Weak Solutions of Mean Field Game Master Equations

Transcription

1 Weak Soluio of Mea Field Game Maer Equaio Cheche Mou ad Jiafeg Zhag March 5, 19 Abrac I hi paper we udy maer equaio ariig from mea field game problem, uder he Lary-Lio moooiciy codiio. Claical oluio of uch equaio ypically require very rog echical codiio. Moreover, ulike he equaio ariig from mea field corol problem, he mea field game maer equaio are o-local ad eve claical oluio ofe do o aify he compario priciple, o he adard vicoiy oluio approach eem ifeaible. We hall propoe wo oio of weak oluio for uch equaio: oe i i he piri of vaihig vicoiy oluio, relyig o he abiliy reul; ad he oher i i he piri of Sobolev oluio, baed o he iegraio by par formula. We hall prove exiece ad uiquee of weak oluio i boh ee. For he crucial regulariy i erm of he meaure, we coruc a mooh mollifer for fucio o Waerei pace, which i ew i he lieraure ad i iereig i i ow righ. I order o focu o he mai idea, i hi paper we coider oly a very pecial cae, ad he more geeral cae will be udied i a accompayig paper. Keyword. Mea field game, maer equaio, forward-backward SDE, weak oluio, Sobolev oluio, Waerei pace AMS Mahemaic ubjec claificaio: 35R15, 6H3, 93E Deparme of Mahemaic, Uiveriy of Califoria, Lo Agele, CA mucheche@mah.ucla.edu. Deparme of Mahemaic, Uiveriy of Souher Califoria, Lo Agele, CA jiafez@uc.edu. 1

2 1 Iroducio Iiiaed idepedely by Caie, Huag, & Malhame 6 ad Lary & Lio 16, mea field game ad he cloely relaed mea field corol problem have received very rog aeio i he pa decade. Such problem coider limi behavior of large yem where he age ierac wih each oher i cerai ymmeric way, wih he yemic rik a a oable applicaio. here have bee umerou publicaio o he ubjec, ee e.g. Cardaliague 7, Beoua, Frehe, & Yam 3, Carmoa & Delarue 9, 1, ad he referece herei. he maer equaio i a powerful ad ieviable ool i hi framework, which play he role of he PDE i he adard lieraure of corol ad game. he mai feaure of maer equaio i ha i ae variable coai probabiliy meaure, ypically he diribuio of cerai uderlyig ae proce, o i ca be viewed a a PDE o Waerei pace. By aure hi i a ifiie dimeioal problem. Maer equaio i alo a coveie ool for (adard) corol problem wih parial iformaio, ee e.g. Badii, Coo, Fuhrma, & Pham 1, ad Saporio & Zhag, ad for ime icoie problem, ee e.g. Wu & Zhag. Due o i ifiie dimeioaliy aure, claical oluio of maer equaio require very rog echical codiio, ee e.g. Buckdah, Li, Peg, & Raier 5, Cardaliague, Delarue, Lary, & Lio 8, Chaageux, Cria, & Delarue 11, ad Saporio & Zhag, a well a Gagbo & Swiech 14, 15 ad Beoua & Yam 4 for fir order maer equaio. here have alo bee ome eriou effor o vicoiy oluio for maer equaio ariig from corol problem, ee e.g. Pham & Wei 19 ad Wu & Zhag. However, mea field game maer equaio have a quie differe aure: i i o-local ad eve claical oluio ypically do o aify he compario priciple. herefore, he vicoiy oluio approach eem ifeaible. here i a cry for a appropriae oio of weak oluio for mea field game maer equaio, which i he mai goal of hi paper. A i adard lieraure, we hall aume he Lary-Lio moooiciy codiio o he coefficie. Noe ha he maer equaio i o characerize he value fucio of he mea field game problem a i equilibrium. he moooiciy codiio i o guaraee he uiquee of he equilibrium ad hu i eeial for he heory. Whe hi codiio i violaed, he mea field game problem may have muliple equilibrium wih muliple value, he he meaig of he value fucio i o clear, o o meio he maer equaio characerizig he value fucio. We remark ha Feiei, Rudloff, & Zhag 13 udy he e of all value for ozero um game problem wih muliple equilibrium, which could be a appropriae approach for mea

