¬ v®¡ ¬ u® PluN0O9qXvu[cKsOikqrqio£br^MZw[]lu[cO*N0O9^v[beZ%ikYYbrik_sZ » » ÐI ¸ " A < ,% P R 6P 6+% P 7P
1 R ¬ C @k®+* 9 5 R ¬ k®
d
dt(g, µt) =
K∑
k=1
k
∫
(A1kg)(x, y1, . . . , yk−1)µt(dx)µt(dy1) · · ·µt(dyk−1),
¬ kk®
A1
k : C∞(X) 7→ Csym(Xk) . # 4?
A1kg(x, y1, . . . , yk−1) = ak(x, y1, . . . , yk−1)g(x)
+ (bk(x, y1, . . . , yk−1),∇g(x)) +1
2(ck(x, y1, . . . , yk−1)∇,∇)g(x) + Γk(y1, . . . , yk−1)g(x),
¬ uv®>1 ∇ (* ! = +*#1 9 P = R P( #F 1 R L
x ∈ Rd >1
Γk(y1, . . . , yk−1)g(x) =
∫
(g(x+ z)− g(x) − χ(z)(z,∇)g(x))νk(x, y1, . . . , yk−1; dz).
¬ u®
" Ak9 5% ¬ C C9® 1 ¬ v® 1 R ¬ C9v® "1* 1 A1
k = (1/k!)Bkπ* >1
PI 1 π 7 +%
πg(x1, . . . , xk) = g(x1)
âã&ä½âÖÞ
C
" 1 R ! (- P ¬ vu® ) ¬ o® / R= µt1 1R R$ 1(
R$PI/ R=+*# &P $ Rµt(dx) = ft(x)dx
ft ∈ L1(X)
d
dtft(x) =
K∑
k=1
k
∫ [
1
2(ck(x, y1, . . . , yk−1)∇,∇)ft(x) + ((∇ck − bk)(x, y1, . . . , yk−1),∇ft(x))
+(ak −∇bk +1
2(∇,∇ck))(x, y1, . . . , yk−1)ft(x) + Γ∗
k(y1, . . . , yk−1)ft(x)
] k−1∏
l=1
ft(yl)dyl,
¬ @v®>1
(∇ck(x, y1, . . . , yk−1))j =
d∑
i=1
∂
∂xicij(x, y1, . . . , yk−1),
(∇,∇ck)(x, y1, . . . , yk−1) =
d∑
i,j=1
∂2
∂xi∂xjcij(x, y1, . . . , yk−1),
>P Γ∗
k(y1, . . . , yk−1) 1 P P F 19 =1 ¬ o® ' ¬ uv®
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X
D 1
A I / 3 1
R/ 1 D
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D A
PP 1 1 1 / µn → µ
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g ∈ D
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C(X)
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£b[]KAig(x1, . . . , xi)) = ai(∆g(x1) + · · · + ∆g(xi)), i = 1, 2, 3,
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K∑
k=1
k
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ck(x, y1, . . . , yk−1)µ(dy1) · · ·µ(dyk−1)¬ u~v®
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[]bvOAxIO<s^Mb[cOm_s^sy¡[]biv^sZ ¦Mikab^1Z\[zlu^sy*Okbr^ y*lkZcO k = K = 3lu^sx
d = 1 ¬ k~k®3N0O9lk^sZ3[]Kslo[
∫
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X ¢ ^y9lkZcOikyiv^v[]b^Y_Miv_sZ4Z\[zlo[cOZcpslky*O X iv^MOy*lu^Z\[zluac[ O9O*qrb^M$[cKsOxIbÀDO*a]O*^sy*OS8O[\£4O*O9^Zw[ca]bey¡[cqrX¨yiv^sxIb[cbrik^slkqqrX p8ivZcb[cbrkO2lk^sx y*ik^sxMb[]biv^sluqrqXp8ivZcb[]bvOAikp8O*azlo[]ikazZ3ik^sqX©£b[cKk = 3 §ZcbrNpsqOSOnMluN0pMqrOiklyiv^sxIb[cbrik^slkqqrX0p1ifZwb[cbrkOvvM_M[ ^Mik[4l(Zw[ca]bey¡[cqrX3yiv^sxIb[cbrik^slkqqrX$p8ivZcb[cbrkOivp1O9a]lu[civa
a]O*psacO;ZwO9^v[zZ [cKMO ivp1O9a]lu[cikaxIO<s^MO;xlvZA3g(x1, x2, x3) = cos(x2 − x3)∆g(x1) + cos(x1 − x3)∆g(x2) + cos(x1 − x2)∆g(x3).
