The Unique Solution of Stochastic Differential Equations. Dietrich Ryter. Midartweg 3 CH-4500 Solothurn Switzerland

Transcription

1 The Unque Soluon of Sochasc Dffeenal Equaons Dech Rye Mdaweg 3 CH-4500 Solohun Swzeland Phone Tme evesal n sysems whou an exenal df sngles ou he an-iô negal. Key wods: Sochasc dffeenal equaons; negaon sense; fowad and bacwad equaons; me evesal

2 2 I. Inoducon Sochasc dffeenal (o ahe negal) equaons ae no unquely defned n he pesence of mulplcave nose. Ths shows up n he me dscezaons of he negal fom, whch nvolve boh a paonng of he me axs and he choce of an evaluaon pon n each neval. The se of hese pons s chaacezed by α ( 0 α 1; α = 0 a he begnnng Iô, α = 1/ 2 n he mddle Saonovch, and α = 1 a he end an-iô ). I s well-nown [1] ha he sum of he ncemens n each neval (of a lengh d ) conveges fo each α, wh a esul dependng on α. The specfcaon of α s hus a demandng exa as of modelng. I wll now be agued ha n he absence of an exenal df he soluon of he SDE s nvaan unde me evesal, so ha he fowad and he bacwad opeaos mus concde. Ths eadly mples ha α = 1, when he couplng wh he nose (hus he dffuson) s no consan. II. Bacgound 2.1 Sochasc dffeenal equaons Le he connuous Maov pocess X () dx a ( X ) d + b ( X ) be gven by = dw (2.1) wh smooh funcons a ( x), b ( x) ; summaon ove double ndces s undesood; a (x) denoes he exenal (nose-ndependen) df. The Wene pocesses W () ae 2 ndependen and obey W ( ) W (0) > = 0 and < [ W ( ) W (0)] >. < = Omng a (x ) empoaly, and consdeng he ncemen X n [ 0, ], wh gven X ( 0) = x, yelds X ( ) = b [ x + X ( τ )] dw ( τ ). Fo small enough s suffcen 0

3 3 o expand b o he fs ode, so ha X ( ) b ( x)[ W ( ) W (0)] + b ( x) X ( τ ) dw m, m τ 0 ( ). Inseng he leadng pa m mn (of he ode O ( ) ) no he negal esuls n X ( τ ) dw ( τ ) = b ( x) Wn ( τ ) dw ( τ ). 0 0 The las negal nvolves α ; fo 2 = n s well-nown o yeld [ W ( ) + (2α 1) ]/ 2, 2 wh he expecaon α and wh he α -ndependen vaance / 2. Fo small enough allows o eplace he negal by he nonandom value α. Snce fo n he expecaon s zeo, he esul s X ( ) = b ( x)[ W ( ) W (0)] + a ( x) α, wh he nose-nduced df [1,2] m a := b, m b. (2.2) Wh a agan ncluded, hs yelds X = b ( x) W + [ a ( x) + α a ( x)] + o( ). (2.3) Clealy X s Gaussan dsbued, wh mean a a ) ( + α and wh covaance D gven by D = b j b j. (2.4) oe ha (2.3) mples he equvalen Iô fom of (2.1) dx fo any α. = b ( X ) dw + [ a ( X ) +α a ( X )] d (2.5) 2.2 The fowad (Foe-Planc) equaon The FPE [1-4] coespondng o (2.5) s w, = { ( a + α a ) w + (1/ 2)( D w), }, : = L w. (2.6) By a = D, / 2, see he Appendx, hs becomes

4 4 w, { a w + (1/ 2)[ D w, + (1 α) D, w]}, =. (2.7) Fo α = 1 educes o w, = [ a w + (1/ 2) D w]. (2.8) Hee only he fs devaves of D ae nvolved, n conas wh (2.7). Tha popey was found fo sysems wh hemal equlbum [5,6]. I s woh nong ha (2.8) can also be obaned whou use of sochasc analyss: Consde pacles movng ndependenly n a space wh coodnaes x. In ems of he densy w( x, ) he pacle cuen J ( x, ) s gven by he exenson of Fc s law J = a w (1/ 2) D w, whee a (x ) s he velocy and D(x) / 2 he dffuson max. By he consevaon law w, + J = 0 hs mmedaely leads o (2.8). One can also specfy he pobably cuen assocaed wh (2.6) : v J = [ a + ( α 1) a ] w (1/ 2) D w, o (2.9) J = a w (1/ 2) D w fo α = 1. (2.10) 2.3 The bacwad equaon The df a mus also be subsued by a + α a n he bacwad equaon u ), = a u, + (1/ 2 D u. Ths s easly seen o yeld he bacwad opeao L + = a + [( D / 2) ] (1 α ). (2.11) a Fo α = 1 he fowad and bacwad opeaos L and L = [ a + ( D / 2) ] + and L = a + [( D / 2) ] whch only dffes by he em wh he exenal df a. + L ae he smple fom, (2.12)

