Stochastic Dynamics of Discrete Curves and Exclusion Processes. Part 1: Hydrodynamic Limit of the ASEP System

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Stochastic Dynamics of Discrete Curves and Exclusion Processes. Part 1: Hydrodynamic Limit of the ASEP

System

Guy Fayolle, Cyril Furtlehner

To cite this version:

Guy Fayolle, Cyril Furtlehner. Stochastic Dynamics of Discrete Curves and Exclusion Processes.

Part 1: Hydrodynamic Limit of the ASEP System. [Research Report] RR-5793, INRIA. 2005, pp.22.

�inria-00001131�

ISRN INRIA/RR--5793--FR+ENG

a p p o r t

d e r e c h e r c h e

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Stochastic Dynamics of Discrete Curves and Exclusion Processes.

Part 1: Hydrodynamic Limit of the ASEP System

Guy Fayolle — Cyril Furtlehner

N° 5793

December 2005

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¡rs¨no_dWªO…cw¨{msvuK†À«€\†„<Ÿ uK}muKbEd…”€Kmdpm…puK}o}dpmoƒ9uK†„Os¨†tx—nuEno_d/}dc‹dc}mos¨qts¨w¨s¨n¹k?u\Ÿ"no_d8ƒt}ul…pdpmm”«

€ `:s¨qtqm`Ÿ¯uK}obϟ¯uK}rno_ds¨†lK€\}osĀ\†lnbEd”€Kmo{t}d/¡¸€Kmxys¨‹dc†?s¨†UJŠ K

49 BC$@A>/G 2 9 C NBD: 4>@ :=BDG->NBDC < CG=@ @ :c9&>U:&<<DG->NB

(=9 /:6GLG %8

O‚m®bEdc†lnosvuK†dp„ €\q9u”‹dy«`¡¸d©€\s¨b €\n—uKqtn€\s¨†ts¨†txº_­kO„O}ul„OkM†€\b®sv…?dpzl{€\nosvuK†mEŸ uK}£€¾…cwĀKmm¤u\Ÿ

dWªt…cw¨{mosvuK†—bEul„tdcwvm”§(^`_dbEdcno_uM„X«M€\w¨no_uK{txy_<}dcw¨kMs¨†tx uK†£…cwĀKmmosv…”€\wƒu¬¡¢dc}=Ÿ¯{twnuMuKwvm

Q

b¤€\}onos¨†O¥

x­€\wvdpm¬«O}dcwĀ\nos¨‹d…puKb®ƒ€K…cno†dpmmu\Ÿ"bEd”€Kmo{t}dpm”«MŸ¯{t†…cnosvuK†€\w€\†€\w¨kMmsvm&RW«M_€KmmuKbEd‚†dc¡ Ÿ¯d”€\no{t}dpm

¡r_tsv…_»mo_uK{twv„E_uKƒ9dWŸP{tw¨w¨k¤ƒt}u”‹dŸP}o{ts¨n=Ÿ¯{tws¨†<uKno_dc}`…puK†­ndWªOnm”§(^`_ddpmmdc†…pdu\ŸÀno_d:€\ƒtƒt}u‹€K…_

svm¸s¨†<ŸÁ€K…cnr…puK†­n€\s¨†dp„»s¨†<no_d€\†€\w¨kMmsvmru\ŸJno_d:ƒuKƒt{twĀ\}r±²œ³´£bEuM„tdcwÁ«Oƒt}dpmdc†­ndp„Žqdcwvu¬¡/§ W¾d

†uKnd no_d®„Os¤…c{tw¨n¹k©nu£Æ†„s¨†Hno_d¼dWªMsvmnos¨†tx£w¨s¨ndc}€\no{t}d—€—…puKb®ƒtwvdcnd¤mono{„Ok©dc†…puKb®ƒ€Kmms¨†tx

y€\}osvuK{m¼moƒ9dp…csĀ\w`…”€Kmdpm

Q

mklb®bEdcno}ok‹«(nuKn€\w€KmokMb®bEdcno}ok‹«Bdcn… RW§µiMuKbEd£ƒt}uMu\Ÿ¯m ¡rs¨w¨w`uK†tw¨k q9d

m½‹dcn…_dp„X«€\†„<no_d/}dcwĀ\ndp„»}dpmo{tw¨nmƒt}dpmdc†­ndp„?€Km…cwĀ\s¨bEmuK}dc‹dc†?…puK†K~odp…cno{t}dpm”§

¿¸uK†msv„tdc}

N

mos¨ndpm`wĀ\q9dcw¨wvdp„<Ÿ¯}uKb

1

^nu

N

«MŸ uK}ob®s¨†tx—€¼„Osvm…c}dcnd8…cwvuymdp„»…c{t}o‹ds¨†»no_dƒtwĀ\†dy«

muno_€\n(no_d`†l{tb8q9dc}os¨†tx8u\Ÿms¨ndpm‰svm‰s¨b®ƒtw¨sv…cs¨now¨k n€\½‹dc†®bEuM„O{twvu

N

^«KsÁ§dy§‰uK†®no_d`„Osvm…c}dcnd`nuK}o{m

G ^{( N )}

= Z /N Z

§¼f°†H_ts¨xy_dc}8„Os¨bEdc†msvuK†À«m€”kuK†Hno_dEwĀ\nonosv…pd

Z ^k

«Xno_d®}dcwĀ\ndp„ mdcn/u\Ÿ`ms¨ndpm

¡¸uK{twv„»q9d„O}€p¡r†©uK†?no_d:nuK}o{m

( Z /N Z ) ^k

^§

W¾d:x­€\no_dc}qdcwvu¬¡)muKbEd:†uKn€\nosvuK†€\wb¤€\ndc}osĀ\wJy€\w¨sv„<no_t}uK{txy_uK{tnno_tsvmmdp…cnosvuK†À§

• R

mon€\†„tmŽŸ uK}no_d }d”€\w w¨s¨†dy§

C ^k [0, 1]

svmno_dµ…puKw¨wvdp…cnosvuK†u\Ÿ—€\w¨w }d”€\wÃ¥ÁK€\w¨{dp„X«

k

^¥

…puK†­nos¨†l{uK{mow¨k „OsÃÂ9dc}dc†lnosĀ\qtwvd<ŸP{t†…cnosvuK†m¤„tdWƆdp„ uK† no_d—s¨†lndc}oK€\w