¬ kv®« dv_1lo[cbrik^ ¬ @v®4br^[]KMbeZy*lvZwO[]lkkO;Zh[]KMOika]N
d
dtft(x) = ∆ft(x)
∫
ft(y)ft(z) cos(y − z)dydz. ¬ 1C;®J%KsOw^Miv^Mqbr^MO;luaxIbÀ8_1Zwbrik^yiYO Gy*bO9^v[beZ^Mik[Zw[ca]bey¡[cqrXp8ivZcb[cbrkOKMO*a]O PSluN0O*qrXksZcbr^syO
∫
f(y)f(z) cos(y − z)dydz =1
2(|f(1)|2 + |f(−1)|2),
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lk_syzKYX3pMa]ikMqrO*N ik ¬ 1C;®³
ft =1√
2πωt
∫
exp
− (x− y)2
2ωt
f0(y)dy,
âã&ä½âÖÞ
C;~
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d
dtft(p) = −1
2p2(|ft(1)|2 + |ft(−1)|2)ft(p),
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ξt = (t+ ξ−10 )−1
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d
dt(g, µt) =
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k=1
1
(k − 1)!(Bkg, µt)‖µt‖(k−1),
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[]KMO y*ikqrqO;y¡[cbrik^ikf[]KMOholkacbeluMqrO9Zxi1 , . . . , xik
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dxI[cKsO(N0O9lvZw_Ma]O
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Csym(X )lu^1x M(X )
y*lk^8O £acb[w[]O*^ªlkZ
(f, ρ) =
∫
f(x)ρ(dx) = f0ρ0 +
∞∑
n=1
(x1, . . . , xn)ρ(dx1 · · · dxn),
f ∈ Csym(X ), ρ ∈ M(X ), ¬ vk®Zci3[]Kslo[ ‖ρ‖ = (1, ρ)
ivaρ ∈ M+(X ) Jilu^Alkacsb[]a]lkacX Y (dx) ∈ M(X)
[]KMO*a]O(y*ika]acO;Zwp8ik^sxsZlN0O;lkZc_MacO
Y ⊗ ∈Msym(X )xIO <s^sO9xYXb[zZyivN0p1iv^MO*^f[]Z
(Y ⊗)n(dx1 · · · dxn) =1
n!Y ⊗n(dx1 · · · dxn) =
1
n!Y (dx1) · · ·Y (dxn). ¬ kv®
JiªO9lvyzKg ∈ Csym(X )
[cKMO9acOyika]a]O9Zcp1iv^sxMZlJ ,% $ = (g, Y ⊗)
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sbg ∈ Csym
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k L
·Y» · " µt ¬ C;k®(*1 νt = (µt)
⊗ ∈ Msym(X ) 1
(- P
d
dtνl
t(dx1 · · · dxl) =l∑
j=1
K∑
k=1
Cll+k−1
∫
xl+1,...,xl+k−1
(Bj,l+1,...,l+k−1k )∗νk+l−1
t (dx1 · · · dxl+k−1),
¬ uf®>1Cl
m
1 RP ; (*B∗
k
P P FBk
(BI
k)∗νt(dx1 · · · dxm) 1 P ,! B∗
k
1I R LR9 4 R I ⊂ 1, . . . ,m
" 1 ' νt ∈ Msym(X )
1( ( 5% ¬ kv®(* P 1 P3 Csym(X)
7 +% 1 (- P
g(x1, . . . , xl) = (LBg)(x1, . . . , xl) =∑
I⊂1,...,l
∑
j 6∈I
(Bj,I|I|+1gI)(x1, . . . , xl),
¬ vk®
>1gI(x1, . . . , xl) = g(xI )
B
j1,...,jk
k
P 1 ! Bk
P R L j1, . . . , jk
R P =L(*(LBg
1)(x1, . . . , xl) = (Bl(g1)+)(x1, . . . , xl).
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µt(dx) =K∑
k=1
1
(k − 1)!B∗
k(µt ⊗ · · · ⊗ µt)(dxdy1 · · · dyk−1),
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d
dt(g, νt) =
∞∑
l=0
K∑
k=1
∑
I⊂1,...,l+k−1,|I|=k−1
∑
j 6∈I
Bj,Ik gI(x1, . . . , xl+k−1)νt(dx1 · · · dxl+k−1)
=
∞∑
m=0
∑
I⊂1,...,m
∑
j 6∈I
Bj,I|I|+1gI(x1, . . . , xm)νt(dx1 · · · dxm),
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gik^ X lu^sx(b^Mb[cbeluqYN0O9lvZw_sacO;Z µ0
ik^XGik^MO4KslvZ½[]KMO4xM_sluqrb[\X a]O*qelo[]biv^
(gt, µ⊗0 ) = (g0, µ
⊗t ) ¬ kv®
brN0pMqrXfbr^M´[]KMOAbr^Yolua]brlk^syOªik [cKMO.yika]a]O9Zcp1iv^sxIbr^M´ZwpslvyOmik wlk^sluqrXf[cbey _M^sy[cbrik^slkqrZiu [cKsOm[\XYp1O(g, Y ⊗)
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" G(t, s, x, y) %8>1/)F #P P /) / 4-P8)
/ (* 7
Bt = 0 $P ∫
G(t, s, x, y)dy = 1
x
t > s
" G(t, s, x, y) 9 P R,% 7 L
t, s, x, y>1
t > s
∣
∣
∣