5 5 III. Pue nose and he value of α 3.1 Tme evesal Fo a ( x) 0,.e. whou an exenal df, he emande dx = b ( X ) dw of (2.1) depends on va W () only. Snce Wene pocesses wh evesed me ae sochascally equvalen, he soluon X () s hus also equvalen wh X ( ). A consequence s + L = L. (3.1) Ths holds fo consan D, by 2, + L w = ( D w), = D w = 2L w, and also fo α = 1, n vew of (2.12). The cucal pon s he fac ha 2,,, ( L L + ) w = (1 α ) [( D w) + D, w ], see (2.7) and (2.11), only vanshes when ( 1 α ) D, 0 (3.2) ( w s abay). Ths man esul excludes any α < 1 n he pesence of mulplcave nose, and s naual o expec ha also holds when a ( x) Smoohng by nose As an mpoan consequence of α = 1, pue nose ( 0 a ) has qualavely he same mpac on a gven pobably densy as n he case of consan D : Consde a pobably densy w( x, ) wh a gven w(x,0 ). By (2.10) he pobably cuen J = (1/ 2) D w s deced agans w (snce w J 0 ), and vanshes whee w = 0. The exema of w( x, ) ae hus no shfed n x, and w( x, ) s smply flaened and spead ou n me (moeove, w, a an exemum s no nfluenced by he x - dependence of D ). These well-nown feaues fo consan D eman hus ue wh any D(x ), ndependenly

6 of (x) a. Wh α < 1 hey would howeve no hold, as (2.9) eadly shows. 6 IV. Seady sae soluons Possble soluons of Lw = 0 exhb he sably popees of he exenal df a (x ) when α = 1. Ths can be nfeed fom he cuen (2.10) : Any w( x, ) wh a maxmum (mnmum) a an aacve (epulsve) pon of a (whee 0 a = ) s cuen-fee. [Fo α < 1 such a w( x, ) would have a nonzeo cuen, whle a cuen-fee soluon has s exemum shfed by ( 1 α)a v away fom he equlbum pon]. In a seady sae wh naual bounday condons (excludng a cuen acoss a gven doman) he densy w(x ) hus vsualzes he sably of a (x ) when α = 1. Ths ncludes he wea-nose analyss [7-9] (wh he nose paamee ε ), whee he quaspoenal φ (x ) n he asympoc expesson w( x) exp[ φ ( x) / ε ] s only a Lapunov funcon of a (x ) when α = 1 (ohewse mus be shfed by ( 1 α)a v whee a ε fo conssency). Concludng emas The geneal fowad and bacwad opeaos ae unque and gven by (2.12). Mnd ha he pesen agumens apply fo scly connuous and Maovan models (by he popees of he Wene pocesses). Tha condon need no eally be me n paccal applcaons. Accodng o [5,6] s a vald appoxmaon n vaous physcal models wh hemal equlbum, bu [10] s appaenly concened wh dffeen suaons. - The appoach by Wong and Zaa [11] s elaed wh α = 1/ 2 and heefoe no confmed. Appendx The nose-nduced df a can be expessed n ems of he dffuson max D, see (2.4).

7 7 Fo a dagonal max B (of he elemens b ) s obvous ha j b, j b = D, / 2, (A.1) and fo symmec B he same follows by dagonalzng B. Each non-symmec B can be symmezed on subsung W () by an equvalen W *( ) gven by dw : = O dw * : oe ha fo any T 1 1T T = B B = B * O O B, B *: = B O he dffuson max D s peseved, D * when O s ohogonal ( de O = 1 s also admed). Ths enals B dw = B * dw *. When B s squae, one can fnd a O whch yelds a symmec B * by whch (A.1) holds agan; a ecangula B can be compleed by zeos. Ths shows ha (A.1) holds n geneal (bu only by sochasc equvalence when B s no symmec) : j a : = b, j b = D, / 2. (A.2) Refeences [1] L. Anold, Sochassche Dffeenalglechungen (Oldenboug, München, 1973), and Sochasc Dffeenal Equaons: Theoy and Applcaons (Wley, ew Yo, 1974) [2] H. Rsen, The Foe-Planc Equaon (Spnge, Beln, 1989) 2nd ed. [3] I.I. Gchman and A.W. Soochod, Sochassche Dffeenalglechungen (Aademe- Velag, Beln, 1971) [4] B. Osendal, Sochasc dffeenal equaons (Spnge, Beln, Hedelbeg, ew Yo, 1992) [5] H. Gabe and M.S. Geen, Phys.Rev.A19, 1747 (1979) [6] Yu.L. Klmonovch, Physca A 163, 515 (1990) [7] M.I. Fedln and A.D. Wenzell, Random Peubaons of Dynamcal Sysems (Spnge, Beln, 1984) [8] D. Rye, J.Sa.Phys.149 (o.6), 1069 (2012)

8 8 [9] D. Rye, Phys.Le.A 377, 2280 (2013) [10].G. van Kampen, Sochasc Pocesses n Physcs and Chemsy (Elseve/oh- Holland, Amsedam, 3 d ed. 2008). [11] E. Wong and M. Zaa, Ann. Mah. Sas. 36, 1560 (1965)