[0, 1]

^«¸€\†„

M

^svm

no_d/moƒ€K…pdu\ŸB€\w¨wXƆts¨ndƒ9uymos¨nos¨‹d:bEd”€Kmo{t}dpmuK†Žno_d:nuK}o{m

G

= [0, 1)

^§

C ₀ ^∞ (K)

svm`no_dmoƒ€K…pdu\Ÿs¨†OƆts¨ndcw¨k»„OsÃÂXdc}dc†­nosĀ\qtwvd:Ÿ¯{t†…cnosvuK†mr¡rs¨no_?…puKb®ƒ€K…cnmo{tƒtƒuK}on s¨†…cw¨{„tdp„Žs¨†

K

^§

•

^TuK}

S

€\†€\}oqts¨no}€\}okbEdcno}osv…Bmƒ€K…pdy«

P ( S )

svmXno_d¢mdcnJu\ŸOƒt}uKq€\qts¨w¨s¨n°k/bEd”€Kmo{t}dpm

D S [0, T ]

svmno_dEmoƒ€K…pd®u\Ÿ`}os¨xy_­n…puK†­nos¨†­{uK{mŸ¯{t†…cnosvuK†m

z : [0, ∞] → S

¡rs¨no_HwvdWŸPn/w¨s¨b®s¨nm€\†„

t → z _t

^§

•

^TuK}

i = 1, . . . , N

^«(wvdcn

A ⁽ _i ^{N )} (t)

^€\†„

B _i ^{( N )} (t)

q9d¤qts¨†€\}okµ}€\†„tuKb K€\}osĀ\qtwvdpm¼}dcƒt}dW¥

mdc†­nos¨†tx¤}dpmoƒdp…cnos¨‹dcw¨k»€¼ƒ€\}onosv…c{twvd/uK}€ _uKwvd€\nmos¨nd

i

«tmu¼no_€\n”«tu”¡rs¨†tx—nu¼no_ddWªO…cw¨{O¥

mosvuK†µ…puK†mno}€\s¨†­n”«

A ⁽ _i ^{N )} (t) + B _i ^{( N )} (t) = 1

«Ÿ¯uK}€\w¨w

1 ≤ i ≤ N

§<^`_l{m

A ^{( N )} (t)

=

A ⁽ _i ^{N )} (t), . . . , A ⁽ _N ^{N )} (t)

, t ≥ 0

svm€€\}o½‹u¬—ƒt}uM…pdpmm”§

• Ω ^{( N )}

¡rs¨w¨w‚„tdc†uKndno_d©x‹dc†dc}€\nuK}<u\Ÿno_d.€\}o½‹u” ƒt}uM…pdpmm

A ^{( N )} (t)

^«€\†„

F _t ^{( N )} = σ A ^{( N )} (s), s ≤ t

svmrno_d/€KmmuM…csĀ\ndp„»†€\no{t}€\wXÆw¨no}€\nosvuK†À§

•

g{t}ƒt{t}oƒuymd:svmrnu¤€\†€\w¨kM·”dno_dmdpzl{dc†…pd/u\Ÿ‰dcb®ƒts¨}osv…”€\w}€\†„tuKbÏbEd”€Kmo{t}dpm

µ ⁽ _t ^{N )} = 1 N

X

i∈ G ^{( N )}

A ⁽ _i ^{N )} (t)δ ⁱ

N ,

^{QO§¨“ R}

¡r_dc†

N → ∞

«€œŸ¯ndc}€»…puK†l‹dc†tsvdc†­n®m…”€\w¨s¨†tx©u\Ÿ`no_dEƒ€\}€\bEdcndc}m u\Ÿrno_d®x‹dc†dc}€\nuK}

Ω ^{( N )}

§^`_d©ƒt}uKq€\qts¨w¨s¨n°k „Osvmono}os¨qt{tnosvuK† €Kmmul…csĀ\ndp„ ¡rs¨no_&no_d©ƒ€\no_ u\Ÿ/no_d.€\}o½‹u”

ƒt}ul…pdpmm

µ ⁽ _t ^{N )} , t ∈ [0, T ]

«OŸ uK}muKbEd:Ætªtdp„

T

«Osvmmos¨b®ƒtw¨k<„tdc†uKndp„Žqlk

Q ^{( N )}

^§

O‚m/{m{€\wÁ«"uK†d—…”€\†µdcbqdp„

G ^{( N )}

^s¨†

G

«‰mu?no_€\n €»ƒ9uKs¨†­n

i ∈ G ^{( N )}

…puK}o}dpmoƒ9uK†„tm8nu?no_d

ƒ9uKs¨†­n

i/N

^s¨†

G

^§ Z‚dc†…pdy«ts¨†ŽMsvdc¡u\Ÿ

Q

O§¨“ RW«ts¨nsvmzl{ts¨nd†€\no{t}€\wJnu¼wvdcnno_d/mdpzl{dc†…pd

Q ^{( N )}

q9d‚„tdWƆdp„»uK†»€8{t†tsvzl{d‚mƒ€K…pd

D M [0, T ]

«O¡r_tsv…_»q9dp…puKbEdpm€8ƒ9uKw¨svmo_<moƒ€K…pd

Q

sÁ§dy§(…puKb®ƒtwvdcnd

€\†„ mdcƒ€\}€\qtwvd R‚MsĀ<no_d¼{mo{€\w¸il½‹uK}uK½‹uM„HnuKƒ9uKwvuKxyk‹«"€Km/muluK† €Km

M

svms¨nmdcwß`|"uKw¨svmo_

Q

mdpd

dy§x§ JˆLK°«X…_€\ƒtndc}

4

^RW§W s¨no_uK{tn:ŸP{t}ono_dc}…puKb®bEdc†­n”«

M

svm/€Kmm{tbEdp„©nu£q9ddc†„tu¬¡¢dp„H¡rs¨no_

no_d<y€\xy{d<ƒt}uM„O{…cnEnuKƒuKwvuKxyk‹«`€Km¤€.…puK†mdpzl{dc†…pd»u\Ÿ:no_d£Ÿ €\bEuK{mEer€\†€K…_O¥ OwĀKuKxywvuº€\†„

^¦kM…_uK†u\¾no_dpuK}dcbEm

Q

mdpddy§x§ J¨“”O«À“”’LK.RW§

Jdcn

φ _a , φ _b

qdn°¡¸uŽ€\}oqts¨no}€\}okŽŸ¯{t†…cnosvuK†m‚s¨†

C ² [0, 1]