∣
∂G
∂x(t, s, x, y)
∣
∣
∣
∣
= O((t− s)−1/α)G(t, s, x, y) ¬ @v®
=P R,% 0 ≤ s < t ≤ t0
*x, y ∈ Rd
ääÉNPORQRSRT
k L
" /% Gs ∈ C∞(X)1 =P-P L R " Gt
C∞(X)
F1 /( % L ! /(-P ¬ ku® PP
t → Gt ∈ C∞(X)P
R ¬ kv® t > s R
Gs t = s
"/
Gt ∈ C1(X)
t > s
‖Gt‖C1(X) = O(t− s)−1/α‖Gs‖C(X); *
σ,A,Bt, φt ∈ Ck(X)*k > 0
*P / Gs 7→ Gt
. = ( P Ck(X)
=P R,% t ∈ [0, t0]
' ¢ ^(y*lvZwO Bt(x) = 0
lu^sxAt(x) = A(x)
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» » ÐI ¸ 1/ =7 P ! 1 P Gs 7→ Gt
4 "1 =(/ 9/ M(X) 7→ M(X)1 P 1 ∗ ) 1L% R/( P 2 8% 171R #% 1(F P / R=!" 1 ; (-P ¬ vk® t > s
F "1 =( / (C1
∞(X))∗ 7→ (C1∞(X))∗1 < P 1 ∗ ) 1L%
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α ∈ (1, 2)*σ(x)
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σ ∈ C1(X)*V ∈ C1(X
" / = J! P # R L< = = =P R R P4F 1I( ?+* PP =
ψ8% 1 =
ψ(x, y; z) ≤ C
1 + |z|β+d¬ u~v®
x, z, y
/9 P β ∈ (0, 2)
*C > 0
P P L8 1 " 5 %I/)F f0 ∈ C∞∩L1(X)
P 4 7I=- P7/)F $L R
ft ∈ C∞ ∩ L1(X) ∩ C1(X)! 1 /( % L F ¬ uv® t 7→ ft
/ PP =
R+ 7→ C∞(X)P R ¬ uf® t > 0
" " 17/ f0 7→ ft
F 99 ,% PP ! 0" )P I / R ! L+
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ft
1R( 17( 1L%?' L1(X)
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4 F F , ! 8 = P P R 1L %(" / R
µ0 7→ µt M+(X)
R/( P ¬ kk®91 3 % g ∈ C2(X) ∩C∞(X)
1I/ R=µt
1 1R #%ft ∈ L+
1 (X) ∩ C∞(X)
t > 0
âã&ä½âÖÞ
" P / µ0 7→ µt ? #P F,% 7 L 1(
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µ0 % =( . P R,% 7 L x1, . . . , xk
' (* σ(x)
*V (x, y)
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L C2(X)
& = ! x=P R2%
P( F P ? R LR(* 1 ( = δkµt
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(x1, . . . , xk;µ0)F1 #
PP R,% 7 Lx1, . . . , xk
PF PL %! 1 7(/ P (C2(X))∗
R 1 =L(*
∥
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g,δ2µt
δµ20
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C(t, ‖µ0‖) # PP 2% 7 D = P
g
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PP PI R F PD%(" / µ0 7→ µt
(C1(X))∗
R/( P ‖µt‖M(X) ≤ C(t)t−1/α‖µ0‖(C1(X))∗ .
/ § Z&beZO;lkZcbqrXZcO*O9^½G[cKsO lu8iokOhN0lkpMpMbr^M t 7→ µtbeZ^siu[&y*ik^f[cbr^Y_Mik_1Z½br^ M+(X) ¬ _s^MqbrkOb[]Za]O9Zw[ca]bry[cbrik^ª[]i
L1(X)®b^2[cKsO$^MivacN []ikp8ikqrikkXªlvZSl_M^sy[cbrik^.iu
tlu[t = 0
iva lua]Mb[]a]lkacXµ0£KsbryzKªbeZ^siu[lksZcikqr_I[cO9qXyiv^v[]b^Y_Miv_sZ£b[cKma]O9Zcp8O9y¡[S[ciÁO*8O9Zck_MO$N0O9lkZc_Ma]O MO*^sy*O[cKsO(^MO;yO9Z]Zcb[\X[]i0£ ivac3br^[cKMO £4O9lk3[civp1ivqivkX3iu
M(X)M£KMbeyzK£brqq1O O*nYpsqivb[]O9xb^[cKsO ^MOnY[J%KMO9ika]O*N
' _sa%Zw[cazlo[]O*vX3b^xIO9lkqbr^M£b[]KO9df_slo[]biv^ ¬ uv® beZ4[cKMOZ]luN0OlkZ%br^[]KMOpMa]O*Ybiv_sZ%ZwO;y¡[]biv^lk^sxbrZ)slkZcO9x$ik^[cKsOiksZcO*a]olo[]biv^[cK1lo[hl Zcikqr_I[cbrik^[ci[]KMbeZ O9df_slu[cbrik^0brZ l <snYO;xp1ivb^f[hikDlN3lupMpsb^Mut 7→ Ft
M£KMO*a]Oikaut ∈ C([0, t0],M+(X))
[cKMO_s^sy¡[]biv^FtZwivqvO9Z [cKMO O;df_slo[]biv^
d
dtFt(x) = −|∆|α/2(σ(x)Ft(x)) − (Vt(x, [u]),∇Ft(x)) − divVt(x, [u])Ft(x)
+
∫
(Ft(x− z)ψt(x− z, z, [u])− Ft(x)ψt(x, z, [u]))) dz¬ kv®
£b[cK[]KMO b^sb[]brlkqxMlo[zlF0 = u0
I£KMO*a]O
Vt(x, [u]) =
∫
V (x, y)ut(dy), ψt(x, z, [u]) =
∫
ψ(x, y; z)ut(dy).