3 field game wihou he moooiciy codiio. We hall propoe wo ew oio of weak oluio for mea field game maer equaio ad eablih heir wellpoede. he fir oe i i he piri of vaihig vicoiy oluio, ad we hall call i a vaihig weak oluio. Roughly peakig, we fir mollify he coefficie of he maer equaio ad le V deoe he claical oluio o he mollified maer equaio. Uder cerai abiliy propery, he limi of V will be he deired weak oluio of he origial maer equaio. here are ome major difficulie i hi approach hough. Fir, for fucio of probabiliy meaure, i mooh mollifier i by o mea eay ad doe o eem o be udied i he lieraure, o he be of our kowledge. hu we hall fir coruc a mooh mollifier for fucio o Waerei pace of probabiliy meaure. Our mai idea i o fir dicreize he uderlyig meaure ad he o mollify he coefficie of he ivolved Dirac meaure. he mai feaure of our mollifier i ha i keep he Lipchiz coa for Lipchiz coiuiy uder 1-Waerei diace (bu o for -Waerei diace). We hall obai he uiform covergece for V ielf ad cerai L 1 -ype of covergece for µ V, provided i exiece ad coiuiy. We believe our mollifier i iereig i i ow righ ad could be ueful for relae field beyod mea field game. A more fudameal difficuly i ha he exiece of claical oluio for he mollified maer equaio over arbirary ime ierval require he moooiciy codiio. Uforuaely, our mollifier doe o maiai hi codiio, ad we are o opimiic for fidig aoher mollifier which could do o. o overcome hi difficuly, we iead uilize he local (i ime) claical oluio for he mollified maer equaio, whoe exiece doe o require he moooiciy codiio. We ex pach he local oluio io a global oe. Such idea ha bee ued uccefully i he lieraure of forward backward SDE, ee e.g. Delarue 1, Zhag 3, ad Ma, Wu, Zhag, & Zhag 17. he key i o obai ome uiform eimae, uder he moooiciy codiio o he coefficie of he origial maer equaio. We hall eablih hee eimae a well a he crucial abiliy reul, uder weaker codiio ha hoe required for claical oluio heory, which will imply he exiece ad uiquee of our vaihig weak oluio. We ex propoe Sobolev oluio for maer equaio, by uig he iegraio by par formula. We remark ha, ulike he mea field corol maer equaio which are ofe oliear o µ V, he parial derivaive of he value fucio V wih repec o he probabiliy meaure, he mea field game equilibrium i defied hrough cerai fixed poi procedure ad coequely he mea field game maer equaio i alway liear i erm of µ V a well a i furher derivaive. hi make i appropriae o apply he iegraio 3

4 by par formula, a lea whe here i oly drif corol o ha he erm xx V i alo liear. We hall alo eablih boh he exiece ad uiquee of Sobolev oluio. We remark ha i our eig he Sobolev oluio require he exiece of µ V. A a by produc, we eablih a poiwie probabiliic repreeaio formula for µ V. A aleraive formula i implied i 8 by uig a forward backward PDE yem whoe iiial codiio i he derivaive of a Dirac meaure. Our repreeaio formula ue he rog oluio o a forward backward SDE yem ad hold uder weak codiio. However, he coecio bewee he wo formulae i o clear. We alo remark ha, while boh he exiece ad uiquee of our Sobolev oluio rely o he moooiciy codiio, for our vaihig weak oluio he moooiciy codiio i required oly for he exiece bu o for he uiquee. Fially, we oe ha he heory ivolve may oaio ad very echical argume. I order o focu o he mai idea, i hi paper we coider oly a very pecial maer equaio. We hall exed our reul o more geeral cae i a accompayig paper. he re of he paper i orgaized a follow. I Secio we coruc a mooh mollifier for fucio of probabiliy meaure. I Secio 3 we iroduce he mea field game ad he aociaed maer equaio, i a heuriic way. Secio 4 i devoed o he uiform regulariy of he value fucio ad he abiliy reul. I Secio 5 we udy claical oluio V of maer equaio over mall ime ierval. I paricular, we provide a poiwie probabiliic repreeaio formula for µ V. Fially, i Secio 6 ad 7 we propoe wo oio of weak oluio for our maer equaio ad eablih heir wellpoede. A mooh mollifier o Waerei pace.1 he baic eig Le, be a fiie ime horizo, := R/Z he (1-dimeioal) oru. Noe ha ay fucio ϕ o ca be viewed a a periodic fucio o R wih period 1: ϕ(x + 1) = ϕ(x) for all x R. hroughou he paper we hall o diiguih hee wo equivale view. Le P deoe he e of all probabiliy meaure o. I paricular, for x, δ x P deoe he Dirac-meaure. Noe ha, ice i bouded, x p µ(dx) < for all p 1 ad µ P. Iroduce he p-waerei diace: { (E X W p (µ, ν) := if Y p ) } 1 p : for all r.v. X, Y uch ha L X = µ, L Y = ν. (.1) 4