€\†„„tdWƆd no_d}d”€\wÃ¥ÁK€\w¨{dp„ƒ9uymos¨nos¨‹d bEd”€Km{t}d

Z _t ^{( N )} [φ _a , φ _b ]

= exp

1

N X

i∈ G ^{( N )}

φ _a i N

A ⁽ _i ^{N )} (t) + φ _b i N

B ⁽ _i ^{N )} (t)

,

^QO§‘-R

¡r_tsv…_ svm8€<ŸP{t†…cnosvuK†€\w¸u\Ÿ

φ _a , φ _b

§STuK}/no_d¤mZ€\½‹d®u\Ÿ`qt}dcMs¨n°k‹«no_dEdWªOƒtw¨sv…cs¨n„tdcƒ9dc†„tdc†…pd¤u\Ÿ

A ⁽ _i ^{N )} (t), B _i ^{( N )} (t), Z _t ^{( N )} [φ _a , φ _b ]

^uK†

N, t, φ

«œ¡rs¨w¨wlqd¦uKb®s¨nondp„8¡r_dc}dc‹dc}"no_dBbEd”€\†ts¨†tx‚}dcb¤€\s¨†m …cwvd”€\}¢Ÿ¯}uKb4no_d‚…puK†­ndWªOnY‰Ÿ uK}¢s¨†mon€\†…pdy«M¡¸du\Ÿ¯ndc†£m_€\w¨wmos¨b®ƒtw¨k¤¡r}os¨nd

A _i , B _i

^uK}

Z _t ^{( N )}

^{§ Owvmu}

Z ^{( N )}

mon€\†„tmrŸ¯uK}no_d:ƒt}uM…pdpmm

{Z _t ^{( N )} , t ≥ 0}

^§

O$mon€\†„€\}„7ƒu¬¡¢dc}=Ÿ¯{twrbEdcno_ul„7nuƒt}u¬‹d<no_d£…puK†l‹dc}ox‹dc†…pd

Q

s¨† €mdc†md£nuqd£moƒ9dp…csÃÆdp„

wĀ\ndc}&R u\Ÿ‚no_d<mdpz­{dc†…pd»u\Ÿ‚ƒt}uKq€\qts¨w¨s¨n°k7bEd”€Kmo{t}dpm¼s¨†lno}ul„O{…pdp„ s¨†

Q

O§¨“ R …puK†mosvmnm Æ}mon®s¨†

m_u”¡rs¨†tx/s¨nm(}dcwĀ\nos¨‹d…puKb®ƒ€K…cno†dpmm”«l€\†„¼no_dc†¤s¨†®‹dc}osßPkMs¨†txno_dr…puKs¨†…csv„tdc†…pdu\ŸÀ€\w¨wtƒuymmos¨qtwvd

w¨s¨b®s¨n ƒ9uKs¨†­nm

Q

mdpd¤dy§x§UJ¨“y“(K°§©uK}dpu¬‹dc} _dc}d¤s¨n m{¤…pdpm8nu?ƒt}u”‹d—no_dpmd¤n¹¡¢u©ƒt}uKƒdc}onosvdpm

Ÿ uK}Bno_dmdpz­{dc†…pdu\Ÿ9ƒt}u\~odp…cndp„—bEd”€Kmo{t}dpm¦„tdWƆdp„£uK†

D R [0, T ]

€\†„¤…puK}o}dpmƒuK†„Os¨†tx nuno_d

ƒt}uM…pdpmmdpm

{Z _t ^{( N )} [φ a , φ _b ], t ≥ 0}

«mos¨†…pdno_d‚Ÿ¯{t†…cnosvuK†m

φ

q9dcwvuK†txEnu

C ² [0, 1]

^§

Jdcn{m†u”¡ s¨†­no}uM„O{…pd zl{€\†­nos¨nosvdpm‚¡r_tsv…_À«9€KmŸÁ€\}‚€Kmm…”€\w¨s¨†tx—svm‚…puK†…pdc}o†dp„X«X€\}d8…c}o{…csĀ\w‰s¨†

uK}„tdc}nu®uKqtn€\s¨†?bEd”€\†ts¨†txKŸ¯{twÀ_­kO„O}ul„OkM†€\b®sv…8dpzl{€\nosvuK†m”§

 



λ(N )

=

λ _ab (N ) + λ _ba (N)

2 ,

µ(N )

=

λ _ab (N ) − λ _ba (N ),

Q

O§-R

¡r_dc}d/no_d/„tdcƒdc†„tdc†…pd8u\Ÿ"no_d}€\ndpmuK†

N

svmdWªOƒtw¨sv…cs¨now¨k<bEdc†­nosvuK†dp„X§

& * ' + c% 7c% &?/c 21>// U;. 4(

 



λ(N )

=

λN ² + o(N ² ), µ(N )

=

µN + o(N ),

Q

O§ R

λ

^<;

µ

¹ 2;://&.% E c6.(' 02 ./E .87" E# ?/

*

'+N./

<; #

log Z _t ^{( N )}

^{&#( ;./ # ; >.87 %} /E&& P.

. . 2 E / E2; 2;X#/ S *( -(c

N ²

^{c<; . 16// -EE &#!}

N ⁻¹

^{" '} 02&&#

% ./S($## E ; .H ?/ &#

log Z ₀ ^{( N )}

^{%c c1.}

t = 0

EH% 7 & . / . "( ; 6 &#

.. 7c% ; H&

ρ(x, 0)

( ./ c

N→∞ lim log Z ₀ ^{( N )} = Z 1

0

[ρ(x, 0)φ _a (x) + (1 − ρ(x, 0))φ _b (x)]dx,

^{/ .}

,

^QO§ˆ-R

( H / ; # (c

φ _a , φ _b ∈ C ₀ ^∞ (K)

^/

K ∈ R

^{SE?// -( 6 cH.87 ./}

.6 %L

[0, 1]

^'

- c %

t > 0

./(&#(# E ; <;-c # &

µ ⁽ _t ^{N )}

^{6% 7 & /; ('}

,P

N → ∞

^{P ; ( &.H &# / %.87 ; .}

(ρ(x, t)

^{. &6/ (}

. &27#H &#,

X./ #H)## ¡¢d”€\½»muKw¨{tnosvuK†?u\Ÿ‰no_d¿`€\{…_lk»ƒt}uKqtwvdcb

Z T

0

Z 1

0

ρ(x, t) ∂θ(x, t)

dt + λ ∂ ² θ(x, t) dx ²

− µρ(x, t) 1 − ρ(x, t) ∂θ(x, t) dx

dxdt

= Z 1

[ρ(x, T )θ(x, T ) − ρ(x, 0)θ(x, 0)] dx,

Q

O§-R

* ' + /;4 cH #

θ ∈ C ^∞ ₀ ([0, 1] × [0, T ])

^'

, % D E# & . 21 E ;

∂ ² ρ(x,0) dx ²

.