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Gt(x) = σ(x)Ft(x)qO;lkxIbr^M[ci[cKsO O9df_slo[]biv^
d
dtGt(x) = −σ(x)|∆|α/2Gt(x) − (Vt(x, [u]),∇Gt(x))
+ [(Vt(x, [u]),∇ ln σ(x)) − divVt(x, [u])]Gt(x)
+
∫
(σ(x)σ−1(x − z)Gt(x− z)ψt(x− z, z, [u])−Gt(x)ψt(x, z, [u]))dzdy.¬ 1C;®
ääÉNPORQRSRT
@ L
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Ft(x) =
∫
σ−1(x)G(t, s, x, y; [u])σ(y)Fs(dy),
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C∞(X)lk^sx£KMO*a]O
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[]i l Zw[ca]ik^MvqXy*ik^f[cbr^Y_Mik_1ZZwO9N0bvaciv_MpsZ0iu beZwivN0O[ca]brO9Z3br^L+
1 (X)lu^sx M+(X)
£KMO9acOmZw[ca]ik^My*ik^f[cbr^Y_Mb[\Xmb^ y*lvZwO M(X)
beZ_M^sxMO*azZ\[]ifiIx2b^.[]KMO3ZwO9^sZwO0ik []KMO0£ O;luª[]ikp8ikqrikkXªik M(X) J%KMOvO*Xpsacivp1O9aw[\X3ik[]KMbrZZcO*N0brka]ik_Mpikqrqrio£Zha]ikN ¬ @f®¡³ Ft ∈W 1(X)lu^1x
‖Ft‖W 1(X) ≤ C1(t0)(t− s)−1/α‖Fs‖M(X)¬ vk®
£KsO*^MO9kO*at > s
lu^sxFs ∈ M(X)
² O(ZcKsluqrqikqrqrio£ ^Mio£¯[cKMOZ]luN0O qrb^sO iu)lua]k_MN0O9^v[lvZ%b^mJ%KMO9ika]O*N CM_M[£b[]K u08O*br^Mlk^
lkacsb[]a]lkacX0O*qrO*N0O9^v[ikL+
1 (X)iva M+(X) ¢ ^1Z\[]O9lkxiu ¬ PC9®4iv^MO KslkZ
‖(L1(s) − L2(s))g‖L1(X) ≤ C1‖u1(s) − u2(s)‖M(X)‖g‖W1(X).¬ kv®
iv^sZcO9df_MO*^f[]qXv ¬ vv®¡ ¬ vu® ¬ vk®4XYbO9qrx
sup0<s≤t
‖Fs([u1]) − Fs([u
2])‖L1(X) ≤ t1−1/αC1C2(t0)‖u0‖M(X) sup0<s≤t
‖u1(s) − u2(s)‖L1(X).
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ut[]iFtbr^[]KMO Zw_ssZwO*[4ik
S0C([0, t0],M+(X))ivI[]lkb^sO9xYX <MnIbr^M
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a]ikN [cKMO([cbrN0Ot1ikM[]lubr^Mbr^Ml3_M^sbrdf_MO(y*ik^f[cbr^Y_slo[]biv^iu[cKMO$Zwivq_I[]biv^[ci3[]KMObr^v[]O*a]oluq
[0, t2]£b[cK
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u1(t), u2(t)lua]O[cKMO<MnIO;xp8ikbr^f[]Z%[]i
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u10, u
20 ∈ M(X)
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[]KMOyivaca]O9Zcp8ik^sxIbr^M$pMacivpsluflo[]ikazZ)[]i ¬ kk® J%KMO9^
u1(t) − u2(t) = U1(t, 0)u10 − U2(t, 0)u2
0 = (U1(t, 0) − U2(t, 0))u10 + U2(t, 0)(u1
0 − u20)
=
∫
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s)U1(s, 0)u1
0 ds+ U2(t, 0)(u10 − u2
0).¬ kv®
iv^sZcO9df_MO*^f[]qX
sup0≤s≤t
‖u1(s) − u2(s)‖M(X) ≤ ω sup0≤s≤t
‖u1(s) − u2(s)‖M(X) + ‖u10 − u2
0‖M(X)
âã&ä½âÖÞ
k~
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(1 − ω) sup0≤s≤t
‖u1(s) − u2(s)‖M(X) ≤ ‖u10 − u2
0‖M(X).