5 A above X, Y are -valued radom variable o arbirary probabiliy pace ad L i he law of he radom variable. I paricular, whe p = 1 we have he dual repreeaio: { } W 1 (µ, ν) = up ϕ(x)µ(dx) ν(dx) : ϕ C 1 (; R) uch ha ϕ() =, ϕ 1.(.) We deoe P p := (P, W p ) whe here i a eed o emphaize he diace W p. Coider a fucio U : P R. By 7, 1, he derivaive of U ake he form µ U : P R aifyig: for all -valued radom variable ξ, η uch ha L ξ = µ, U(L ξ+η ) U(µ) = E µ U(µ, ξ)η + o( η ). (.3) Remark.1 (i) For ay p 1, P p i compac, coequely ay µ C (P p ) i bouded ad uiformly coiuou. However, hi i o rue whe i replaced wih R. (ii) For fixed µ, µ U(µ, ) i uique µ-a.. However, for U C 1 (P ), amely µ U exi ad i coiuou o P, he µ U(µ, x) i uique for all (µ, x). Give U C 1 (P ), we may defie x µ U ad µ µ U i obviou ee, ad we may defie higher order derivaive i he ame maer. Oe crucial propery of mooh U i he Iô formula, which play a impora role i he heory. o be precie, deoe Θ :=, P, ad le U C 1,, (Θ), amely U : Θ R i coiuou, U, x U, xx U, µ U(, x, µ, x), ad x µ U(, x, µ, x) exi ad are coiuou. he for ay dx = b d + σ db, where B i a Browia moio, he followig Iô formula hold (cf., e.g., 5, 11): du(, X, L X ) = U + x Ub + 1 xxuσ (, X, L X )d + x U(, X, L X )σ db +Ẽ µ U(, X, L X, X ) b + 1 x µ U(, X, L X, X ) σ d, (.4) where ( X, b, σ) i a idepede copy of (X, b, σ), ad wih repec o ( X, b, σ) for give X. Ẽ i he codiioal expecaio. Corucio of a mooh mollifier he mai purpoe of hi ubecio i o coruc a mooh mollifier for U C (P 1 ), which i ew i he lieraure, o our be kowledge. he idea work for C (P 1 (R d )), bu for impliciy we focu o C (P 1 ()) here. We proceed i wo ep. Sep 1. Dicreizaio of µ P. Fix 3 ad deoe x i := i, i =,,. Oe aive corucio i o approximae µ wih 1 ψ i= i (µ)δ xi, where ψ i (µ) := µ(x i, x i+1 )). 5

6 However, ice 1 xi,x i+1 ) i dicoiuou, oe ca eaily check ha ψ i / C 1 (P ), which i a obacle for he moohe of U we will coruc. o ge aroud of hi difficuly, we iroduce a fucio I = I C (, 1) uch ha I(x) = 1 for x 1 3, I(x) = 1 x for 1 x 1 1, I(x) = for x ; I 1, I, ad I(x) + I(1 x) = 1. We ex defie, for i =,,, ad x, (.5) φ i (x) := I ( x x i ) 1 xi 1,x i+1 (x). (.6) he oe ca verify raighforwardly ha φ i C (), φ i 1, ad for ay x x i, x i+1 : φ i (x) + φ i+1 (x) = 1, φ j (x) = for j i, i + 1. (.7) For each µ P, deoe µ := ψ i (µ)δ xi, where ψ i (µ) := φ i (x)µ(dx). (.8) Noe ha µ ψ i (µ, x) = φ i (x), he clearly ψ i C (P ). Moreover, µ () = φ i (x)µ(dx)δ xi () = φ i (x)µ(dx) = µ() = 1. hi implie ha µ P. Sep. Mollificaio of U. We fir defie Ũ (µ) := U(µ ) = U ( ) ψ i (µ)δ xi. (.9) Sice µ i a dicree meaure, oe ca mollify Ũ hrough he coefficie ψ i (µ). However, oe ha {ψ i (µ)} 1 i i a (dicree) probabiliy ad U i defied oly o probabiliy meaure, we eed ome pecial reame for he mollificaio. o be precie, deoe := {y = (y 1,, y 1 ) : y i 1 } R 1, ad we alway deoe y 3 := 1 y i. he y i = ad y 1. Defie µ (y) := ψ i (µ, y)δ xi, where ψi (µ, y) := ψ i(µ) + y i. (.1) Noe ha ψ i (µ, y), ψ i (µ, y) = ψ i (µ) + y i = = 1. ha i, µ (y) P for all y. Now le ζ be a mooh deiy fucio wih uppor, ad defie U (µ) := ζ (y)u ( µ (y) ) dy. (.11) 6