*

' + 2;# E& M &&

= #7 &## c

∂ρ(x, t)

dt = λ ∂ ² ρ(x, t)

dx ² + µ[1 − 2ρ(x, t)] ∂ρ(x, t) dx .

^`_d¼mo½‹dcn…_¾u\Ÿ¸no_d ƒt}ulu\Ÿ¦svmmƒt}d”€K„u¬‹dc}no_t}dpd b¤€\s¨†¾mo{tqmdp…cnosvuK†m”«9}dWŸ dc}o}dp„.nu

_dc}d”€œŸ¯ndc}€Km «?€\†„À§

!"!$#%&'(*)+ -,/.012 3#145-06 OmE{mo{€\ws¨†&ƒt}uKqtwvdcbEm

„td”€\w¨s¨†tx7¡rs¨no_ …puK†­‹dc}ox‹dc†…pd u\Ÿmdpzl{dc†…pdpm£u\Ÿƒt}uKq€\qts¨w¨s¨n°k bEd”€Kmo{t}dpm”«uK{t}<‹dc}ok&mn€\}onos¨†tx

ƒ9uKs¨†­n¢¡rs¨w¨wq9drnu8dpmn€\qtw¨svmo_¤no_d¡¢d”€\½®}dcwĀ\nos¨‹d‚…puKb®ƒ€K…cno†dpmm¢u\ŸÀno_dmdcn

{log Z _t ^{( N )} , N ≥ 1}

^§

iMuKbEd¸u\Ÿno_d¢ƒt}uKq€\qts¨w¨svmonosv…€\}oxy{tbEdc†­nm(dcb®ƒtwvu¬k‹dp„¼s¨†¼no_tsvmJƒ€\}€\xy}€\ƒt_¤€\}d¢s¨†¼€¡¸€”k…cwĀKmmsv…”€\w

€\†„?…”€\†»qd‚Ÿ uK{t†„»s¨†»x‹uMul„<q9uluK½Om”«tdy§x§*J¨“”ˆt«X“y“(K°«€\w¨no_uK{txy_ŽŸ uK}mos¨b®ƒtwvdc}bEuM„tdcwvm”§

^`_d/ƒt}ul…pdpmm

U _t ^{( N )}

= Z _t ^{( N )} − Z ₀ ^{( N )} − Z _t

0

Ω ^{( N )} [Z _s ^{( N )} ]ds

^QO§ŠcR

svm:€—quK{t†„tdp„

{F _t ^{( N )} }

¥Áb¤€\}onos¨†tx­€\wvdy§87‚mos¨†tx£no_ddWªOƒuK†dc†lnosĀ\w‰Ÿ¯uK}obu\Ÿ

Z _t ^{( N )}

nuKx‹dcno_dc}:¡rs¨no_

…cwĀKmmosv…”€\wÀmonuM…_€Kmonosv…/…”€\wv…c{tw¨{m

Q

mdpd/dy§x§*JˆK°«t…_€\ƒÀ§O«ƒ€\x‹d8Œy-RW«s¨n`Ÿ¯uKw¨wvu¬¡mno_€\n

[V _t ^{( N )} ]

= (U

_t ^{( N )} ) ² − Z _t

0

Ω ^{( N )} [(Z _s ^{( N )} ) ² ] − 2Z _s ^{( N )} Ω ^{( N )} [Z _s ^{( N )} ]

ds

^{QO§^]-R}

svm€\wvmuE€ quK{t†„tdp„Ž}d”€\wÀb¤€\}onos¨†tx­€\wvdy§

T}uKb

A ⁽ _i ^{N )} (t) + B _i ^{( N )} (t) = 1, ∀1 ≤ i ≤ N

«uK†©mdpdpmno_€\n

Z _t ^{( N )}

svmb¤€\s¨†tw¨kŽ€ Ÿ¯{t†…cnosvuK†€\wu\Ÿ

no_d:muKwvdŸ¯{t†…cnosvuK†

ψ _xy

=

φ _x − φ _y = −ψ _yx

«O{tƒ<nu®€¼…puK†mon€\†lnr{t†ts߯uK}ob®w¨k£quK{t†„tdp„£s¨†

N

^§

Z‚dc†…pdy«mdcnonos¨†tx

∆ψ xy

i N

= ψ xy

i + 1 N

− ψ xy

i N

, e λ _xy (i, N )

=

λ _xy (N )

exp

1

N ∆ψ _xy i N

− 1

, xy = ab

^uK}

ba,

¡¸d_€p‹d

Ω ^{( N )} [Z _t ^{( N )} ] = L ⁽ _t ^{N )} Z _t ^{( N )} ,

^QO§Œ-R

¡r_dc}d

L ⁽ _t ^{N )} = X

i∈ G ( N )

e λ _ab (i, N)A i B i+1 + e λ _ba (i, N )B i A i+1 .

^{QO§¨“”’-R}

e¸k {mos¨†txµno_d©dWªt…cw¨{mosvuK† ƒt}uKƒdc}on¹k‹«€¾mno}€\s¨xy_­n=Ÿ uK}o¡¸€\}„ …”€\wv…c{twĀ\nosvuK† s¨† dpzl{€\nosvuK†

Q

O§¨“”’-R

€\w¨wvu¬¡mnu¼}dc¡r}os¨nd

Q

O§^]-R¸s¨†»no_d:Ÿ uK}ob

[V _t ^{( N )} ] = (U _t ^{( N )} ) ² − Z _t

0

(Z _s ^{( N )} ) ² R ⁽ _s ^{N )} ds,

^{QO§¨“y“ R}

¡r_dc}d/no_dƒt}ul…pdpmm

R ⁽ _t ^{N )}

svmmono}osv…cw¨k£ƒuyms¨nos¨‹d8€\†„»xys¨‹dc†Žqlk

R ⁽ _t ^{N )} = X

i∈G ^{( N )}

[ e λ _ab (i, N )] ²

λ _ab (N ) A _i B _i+1 + [ e λ _ba (i, N)] ²

λ _ba (N ) B _i A _i+1 .