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‖(L1(s) − L2(s))g‖(C1(X))∗ ≤ C1‖u1(s) − u2(s)‖(C1(X))∗‖g‖L1(X)lk^sxacO;Zwp8O9y[cbrkO*qrX
sup0<s≤t
‖Fs([u1]) − Fs([u
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≤ t1−1/αC1C2(t0)‖u0‖(C1(X))∗ sup0<s≤t
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k£KMbeyzKbeZ)lZc_MI¤ZcO[br^ M+(X)
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C0(M+M (X))
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C(M+M (X))
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L∗t g = −σ(x)|∆|α/2g + (Vt(x, [u]),∇F(t)) +
∫
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0
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0
)
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0 − u0).
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0
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x,z|δn
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' ¦sikqrqio£Z4brN0NO;xIbelo[cO9qXYXxIbÀ8O9acO9^v[]brlu[cbr^M ¬ @v~v®4_sZcbr^M[cKMO pMa]iIxI_sy[%a]_MqrO ² Oy9lu^ Zw[]luac[^Mio£ [cKsOlu^slkqXIZcbrZiu%[cKMOpMaciIy*O9Z]Z iu%¾s_sy[c_slu[cbrik^sZ$lua]ik_M^1xA[]KMOO9kivq_I[]biv^´xIO¤Z]ya]br1O;xYX3[]KMO Yb^MO*[cbeyO9df_slu[cbrik^sZ
»h» Ð » F (Y )
5% ¬ o® " g = gm ∈ C2,sym
∞ (Xm)
µt =
µt(B) 1 (- P ¬ C9k® ΛB,h
t
1 ' 1 L R h−1/2(Zh
B(t) − µt) >1 ZhB(t)
1( ( +% 1 ¬ u~k®(* ¬ @kk® " 5% ¬ ~kk® a = h1/2
L = LhB
1ΛBk,h
t F 1,% = ' P R
ΛBkht F (Y ) =
m∑
l=2
h−1+l/2
∫
Xm−l
Φkh[Bl
kgz](√hY + µt)Y
⊗(m−l)(dz)
+ h−1/2
∫
Xm−1
(Φkh[B1
kgz](√hY + µt) − (Bkg
+z, µ⊗k
t ))Y ⊗(m−1)(dz). ¬ ~1C;®
P =L(*
ΛB1,ht F (Y ) =
∫
Xm−l
∫
X
B1gz(y)(dy)Y⊗(m−1)(dz)
+
m∑
l=2
h−1+l/2
∫
Xm−l
∫
X
Bl1gz(y)(
√hY + µt)(dy)Y
⊗(m−l)(dz). ¬ ~vk®
' § pMpMqrXYb^M ¬ u® [ci
ΩtF (Y ) = F ((Y − µt)/√h) = h−m/2
m∑
p=0
(−1)m−p(g, Y ⊗p ⊗ µ⊗(m−p)t )
XYbrO*qexMZ
hk−1LhkΩtF (Y ) = h−m/2
m∑
p=1
(−1)m−p
p∑
l=1
hl−1
×∫
Xm−p
∫
Xp−l
(
Φkh[Blgw,u](Y )
)
Y ⊗(p−l)(dw)µ⊗(m−p)t (du),
âã&ä½âÖÞ
C
lk^sxyik^1ZwO;dv_sO*^f[cqrX
hk−1Ω−1t Lh
kΩtF (Y )) = h−m/2m∑
l=1
hl−1m∑
p=l
(−1)m−p
×∫
Xm−p
∫
Xp−l
(
Φh[Blgw,u](√hY + µt)
)
(√hY + µt)
⊗(p−l)(dw)µ⊗(m−p)t (du)
= h−m/2m∑
l=1
hl−1m−l∑
r=0
m∑
p=r+l
Cm−pm−l−r(−1)m−p
×∫
Xr
∫
Xm−l−r
(
Φkh[Blgz,y](
√hY + µt)
)
hr/2Y ⊗r(dz)µ⊗(m−l−r)t (dy),
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p=n
(−1)m−pCm−pm−n =
1, n = m
0, n < m.