7 .3 he covergece reul heorem. Le U C (P 1 ) ad U be defied by (.11). he (i) U C (P ) ad lim U U =, where U := up µ P U(µ). (ii) If U i Lipchiz coiuou uder W 1 wih Lipchiz coa L, he U i uiformly Lipchiz coiuou uder W 1 wih he ame Lipchiz coa L. (iii) If U C 1 (P ), he lim up µ U (µ, x) µ U(µ, x) dx =. µ P Remark.3 (i) If U i Lipchiz coiuou uder W wih Lipchiz coa L, we are o able o how ha U i uiformly Lipchiz coiuou uder W wih he ame L. Neverhele, ice W 1 W, o uder he codiio i heorem. (ii) we ee ha U i alo uiformly Lipchiz coiuou uder W wih he ame Lipchiz coa L. (ii) I heorem. (iii), our U doe o aify (recallig Remark.1 (ii)) lim up (µ,x) P µ U (µ, x) µ U(µ, x) =. See Example.4 below. I will be iereig o kow if here exi a aleraive mollifier uch ha he above uiform covergece hold for U C 1 (P ). Neverhle, he covergece of µ U we obai here i ufficie for our udy of Sobolev oluio i Secio 7. Example.4 Le U(µ) = g(x)µ(dx) for ome mooh fucio g. he U (µ) = = ζ (y) + 1 Noe ha µ U(µ, x) = g (x) ad µ U (µ, x) = ψ i(µ) + y i g(x i )dy 1 + ψ i(µ) g(x i ) + ζ (y) + 1 R 1 µ ψ i (µ, x)g(x i ) = + 1 y i g(x i )dy. φ i(x)g(x i ) By (.5) ad (.6), we ee ha φ i (x j) = for all i, j, ad hu µ U (µ, x j ) =. herefore, aumig g 1, µ U(µ, x j ) µ U (µ, x j ) = g (x j ) 1. Proof of heorem.. (i) For z R 1, deoe κ i (µ, z) := z i ψ i (µ) 1, κ(µ, z) := (κ 1(µ, z),, κ 1 (µ, z)), z := + 1 z i. 1 7

8 Le := {z R 1 : κ(µ, z) }. I i clear ha +1 z iδ xi P for z, ad U (µ) = ζ (κ(µ, z))u( z i δ xi )dz. (.1) + 1 Sice φ i (x) i mooh, he clearly ψ i C (P ) ad hu κ i (, z) C (P ) for ay z. Now ice ζ i alo mooh, we ee ha U C (P ). Nex, recall (.) ad le ϕ C 1 (; R) aify ϕ() =, ϕ 1. he, for ay y, ϕ(x)µ (y)(dx) µ(dx) = ψ i (µ, y)ϕ(x i ) ϕ(x)µ(dx) = 1 ψ i (µ)ϕ(x i ) ϕ(x)µ(dx) + ψ i (µ)ϕ(x i ) ψ i (µ)ϕ(x i ) y i ϕ(x i ) xi+1 ϕ(x i ) φ i (x)µ(dx) ϕ(x)µ(dx) + C x i 1 xi+1 ϕ(x i )φ i (x) + ϕ(x i+1 )φ i+1 (x)µ(dx) φ i (x) + φ i+1 (x)ϕ(x)µ(dx) + C x i xi+1 x i xi+1 ϕ(x i ) ϕ(x) φ i (x) + ϕ(x i+1 ) ϕ(x) φ i+1 (x)µ(dx) + C x i φ i (x) + φ i+1 (x)µ(dx) + C C. hi implie ha W 1 (µ, µ) C. By Remark.1 (i), U i uiformly coiuou i P 1, he we ee ha U (µ) = U(µ ) coverge o U(µ) uiformly. (ii) Le µ, ν P 1. For ϕ C 1 (; R) aifyig ϕ() =, ϕ 1 ad y, we have ϕ(x)µ (y)(dx) ν (y)(dx) = ψ i (µ, y) ψ i (ν, y)ϕ(x i ) = + 1 ψ i (µ) ψ i (ν)ϕ(x i ) = ϕ(x)µ(dx) ν(dx) (.13) where ϕ(x) := +1 φ i(x)ϕ(x i ). Noe ha, for x x i, x i+1, we have ϕ(x) = φi (x)ϕ(x i ) + φ i+1 (x)ϕ(x i+1 ) = φi (x)ϕ(x i ) + 1 φ i (x)ϕ(x i+1 ) he, by (.5), ϕ (x) = φ + 1 i(x)ϕ(x i ) ϕ(x i+1 ) φ i(x) + 1 I ((x x i )) + 1 (1 + 1 ) = 1. 8