^`_ds¨†­ndcxy}€\wndc}obs¨†

Q

O§¨“y“ R‰svm¢†uKno_ts¨†txdcwvmdqt{tn¦no_ds¨†…c}d”€Kms¨†tx8ƒt}uM…pdpmm¢€Kmmul…csĀ\ndp„¤¡rs¨no_

:uMuKqm„tdp…puKb®ƒ9uymos¨nosvuK†?u\Ÿ‰no_d/mo{tqtb¤€\}onos¨†tx­€\wvd

(U _t ^{( N )} ) ²

^§

^`_dŸ¯uKw¨w¨wvu¬¡rs¨†tx¤dpmonos¨b¤€\ndpm€\}d8…c}o{…csĀ\wÁ§

1

L ⁽ _t ^{N )} = O(1),

^{QO§¨“”‘-R}

R ⁽ _t ^{N )} = O 1 N

.

^{QO§¨“”-R}

W¾d8¡rs¨w¨w„tdc}os¨‹d

Q

O§¨“”‘-Rrq­kŽdpmonos¨b¤€\nos¨†tx—no_d/}os¨xy_ln=¥Á_€\†„©mosv„td/bEdcb8q9dc}‚u\ŸBdpzl{€\nosvuK†

Q

O§¨“”’-RW§

¿¢wvd”€\}ow¨k‹«

∆ψ _xy

i N

= _N ¹ ψ ⁰ _xy

i N

+ O

1 N ²

«‰¡r_dc}d

ψ ⁰

„tdc†uKndpm¼no_d£„tdc}os¨K€\nos¨‹d<u\Ÿ

ψ

^§

^`_dc†À«On€\½Ms¨†tx¤€ mdp…puK†„»uK}„tdc}dWªOƒ€\†mosvuK†Žu\Ÿno_ddWªOƒuK†dc†lnosĀ\w9ŸP{t†…cnosvuK†?€\†„<{ms¨†txE„tdWƆtsÃ¥

nosvuK†m

Q

O§-Rr€\†„

Q

O§ RW«t¡¸d/…”€\†Ž}dc¡r}os¨nd

Q

O§¨“”’-Rr€Km

L ⁽ _t ^{N )} = µ(N ) N

X

i∈ G ^{( N )}

A _i + A _i+1

2 − A _i A _i+1

∆ψ _ab i N

+ λ(N ) N

X

i∈ G ^{( N )}

(A _i − A _i+1 )∆ψ _ab i N

+ O 1 N

.

^{QO§¨“ R}

^`_dÆ}mon¸mo{tb4s¨†

Q

O§¨“ RBsvm¦{t†tsß uK}ob®w¨kEq9uK{t†„tdp„—qlk¤€8…puK†mn€\†­nr„tdcƒdc†„Os¨†tx¼uK†

ψ

§(f°†„tdpdp„X«

|A i | ≤ 1

^€\†„

ψ ∈ C ² [0, 1]

^{«mu¼no_€\n}

ψ ⁰

svmu\Ÿ‰quK{t†„tdp„ŽK€\}osĀ\nosvuK†À§

O‚m¸Ÿ uK}rno_d/mdp…puK†„Žmo{tbυpuKb®s¨†tx—s¨†

Q

O§¨“ RW«t¡¢d/_€p‹d

X

i∈ G ^{( N )}

(A _i − A _i+1 )∆ψ _ab i N

= X

i∈ G ^{( N )}

A _i+1

∆ψ _ab i + 1 N

− ∆ψ _ab i N

.

^`_dc†?no_d„Osvm…c}dcnd J€\ƒtwĀK…csĀ\†

∆ψ _ab i + 1 N

− ∆ψ _ab i N

≡ ψ _ab i + 2 N

− 2ψ _ab i + 1 N

+ ψ _ab i N

€K„Ob®s¨nmu\Ÿ"no_dms¨b®ƒtwvd:Ÿ¯uK}ob

∆ψ _ab i + 1 N

− ∆ψ _ab i N

= 1

N ² ψ _ab ⁰⁰ i N

+ O 1 N ²

,

^{QO§¨“”ˆ-R}

¡r_dc}d

ψ ⁰⁰

„tdc†uKndpmno_d/mdp…puK†„»„tdc}os¨y€\nos¨‹d8u\Ÿ

ψ

^§

e¸k

Q

O§ RW«

λ(N ) = λN ² + o(N ² )

^{«mu¼no_€\n Q}O§¨“”ˆ-Rrs¨b®ƒtw¨svdpm

λ(N) N

X

i∈ G ^{( N )}

(A _i −A _i+1 )∆ψ _ab i N

= X

i∈ G ^{( N )}

λA _i+1 N ψ _ab ⁰⁰ i

N

+o 1 N

= O(1),

^{QO§¨“ -R}

¡r_tsv…_©…puK†…cw¨{„tdpmno_dƒt}uMu\Ÿ(u\Ÿ

Q

O§¨“”‘-RW§¸^`_d/…puKb®ƒt{tn€\nosvuK†.u\Ÿ

R ⁽ _t ^{N )}

wvd”€K„Os¨†tx¤nu

Q

O§¨“”-R…”€\†

q9duKqtn€\s¨†dp„»MsĀ®mos¨b®s¨wĀ\}€\}oxy{tbEdc†lnm”§

^u8mo_u¬¡ no_d}dcwĀ\nos¨‹d…puKb®ƒ€K…cno†dpmm¸u\Ÿ9no_drŸ €\b®s¨w¨k

Z ^{( N )}

«y¡r_tsv…_—_dc}dy«­q­kEmdcƒ€\}€\qts¨w¨s¨n¹k¤€\†„

…puKb®ƒtwvdcndc†dpmm8u\Ÿ`no_d®{t†„tdc}ow¨kls¨†txmoƒ€K…pdpm”«Xsvm8dpz­{ts¨y€\wvdc†­nnu»nos¨xy_­no†dpmm¬«À¡¢dEƒt}ul…pdpdp„ €Km8s¨†