VI_MI[cazlky[cbr^M
h−1/2
(
δF
δY(Y ), µt
)
= h−1/2
∫ ∫
(Bkg+z
(y)Y ⊗(m−1)(dz)µ⊗kt (dy)
XYbrO*qexMZ ¬ ~PC9® § Z%[cKMO[]O*a]N3Z%iuikazxIO*a h−1/2 y9lu^sy*O*qbr^ ¬ ~1CG®4ik^MOikM[]lubr^sZ
» » ÐI ¸
ΛBk,ht F (Y ) = ΛBk
t F (Y ) +O(√h) ¬ ~kv®
ΛBk,ht F (Y ) =
∫
Xm−1
∫
Xk
(Bk(g+w
))(v)(Y ⊗ µ⊗(k−1)t )(dv)Y ⊗(m−1)(dw)
+
∫
Xm−2
∫
Xk
(B2k(g+
w))(v)(µ⊗k
t )(dv)Y ⊗(m−2)(dw) ¬ ~uf®
1 4 R 1 m = 1
" *9O(
√h)
, P,% = ! Y
'
m+k−1 = 4=5% 1 /5 = ! P )FR ' 1 = P
Blkg, l = 1, . . . ,m
*#=P R,% =(µt
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ääÉNPORQRSRT
Y L
» » ÐI ¸ = F' R ¬ u® P ! P F
ΛBk
t
¬ ~uv®I +%
ΛBt F (Y ) =
(
B1k
δF
δY, Y ⊗ µ
⊗(k−1)t
)
+
(
B2k
δ2F
δY 2, µ⊗k
t
)
, ¬ ~vk®9 ,%
ΛBk
t F (Y ) =
(
Bk
(
δF
δY
)+
, Y ⊗ µ⊗(k−1)t
)
+
1
2Bk
k∑
i,j=1
δ2F
δY (yi)δY (yj)−
By1,...,yk
k
k∑
i,j=1
δ2F
δY (zi)δY (yj)
|∀i zi=yi, µ⊗k
t
. ¬ ~kv®
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g(x) = −σ(x)|∆|α/2g(x) +
∫
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1
supt≤T
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(
g2,
(
ZhB(t) − µt(µ0)√
h
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hB(0))√
h
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v L
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)
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(
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LBf(x) =K∑
k=1
Lkf(x) =K∑
k=1
∑
I⊂1,...,l:|I|=k
[(BIkf)(x) +
∫
(f(y,xI ) − f(x))Pk(x, dy)] ¬ C;kv®
ivax = (x1, . . . , xl)
£b[cKlkacMb[cazlua]Xl J%KMOSZ]y*lkqO;x3pMa]iYy*O9Z]Z)iv^ M+
δ,h(X)beZ lkvlubr^3kbrkO9^3fX ¬ of®¡
¬ Gv® _sa$y*lkqry*_Mqelo[cbrik^1Zbr^ VYO9y[cbrik^sZ0ªlk^sx m£4O*a]Oy9lua]acbrO9x.ik_M[$br^ Zc_syzK lm£%lGXm[]Kslo[[cKMO9X´lua]OZw[cazlubrkKf[cika]£4lka]xIqrXkO*^sO*azluqrbLK9O9x3[]i[cKMO^MO9£¯N0iIxIO*qik[]KMbrZZwO;y¡[cbrik^ ¦sikabr^sZw[]lu^1yOk ¬ o® ¬ @k®%Zw[cbrqqKsikqexI£KMO*a]Oika%[]KMO b^f[]O*kazluqivp1O9a]lu[civaBk : Csym(X ) 7→ Csym(Xk)
iu[cKMOivacN
Bf(x) =
∫
X
(f(y) − f(x))P (x, dy), x ∈ Xk,
âã&ä½âÖÞ
Y
[]KMO ikp8O*azlo[]ikaBl
k : Csym(X l) 7→ Csym(Xk)beZkbrkO9^YX
Blkg(x) =
l∑
q=1
(−1)l−q
∫ ∫ ∫
g(y, z)(δu − δx)⊗q(dy)P (x, du)δ⊗(l−q)x
(dz). ¬ C;1C;®
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b^ ¬ C9vk®lu^sx.Ybr^MO[]bry0O;dv_1lo[cbrik^ ¬ C;k®[zlukO;Z[cKMOivacNd
dt(g, µt) =
K∑
k=1
∫ ∫
(g+(y) − g+(x1, . . . , xk))Pk(x1, . . . , xk; dy)µ⊗kt (dx). ¬ C;vk®
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ΛBk
t F (Y ) =
∫ ∫
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+1
2
∫ ∫
(δ2F (z, z) + δ2F (y,y) − 2δ2F (y, z))Pk(y, dz)µ⊗kt (dy) ¬ C;kf®
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brZ4kbrkO9^YX0[]KMO O9df_slu[cbrik^
g(x) =
K∑
k=1
N∑
n=1
∫
[
n∑
i=1
g(zi) − g(x) −k−1∑
i=1
g(yi)
]
Pk(x, y1, . . . , yk−1; dz1 · · · dzn)µ⊗(k−1)t (dy).