9 hi, ogeher wih (.13), implie ha W 1 (µ (y), ν (y)) W 1 (µ, ν). he U (µ) U (ν) ζ (y) U(µ (y)) U(ν (y)) dy ζ (y)lw 1 (µ (y), ν (y))dy ζ (y)lw 1 (µ, ν)dy = LW 1 (µ, ν). hi i he deired uiform Lipchiz coiuiy. (iii) We fir expre µ U i erm of µ U. Recall (.3). Fix ξ ad η uch ha L ξ = µ. For each y ad > mall, le ξ (y) be a dicree radom variable uch ha L ξ(y) = µ (y), amely P(ξ (y) = x i ) = ψ i (µ, y). We hall coruc a radom variable η (, y) uch ha P(ξ (y) + η (, y) = x i ) = ψ i (L ξ+η, y). For hi purpoe, we hall fir coruc ome fucio p i (, y) aifyig p i (, y) ψ i (µ, y), ad he e η (, y) akig wo value: ad 1 wih P(ξ (y) = x i, η (, y) = 1 ) = p i(, y). Noe ha ψ i (L ξ+η, y) = P(ξ (y) + η (, y) = x i ) = P(ξ (y) = x i, η (, y) = ) + P(ξ (y) = x i 1, η (, y) = 1 ) = ψ i (µ, y) p i (, y) + p i 1 (, y). Le C be a coa which will be pecified laer. For i = 1,,, e p i (, y) := C i j=1 ψ j (L ξ+η, y) ψ j (µ, y) = C + 1 i ψ j (L ξ+η ) ψ j (µ),(.14) Oe ca eaily how ha i j=1 ψ j(l ξ+η ) ψ j (µ) C, for a coa C > which may deped o ad η, bu idepede of. he p i (, y). Moreover, oe ha y i 1 3 for i = 1,, 1 ad y 1 y i 1, by (.1) we have ψ i (µ, y) = j=1 1 ( + 1), he p i (, y) C ψ i (µ, y) for mall. hu he deired η (, y) ca be coruced. Now by (.3), we have U (L ξ+η ) U (L ξ ) = ζ (y)u(l ξ(y)+η(,y)) U(L ξ )dy 1 = ζ (y) E µ U(L ξ(y)+θη(,y), ξ + θη (, y))η (, y) dθdy = 1 1 ζ (y) µ U(L ξ(y)+θη(,y), x i + θ )p i(, y) dθdy 9

10 By he uiform coiuiy of µ U, we have Noe ha, by (.14), E µ U (µ, ξ)η 1 = lim U (L ξ+η ) U (L ξ ) = 1 1 ζ (y) µ U(L ξ(y), x i + θ ) lim p i (, y) dθdy. p i (, y) lim = C + 1 i E µ ψ j (µ, ξ)η = C + 1 j=1 i E φ j(ξ)η. j=1 he E µ U (µ, ξ)η = C 1 ζ (y) + 1 ζ (y) 1 1 µ U(L ξ(y), x i + θ ) dθdy µ U(L ξ(y), x i + θ i ) E φ j(ξ)η dθdy. j=1 Noe ha he lef ide ad he ecod lie of he above equaliy do o deped o C, ad followig our argume he choice of C i o uique, he we mu have ad hece ζ (y) 1 µ U(L ξ(y), x i + θ ) dθdy =, E µ U (µ, ξ)η = 1 1 ζ (y) + 1 hi implie µ U (µ, x) = 1 1 ζ (y) + 1 µ U(L ξ(y), x i + θ i ) E φ j(ξ)η dθdy. j=1 µ U(L ξ(y), x i + θ i ) φ j(x) dθdy. Moreover, for x x k, x k+1 ), we have i j=1 φ j (x) = φ k (x)δ i,k. he µ U (µ, x) = 1 1 ζ (y) µ U(L + 1 ξ(y), x k + θ )φ k (x) dθdy (.15) = µ U(µ, x) φ k (x) + o(1), for large. Recall (.5) ad (.7), we ee ha 1 φ k (x) = I ((x x k )) ad hu 1 φ k (x) = 1 for 1 (x x k) 1 1, ad φ k. 1 j=1