J¨“y“Kq­k£bEd”€\†mu\Ÿ‰no_d‚Ÿ¯uKw¨wvu¬¡rs¨†tx¤{mdWŸ¯{twÀ…c}os¨ndc}osvuK†À§

# !0 .80 '5-0 ! / ) 6 0 ($## E

{X ^{( N )} }

^{;1 '}

; >? 6. ;

D _R [0, T ]

^{7 * '.'X. ;# ; ./}

{X ^{( N )} }

^{c 7 +}

. (M

.87"<; 1; ,

*

+

a→∞ lim lim sup

N

P [||X ^{( N )} || ≥ a] = 0,

^{QO§¨“¬ŠcR}

/

||X ^{( N )} ||

= sup

t≤T

|X _t ^{( N )} |

^'

*

.+ X -

, η

^{2./ 21c2}

δ ₀

^<;

N ₀

^{#/ ./ c}

δ ≤ δ ₀

^<;

N ≥ N ₀

<;

τ

^{. ,/ /.87S ..}

τ + δ ≤ T

^4./

P

|X _τ ^{( N} _+δ ⁾ − X _τ ^{( N )} | ≥

≤ η.

^{QO§¨“]-R}

c . c <;

*

'.+ +

EE c&2c7&E'

W¾dŽ¡rs¨w¨wr†u”¡ €\ƒtƒtw¨k Àdcb®b¤€ O§‘.nu¾dpzl{€\nosvuK†m

Q

O§ŠcRE€\†„

Q

O§¨“y“ RW«¸no_d»}uKwvdŽu\Ÿ

X ^{( N )}

^s¨†

|B}uKƒuymos¨nosvuK†O§¼qdcs¨†tx®ƒtwĀ”k‹dp„Žqlk

Z ^{( N )}

^§

gqmdc}o‹d8no_€\n”«q­kŽno_d8{t†ts߯uK}ob q9uK{t†„tdp„O†dpmm:u\Ÿ

Z _t ^{( N )}

«…puK†„Os¨nosvuK†

Q

O§¨“¬ŠcRrsvms¨b®bEdp„OsĀ\ndcw¨k

‹dc}osÃÆdp„X§

^uE…_dp…½»…puK†„Os¨nosvuK†

Q

O§¨“]-RW«t}dc¡r}os¨nd

Q

O§ŠcRr€Km

Z _t+δ ^{( N )} − Z _t ^{( N )} = U _t+δ ^{( N )} − U _t ^{( N )} + Z t+δ

t

Ω ^{( N )} [Z _s ^{( N )} ]ds.

^{QO§¨“”Œ-R}

^`_d¤s¨†­ndcxy}€\w¸ndc}ob s¨†

Q

O§¨“”Œ-Rsvm8quK{t†„tdp„µs¨† bEuM„O{tw¨{m8qlk

Kδ

^J¡r_dc}d

K

svm €»…puK†mn€\†­n {t†tsß uK}ob®w¨k7quK{t†„tdp„ s¨†

N

^€\†„

ψ

^K€\†„ _dc†…pd?mZ€\nosvm=Ædpm

Q

O§¨“]-RW§ W¾d?€\}dŽwvdWŸPn®¡rs¨no_ no_d

€\†€\w¨kOmosvm¸u\Ÿ

U _t ^{( N )}

§(e¸{tn”«­ŸP}uKb

Q

O§¨“y“ RW«

Q

O§¨“”-R¸€\†„—‚uMuKqm¢s¨†dpz­{€\w¨s¨n¹kEŸ uK}¢mo{tqO¥Áb¤€\}onos¨†tx­€\wvdpm”«

¡¸d_€p‹d

E

(U _t+δ ^{( N )} − U _t ^{( N )} ) ²

= E Z t+δ

t

(Z _s ^{( N )} ) ² R ⁽ _s ^{N )} ds

≤ C N ,

P

"

sup

t≤T

|U _t ^{( N )} | ≥

#

≤ 4 ² E

Z _t

0

(Z _s ^{( N )} ) ² R ⁽ _s ^{N )} ds

≤ 4C

N ² ,

^{QO§‘y’-R}

¡r_dc}d

C

svm:€Eƒuyms¨nos¨‹d…puK†mon€\†ln„tdcƒdc†„Os¨†tx»uK†tw¨k?uK†

ψ

^{§^`_­{m}

U _t ^{( N )} → 0

€\w¨bEuymon:mo{t}dcw¨k

€Km

N → ∞

§^`_tsvmwĀKmonƒt}uKƒ9dc}on°kŽnuKx‹dcno_dc}‚¡rs¨no_H€Kmmo{tb®ƒtnosvuK†

Q

O§ˆ-R`kMsvdcwv„

Q

O§¨“]-R€\†„Žno_d

€\†t†uK{t†…pdp„

Q

¡¢d”€\½HR}dcwĀ\nos¨‹d®…puKb®ƒ€K…cno†dpmm/u\Ÿ¢no_d®mdpz­{dc†…pd

Z _t ^{( N )}

^§ Z‚dc†…pdy«Xno_d¼mdpzl{dc†…pd u\Ÿ"ƒt}uKq€\qts¨w¨s¨n°k£bEd”€Kmo{t}dpm

Q ^{( N )}

«t„tdWƆdp„»uK†

D M [0, T ]

€\†„»…puK}o}dpmoƒuK†„Os¨†tx®nu¼no_d‚ƒt}uM…pdpmm

µ ⁽ _t ^{N )}

«‹svm¦€\wvmu‚}dcwĀ\nos¨‹dcw¨k®…puKb®ƒ€K…cnYno_tsvmBsvm¦€‚…puK†mdpz­{dc†…pdu\Ÿ9…cwĀKmmsv…”€\wtƒt}u\~=dp…cnosvuK†Eno_dpuK}dcbEm