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[cKMOy*ika]acO;Zwp8ik^sxMb^MO*vikqr_I[cbrik^.ikp8O*azlo[]ikazZiv^
gbr^b^YvO*azZwOS[cbrN0O ¬ slvyzf£%luazxpsacivpsluflo[civaz® ² OZcKslkqq yiv^sZcbrxIO9aS^sio£°[cKMO0Z\[zlu^sxsluazxªN0iYxMO*qiuhy*ivluv_Mqelo[cbrik^mlk^sxªazluvN0O*^f[]lu[cbrik^myivN(Mbr^MO;x£b[cK.p8ivZ]ZwbrMqrOy*ikqrqbeZwbrik^AMacO;luoluvOkD£KMO9acO
X = R+PkxIiYO9Z^Mik[ Glk^MbeZwK2ik^MqrXika
k8O*br^M[\£4i
lk^sxik^MOvMZwi[cK1lo[P2(x1, x2; dy) = K(x1, x2)δ(x1 + x2 − y)dy + C(x1, x2, y1)δ(x1 + x2 − y1 − y2)dy1dy2,lk^sxP1(x; dy1dy2) = F (x, y1)δ(y1 + y2 − x) dy1dy2 J%KsOyik^f[]b^Y_Miv_sZ_M^sy[cbrik^sZ K,C, F lkacOy9luqrqO;x[cKMOyivlkk_Mqelo[]biv^½MyivqqrbeZwbrik^ªlu^1xa]lkkN0O*^f[]lu[cbrik^kO9ac^sO*qeZacO;Zwp8O9y[cbrkO9qX J%KMO$yika]a]O9Zcp1iv^sxIbr^MqelG£ ikqelua]kO^Y_MN(8O*azZ ¬ C;vu® [zlukO;Z4[cKMOivacN
d
dt(g, µt) =
∫ ∞
0
∫ ∞
0
(g(x1 + x2) − g(x1) − g(x2))K(x1, x2)µ⊗2t (dx1dx2)
+
∫ ∞
0
∫ ∞
0
∫ x1+x2
0
dz(g(z) + g(x1 + x2 − z) − g(x1) − g(x2))C(x1, x2, z)µ⊗2t (dx1dx2)
+
∫ ∞
0
∫ x
0
dz(g(z) + g(x− z) − g(x))F (x, z)µt(dx)¬ C;vv®
ääÉNPORQRSRT
f L
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supx1,x2,x
max(K(x1, x2),
∫
C(x1, x2, z)dz,
∫
F (x, z)dz) <∞. ¬ C9vv®J%KsO*^ b[$beZ(£4O*qrqhY^Mio£^ []Kslo[ ¬ C;vu®brZ$£ O9qq p1ifZwO;x ¬ ZcO*OO >ÏkA4ika0yifluv_Mqrlu[cbrik^sZ lk^sx'> vA%br^vO*^MO9a]lkqMy*lvZwOG®¡ub O ikahlk^fX <s^Mb[cO µ0
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xµt(dx) ¢ ^ > fAlk^sx,> k~ Ab[SbrZZwKsio£^[]Kslo[S[cKMbeZSO9df_slo[]biv^KsikqexMZ lkqrZcibr^[]KMOZw[ca]ik^s(ZwO9^sZcO£b[cK[]KMOxIO*a]bolu[cbrkO8O*br^M_M^sxIO9a]Zw[ciYiIx3br^[cKsOSZcO*^sZcOik½[cKMO^MivacNik M(X) J%KMbeZlkqqrio£Z%[cixMbÀDO*a]O*^f[cbelo[]O [cKsbrZO9df_slo[]biv^£b[]KmacO;Zwp8O9y[[ci0[]KMO(br^Mb[cbeluqNO;lkZc_Ma]O µ0 « dv_1lo[cbrik^ikalkqqxIO*a]bolu[cbrkO9Z%lkacOikYYbrik_sZcqX3qrbr^MO9lka%lk^sxYXl0ZwbrN0pMqO br^sxI_1y¡[cbrik^iv^MO ZwO9O9Z4[cK1lo[
supx1,...,xl
∥
∥
∥
∥
δlµt
δµl0
(x1, . . . , xl)
∥
∥
∥
∥
≤ Cl(µ0)tll! ¬ C; k®
ivaluqrql£b[]KZcikN0O yiv^sZw[]lu^f[
C(µ0) VYbrN0bqelua [ci$J%KMO*ivacO9N £4OvO[h^Mio£[]KMOivqqrio£b^sacO;Zw_sq[hiv^0[cKMOSa]lu[cOikyiv^fvO*a]kO9^syO4[ci([cKMOqelG£ ikqelua]kO^Y_MN(8O*azZ ¬ C;vu® ·Y» · R=/ ¬ C9vk®? 1 =L )' R ), / / ¬ C9fu® gm ∈ C∞(Xm)
F (µ) = (gm, µ⊗m)
R= µ = hδx
x = (x1, . . . , xn)
1supt≤T
‖T ht F (µ) − TtF (µ)‖ ≤ hC(m,T, ‖µ‖)‖g‖C(Xm)
1 F C
FP m,T, ‖µ‖
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1
k!