11 hu 1 µ U (µ, x) µ U(µ, x) dx = 1 = k= 1 C x k + 1 xk+1 + x k x k+1 1 k= x k + 1 x k + hi complee he proof. xk+1 x k+1 1 k= xk+1 x k µ U(µ, x)1 + φ k (x) dx + o(1) C µ U(µ, x)1 + φ k (x) dx + o(1) dx + o(1) + o(1) = o(1). 3 A mea field game ad he maer equaio Le (Ω, F, F, P) be a filered probabiliy pace, B i a F-Browia moio, ad F = F F B. We aume F i rich eough o uppor ay µ P. Give (, µ), P ad a Markovia (for impliciy) corol α A := L (, R), coider he followig SDE: X,µ,α = ξ + α r (X,µ,α r )dr + B,,, (3.1) where B := B B ad ξ L (F ) wih L ξ = µ. By Giraov heorem, he above SDE ha a uique weak oluio. Whe µ = δ x, we may alo deoe i a X,x,α. Now fix θ = (, x, µ) Θ ad α A, coider he followig opimizaio problem: J(θ; α, α ) := E P G(X,x,α V (α; θ) := up J(θ; α, α ),, L X,µ,α) + α A F (X,x,α where F, G : P R are meaurable i all variable. where, L X,µ,α ) 1 α (X,x,α ) d, (3.) Defiiio 3.1 We ay α A i a mea field equilibrium (MFE) of (3.) a (, µ) if V (α ;, x, µ) = J(, x, µ; α, α ) for µ-a.e. x. We remark ha a MFE i local i (, µ), bu i global i x. Whe here i a uique MFE for each (, µ), deoed a α (, µ), he clearly he game problem lead o a value fucio: V (, x, µ) := V (α (, µ);, x, µ). (3.3) he above value fucio i aociaed wih he followig maer equaio: LV (θ) := V + 1 xxv 1 xv Ẽ + F + MV =, V (, x, µ) = G(x, µ), 1 where MV (θ) := x µ V (, x, µ, ξ) µ V (, x, µ, ξ) x V (, ξ, µ). (3.4) 11

12 Here L ξ = µ ad Ẽ i wih repec o ξ. We emphaize ha he erm x V i MV i global i x, ad hu he maer equaio i o-local. I paricular, we cao expec a compario priciple for i oluio. he followig reul i well kow, ee e.g. 9. Propoiio 3. Aume F, G are coiuou ad he maer equaio (3.4) ha a claical oluio V C 1,, (Θ). he α (, µ; x) := x V (, x, µ) i a MFE a (, µ), for all (, µ). 3.1 Characerizaio via forward backward PDE ad SDE he maer equaio (3.4) i aociaed wih he followig yem of forward backward PDE: for ay fixed (, µ), P ad coider he PDE o,, ρ(, x) 1 xxρ(, x) div(ρ(, x) x u(, x)) = ; u(, x) + 1 xxu(, x) 1 xu(, x) + F (x, ρ ) = ; ρ(, ) = µ, u(, x) = G(x, ρ ). (3.5) Here ρ(, ) i uderood i meaure ee, while ρ(, ) i a deiy fucio for >. We hall deoe by ρ he meaure wih deiy ρ(, ). Propoiio 3.3 Aume he maer equaio (3.4) ha a claical oluio V C 1,, (Θ). he PDE (3.5) ha a claical oluio (ρ, u) ad i aifie u(, x) = V (, x, ρ ). Nex, give (, µ) ad ξ L (F ) wih L ξ = µ. Coider he followig forward backward SDE o, : X,ξ = ξ Z,ξ d + B Y,ξ = G(X,ξ, L X,ξ) + X,x,ξ Y,x,ξ = x Z,x,ξ d + B = G(X,x,ξ, L X,ξ) + F (X,ξ, L X,ξ F (X,x,ξ, L X,ξ ) + 1 Z,ξ d Z,ξ db ; ) + 1 Z,x,ξ d Z,x,ξ db. Propoiio 3.4 Aume he PDE (3.5) ha a claical oluio (ρ, u). he FBSDE (3.6) ha a uique rog oluio ad i aifie L X,ξ = ρ ad Y,ξ = u(, X,ξ ), Z,ξ = x u(, X,ξ ); Y,x,ξ = u(, X,x,ξ ), Z,x,ξ = x u(, X,x,ξ ). (3.6) 1