Q

mdpdŸ uK}¢s¨†mn€\†…pd‚^`_dpuK}dcb “ O§‘‹Š8s¨† J¨“”’LK.RW§ W¾d€\}d‚†u”¡ s¨†Ž€/ƒ9uymos¨nosvuK†<numn€\nd:€/Ÿ¯{t}ono_dc}

s¨b®ƒ9uK}on€\†­nƒt}uKƒdc}on¹k‹§

Jdcn

Q

no_d¼w¨s¨b®s¨nƒuKs¨†ln/u\Ÿ¸muKbEd®€\}oqts¨no}€\}okHmo{tqmdpzl{dc†…pd

Q ^{(n k )}

^«J€Km

n _k → ∞

^«À€\†„

Z _t

=

lim _n →∞ Z _t ^{(n k )}

§^`_dc†®no_d`mo{tƒtƒ9uK}onBu\Ÿ

Q

svmB€‚mdcn(u\Ÿm€\b®ƒtwvd¸ƒ€\no_m¦€\qmuKw¨{tndcw¨k …puK†lnos¨†­{uK{m

¡rs¨no_£}dpmoƒ9dp…cn¢nuno_d Àdcqdpmxy{dbEd”€Km{t}dy§(f°†„tdpdp„X«Mno_d:€\ƒtƒtw¨sv…”€\nosvuK†

µ _t → sup _t≤T log Z _t

^svm

…puK†lnos¨†­{uK{m:€\†„»¡¢d/_€p‹d/no_d:s¨b®bEdp„OsĀ\nd8quK{t†„

sup

t≤T

log Z _t ≤ Z ₁

0

[|φ _a (x)| + |φ _b (x)|]dx,

¡r_tsv…_¾_uKwv„tm:Ÿ uK}8€\w¨w

ψ _a , ψ _b ∈ C ² [0, 1]

^§ Zdc†…pdy«Jq­k¡¢d”€\½.…puK†­‹dc}ox‹dc†…pdy«(€\†­kw¨s¨b®s¨n/ƒ9uKs¨†­n

Z _t

_€Kmrno_d:Ÿ uK}ob

Z _t [φ _a , φ _b ] = exp hZ ¹

0

[ρ(x, t)φ _a (x) + (1 − ρ(x, t)φ _b (x)]dx i

,

^{QO§‘O“ R}

¡r_dc}d

ρ(x, t)

„tdc†uKndpmBno_dw¨s¨b®s¨n¦„tdc†mos¨n°k

Q

€:ƒt}osvuK}os}€\†„tuKbSR(u\Ÿ9no_dmdpzl{dc†…pdu\ŸXdcb®ƒts¨}osv…”€\w

bEd”€Km{t}dpm

µ ^(m _t ^{k )}

s¨†­no}uM„O{…pdp„Žs¨†

Q

O§¨“ RW§

.2401% 2&' 12 !#1' 3 1 14 !%" "! #%((6 ^`_tsvm¦svm¢muKbEdc_u”¡

no_d`:uK}„OsĀ\†<½M†uKn¸u\ŸÀno_dƒt}uKqtwvdcbŽ§Bdcw¨kMs¨†tx¼uK†—no_d:€\qu¬‹d‚¡¢d”€\½¤…puKb®ƒ€K…cno†dpmm`ƒt}uKƒ9dc}on°k‹«

uK{t}8†dWªMn/}dpm{tw¨nmo_u¬¡m/no_€\n €\†lk¾€\}oqts¨no}€\}okHw¨s¨b®s¨n8ƒ9uKs¨†­n

Q

svm8…puK†…pdc†­no}€\ndp„7uK†µ€Žmdcn8u\Ÿ

no}€œ~odp…cnuK}osvdpm¡r_tsv…_€\}d:¡¸d”€\½<muKw¨{tnosvuK†mu\ŸB€

&#( =.6 7c; N&## c

Qf T Î RW§

T"s¨}mon”«Oqlk

Q

O§ŠcRW«

Q

O§Œ-Rr€\†„

Q

O§¨“”’-RW«t¡¸d/uKqtn€\s¨†©€\nuK†…pd

∂(Z _t ^{( N )} − U _t ^{( N )} )

∂t =

N ² X

i∈ G ( N )

λ e _ab (i, N ) ∂ ² Z _t ^{( N )}

∂φ _a ( _N ⁱ )∂φ _b ( ⁱ⁺¹ _N ) + e λ _ba (i, N) ∂ ² Z _t ^{( N )}

∂φ _a ( ⁱ⁺¹ _N )∂φ _b ( _N ⁱ ) .

Q

O§‘y‘-R

f¹n(svm(¡¸uK}ono_¤}dcb¤€\}o½Ms¨†tx8no_€\n

Q

O§‘y‘-RBmo_uK{twv„®q9d`¡r}os¨nondc†À«lmono}osv…cnow¨k®moƒ9d”€\½Ms¨†tx«­€Km¦€monuM…_€Kmnosv…

„OsÃÂXdc}dc†­nosĀ\wdpz­{€\nosvuK†À«­¡r_tsv…_Esvm(¡¸dcw¨wÃ¥°„tdWƆdp„¤mos¨†…pd`s¨†„tdpdp„—€\w¨wtno_d`{t†„tdc}ow¨kls¨†tx/ƒt}uKq€\qts¨w¨s¨n¹k

mƒ€K…pdpmdcb¤€\†€\ndŸ¯}uKb$€Ÿ €\b®s¨w¨svdpmu\Ÿ"s¨†lndc}€K…cnos¨†tx—|"uKsvmmuK†©ƒt}ul…pdpmmdpm”§

dcƒtwĀK…cs¨†tx.Ÿ¯uK}¤€©¡r_ts¨wvd»no_d»zl{€\†­nos¨nosvdpm

φ _a ( _N ⁱ )

^€\†„

φ _b ( _N ⁱ )

q­kºK€\}osĀ\qtwvdpm

x ⁽ _i ^{N )}

^€\†„

y ⁽ _i ^{N )}

}dpmƒdp…cnos¨‹dcw¨k‹«

Q

O§‘y‘-R`qdp…puKbEdpm

∂(Z _t ^{( N )} − U _t ^{( N )} )

∂t = N ² X

i∈ G ^{( N )}

α _xy (i, N ) ∂ ² Z _t ^{( N )}

∂x ⁽ _i ^{N )} ∂y _i+1 ^{( N )} + α _yx (i, N ) ∂ ² Z _t ^{( N )}

∂y ⁽ _i ^{N )} ∂x ⁽ _i+1 ^{N )} ,

^{QO§‘y-R}

¡r_dc}d/¡¢d_€”‹dƒt{tn

α _xy (i, N ) = λ _ab (N )