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l=2
hl−1
∥
∥
∥
∥
Blk
δlFt
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∥
∥
∥
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BPh)TsF (µ)beZikiva]xIO9a
hbr^ ¬ @ @k®¡£KsbryzK©brNpsqbrO9Z[cKMOAZw[]lu[cO9NO9^f[iu[cKMOJ%KsO*ika]O*N
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(
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∥
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h
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)
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g ∈ C∞(X)1
supt≤T
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∣
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E
(
g,Zh
B(t) − µt(µ0)√h
)
−(
Gt,0g,Zh
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)∣
∣
∣
∣
≤ C(T, ‖ZhB(0)‖)
√h‖g‖
(
1 +
∥
∥
∥
∥
ZhB(0) − µ0√
h
∥
∥
∥
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2
M(X)
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Blu^sx
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£b[cKg1, . . . , gm ∈ B
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£b[cKX8O*br^M
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iu[]KMO qrb^MO;luay*ik^f[cazlky[cbrik^sZ4br^Biu&[cKsOika]N
Pjv =
Lj∑
l=1
(wlj , v)φ
lj ,
¬ C;k~f®
£KsO*a]Oφl
j
lk^sxwl
j
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1
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j
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j
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j
lk^sx´olu^sbrZcKMO9Zika‖x− xl
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Pjf(x) =
Lj∑
l=1
f(xlj)φ
lj(x) =
Lj∑
l=1
(f, δxlj)φl
j(x)
ivacN lu^lupMpMa]iGnIbN3lu[cbrkObrxIO9^f[cb[\Xb^B = C∞(X)
»h» Ð » P /$ ,%P1, P2, · · ·
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"3 % F ∈ C(B∗1 )
1 ,%Fj = F (P ∗
j ) R F
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C(B∗
1 )
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l=1
(φlj , µ− ν)2
, ν ∈ B∗,
R/4 P Πj(F )
P F=P R ,%
B∗1
* " F
k /R PP R,%. 7 L 1 1 P
δkF (µ)(v1, . . . , vk)4
∗ ) 2%: PP = ! k + 1
R LR(* 1 P6 R R?' P
k!
Πj(F ) 1 R 1 .R !
F= R ,%?
B∗1
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Lj∑
l=1
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j .
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j (µ))yqrO9lua]qrXy9lu^8O4£acb[w[]O*^(br^[]KMO ika]N
Fj(µ) =
fj(y(µ))£b[cK
y(µ) = (φ1j , µ), . . . , (φ
Lj
j , µ) lk^sx fj1O9b^M8ik_M^sxMO9xAy*ik^f[cbr^Y_Mik_1Z_M^1y¡[cbrik^1ZSik
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exp−εj∑Lj
l=1(yj − ξj)2 £ O[]KMO*^ªxIO<s^MO
Πj(F (µ)) = hj(y(µ))1£KMbryzKO9^oj\ioXIZh[]KMO a]O9df_MbracO;xpMa]ikp8O*ac[\X ¬ bbrbe® ^MOik^MqrX^MO9O9xMZ4[]i3^Miu[]bry*OS[]Kslo[b F beZ k []bN0O;Zyiv^v[]b^Y_Miv_sZcqXxIbÀDO*a]O*^f[cbeluMqrOkY[]KMO*^
δkFj(µ)(v1, . . . , vk) = δkF (P ∗j (µ))(P ∗
j v1, . . . , P∗j vk)
lk^sx[cKMO9^[]KMO a]O9Zc_Mq[%ivqqrio£Z4acivN ¬ bÖ® '4( # »h» Ð »< /% =
k*19 7 7/
f 7→ Φk[f ] = Φ[f ]
1I 1 Csym(Xk)
F 1 1,% = / R=R ' I kR/( 1
∑
I⊂1,...,n,|I|=k
f(xI ) = Φk[f ](δx) ¬ C C9f®
,5 %P x = (x1, . . . , xn)
X I/ =P 1 R
Φk[f ](Y ) = (f, Y ⊗k) +
k−1∑
l=1
(−1)l(Φkl [f ], Y ⊗(k−l)) ¬ C CCG®
Φk
l [f ] 31R %7 RR = (I 1
Csym(Xk)F1 )
P = P ! k − l
R DR /P =L(*
Φ2[f ](Y ) =1
2
∫ ∫
f(y1, y2)Y (dy1)Y (dy2) −1
2f(y, y)Y (dy). ¬ CCGk®
/ H4ik[cKZwbexIO9Z%ik ¬ CC;k®4olu^MbeZcKbr^ªy*lkZcO n < k ' 1ZwO9acYbr^M([]Kslo[
∑
I⊂1,...,n,|I|=k
f(x1) = (f, (δx)⊗k) − σ,
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k − 1xIbÀ8O9acO9^v[Sp1ivb^f[]Z9
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I⊂1,...,n,|I|=k
f(x1) = (f, (δx)⊗k) − 1
k!
k−1∑
j=1
jj!∑
J⊂1,...,n,|J|=j
P jf(xJ , z)δ⊗(k−j−l)x
(dz),
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Pf(x1, . . . , xn−1) = f(x1, x1, x2, . . . , xn−1).
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Φkl (f) =
k−1∑
il=l
il
il−1∑
il−1=l−1
il−1 · · · i2i2−1∑
i1=1
i1Pi1 · · ·P i1f.
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Φkl [f ](Y ) = hkΦk[f ](Y/h). ¬ CC;v®
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hk∑
I⊂1,...,n,|I|=k
f(xI) = Φkh[f ](hδx) = (f, (hδx)⊗k) +
k−1∑
l=1
(−h)l(Φkl [f ], (hδx)⊗(k−l)). ¬ C C*Y®
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Unité de recherche INRIA RocquencourtDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)
Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)
Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)
Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)
Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399
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