"

exp x ⁽ _i+1 ^{N )} − x ⁽ _i ^{N )} + y ⁽ _i ^{N )} − y _i+1 ^{( N )} N

− 1

# ,

α _yx (i, N ) = λ _ba (N )

"

exp y _i+1 ^{( N )} − y ⁽ _i ^{N )} + x ⁽ _i ^{N )} − x ⁽ _i+1 ^{N )} N

− 1

# .

W¾dm_€\w¨wÀ}dc¡r}os¨nd

Q

O§‘y-R¸s¨†Žno_d8uKƒdc}€\nuK}rŸ uK}ob

− ∂U _t ^{( N )}

∂t = L ⁽ _t ^{N )} [Z _t ^{( N )} ],

^{QO§‘ R}

}dcb¤€\}o½Ms¨†tx?s¨† no_dEƒt}dpmdc†lnmdcnonos¨†txŽno_€\n”«Ÿ uK}/d”€K…_µÆ†ts¨nd

N

^«

L ^{( N )}

^{€K…cnm8uK†} no_d®Ÿ¯{t†…cnosvuK†

mƒ€K…pd

C ^p

−|φ|, |φ|] ^{2 N}

^«O¡r_dc}d

|φ|

= sup

z∈[0,1]

|φ _a (z)|, |φ _b (z)|

,

^{QO§‘yˆ-R}

€\†„

p

svm€\†€\}oqts¨no}€\}ok:ƒuymos¨nos¨‹d‰†l{tb8q9dc}”«y€Km

Z _t ^{( N )}

svmJ€\†€\w¨kMnosv…(¡rs¨no_}dpmoƒ9dp…cnÀnu

{φ _a (.), φ _b (.)}

^§

^`_duKƒdc}€\nuK}

L ⁽ _t ^{N )}

svmBu\Ÿ9ƒ€\}€\quKw¨sv…n°kMƒdy«‹qt{tnBno_dc†Es¨†®no_dr¡rsv„tdmdc†mdy«‹mos¨†…pdr_dc}duK†dr…”€\†

…_dp…½©no_dzl{€K„O}€\nosv…/Ÿ uK}ob {mo{€\w¨w¨k©€KmmuM…csĀ\ndp„Ž¡rs¨no_no_d mdp…puK†„©uK}„tdc}„tdc}os¨y€\nos¨‹d8ndc}obEm

svm†uK†Ž„tdWƆts¨ndy«mdpddy§x§ JK.RW§

^`_dE½‹dckƒ9uKs¨†­n8¡rs¨w¨w¦q9d¼nuŽmo_u”¡4no_€\n€\†­kHw¨s¨b®s¨n8ƒuKs¨†ln

Z _t ^Law = lim

n k →∞ Z _t ^{(n k )}

m€\nosvm=Ædpm€\†

faT Î «OuKqtn€\s¨†dp„<q­k£mono{„Okls¨†tx®no_d:mdp…puK†„»uK}„tdc}`w¨s¨†d”€\}rƒ€\}onosĀ\wÀ„OsÃÂXdc}dc†­nosĀ\wJuKƒ9dc}€\nuK}m

L ⁽ _t ^{N )}

€\wvuK†tx¤no_d/mdpzl{dc†…pd

n _k → ∞

^§

^u…”€\}o}okuK{tnÀno_d¦€\†€\w¨kOmosvmÀu\Ÿtno_d(w¨s¨b®s¨nJm{tb…puKb®s¨†txs¨†

Q

O§‘y-R

Q

¡r_tsv…_8svm"€¸ƒt}osvuK}os­s¨†­no}osv…”€\nd RW«

¡¸d‚ƒt}uKƒuymd€x‹dc†dc}€\wJ€\ƒtƒt}uM…_À«t¡r_tsv…_?€\s¨bEm€\n¸ƒt}u¬ls¨†tx¼Æ}mn¸no_€\n

Z _t

^{svm€ % #/c}

Q

uK} ; # s¨†no_d¼mdc†mdu\ŸiM…_­¡`€\}ono·LR:u\Ÿ¸€»¿`€\{…_lk©n¹klƒ9d¼uKƒdc}€\nuK}”§8^`_dw¨s¨†d¼u\Ÿ

€\}oxy{tbEdc†ln¡rs¨w¨wÀq9dmo½‹dcn…_dp„»qdcwvu¬¡/§

e`dWŸ uK}dc_€\†„X«Ÿ¯uK}/no_d—m€\½‹dEu\Ÿmo_uK}ono†dpmm¬«Js¨n¡rs¨w¨w¦q9dE…puK†­‹dc†tsvdc†ln nu?„tdWƆd¤no_d¼Ÿ uKw¨wvu”¡rs¨†tx

…ckMw¨s¨†„tdc}mdcnm¬«MŸ¯uK}

p = 1, 2 . . .

^«

U _t ^p

= [−|φ|,

|φ|] ^p × [0, t], U ^p

= [−|φ|,

|φ|] ^p .

f¹†­no}uM„O{…pd no_d®uKƒdc}€\nuK}

L e ⁽ _t ^{N )}

«9¡r_tsv…_Hsvm‚no_d®€K„\~=uKs¨†ln/u\Ÿ

L ⁽ _t ^{N )}

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h ∈ C ₀ ^∞ (U _t ^{2 N} )

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= ∂h

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g ∈ L ₂

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h ∈ C ₀ ^∞ (U _T ^{2 N} )

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g L e ⁽ _t ^{N )} [h] d~u dt = 0,

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h ∈ C ₀ ^∞ (U _T ^{2 N} )

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Z

U T ^{2 N}

(Z _t ^{( N )} − U _t ^{( N )} ) ∂h

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d~u dt = Z

U ^{2 N}

(Z _T ^{( N )} − U _T ^{( N )} )h(~u, T ) − Z ₀ ^{( N )} h(~u, 0) d~u .

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1 - /4 M

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∂(Z _t ^{( N )} − U _t ^{( N )} )

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∂x ⁽ _i ^{N )} ∂x ⁽ _i+1 ^{N )}

#

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φ _a (.), φ _b (.)

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C ₀ ^∞ (U _T ^{( N )} )

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"

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,

Q

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C ₀ ^∞ (W × [0, T ])

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Q

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∂t

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0

"

µ ∂ ² k φ(.), t

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 

 

 



∂Y _t

∂φ(.) = ρ(., t)Y _t ,

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∂φ ² (.) = ρ ² (., t)Y _t .

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