13 3. echical codiio I hi ubecio, we collec ome echical codiio which will be ued i he paper. We fir aume ome regulariy codiio o F, G. Aumpio 3.5 F, G : P 1 R are uiformly Lipchiz coiuou i boh x ad µ (uder W 1 ) wih Lipchiz coa L 1. For coveiece, we hall alo deoe by L > a upper boud of F, G. Aumpio 3.6 F, G are differeiable i x, ad x F, x G are alo uiformly Lipchiz coiuou i boh x ad µ (uder W 1 ) wih Lipchiz coa L. Aumpio 3.5 i adard, excep ha i would look more aural o aume he Lipchiz coiuiy uder W, i ligh of (.3). We ue W 1 here maily becaue of he iue meioed i Remark.3 (i), ee alo Remark 4.3 below. We emphaize agai ha Aumpio 3.5 implie F ad G are uiformly Lipchiz coiuou uder W. Aumpio 3.6 i omewha roger ha wha we expeced ad i will be ideal o remove i. However, we hould poi ou ha i i ill much weaker ha he echical codiio required i he lieraure for he exiece of claical oluio o he maer equaio, ee e.g. 11. A a direc coequece of he above aumpio we have he regulariy of PDE (3.5). Propoiio 3.7 (i) Le Aumpio 3.5 hold. he PDE (3.5) admi a claical oluio u C 1, (, ) ) C (, ) ad ρ C 1/ (,, P ), where he equaio for ρ hold i he ee of diribuio. Moreover, here exi coa C = C (, L ) ad C 1 = C 1 (, L, L 1 ), for he L, L 1 i Aumpio 3.5, uch ha u(, x) C, x u(, x) C 1, W (ρ, ρ ) C 1 1/ ; xxu(, x) C 1, ρ(, x) C 1. (3.7) (ii) Aume furher ha Aumpio 3.6 hold wih coa L. he u C,1 (, ) ad for a C = C (, L, L 1, L ), we have u(, x) + xxu(, x) C. (3.8) Proof he exiece of claical oluio i adard, ee e.g. 7. o ee he uiform eimae, deoe v := e u. he v aifie he followig liear PDE: v(, x) + 1 xv(, x) v(, x)f (x, ρ ) =, v(, x) = e G(x,ρ ). 13

14 I i raighforward o derive he eimae for v, x v, xxv, which implie he eimae for u, x u, xxu. he boudede of u follow from he PDE for u. Moreover, oe ha ρ = L X, where L ξ = µ ad X i he (rog) oluio of X = ξ x u(, X )d + B, he, ice x u C 1, oe ca eaily ee ha W (ρ, ρ ) E X X C 1. Fially, o derive he eimae for ρ(, x), for oaioal impliciy we aume = ad =. he he Malliavi derivaive aifie: D X = 1 hu D X = exp ( xxu(, X r )dr ), ad xxu(r, X r )D X r dr,. DX := D X d = exp ( exp ( C 1 r dr ) d = xxu(, X r )dr ) d exp ( C 1 ) d C 1. Now for ay mooh (i he ee of Malliavi calculu) radom variable η, applyig he iegraio by par formula (cf 18 Lemma 1..1), E D ηd X d = E η Eη hi implie ha he Skorohod iegral aifie: D X db Eη E exp ( E δ(dx ) C 1. Now by 18 Propoiio.1.1 we have ρ(, x) = E 1 {X >x}δ(dx ) DX E δ(dx ) DX C 1 hi complee he eimae for ρ i (3.7). D X d C 1 r dr ) d C 1 Eη. = C 1. hroughou he paper, we hall ue C o deoe a geeric coa depedig oly o, ad C i if i deped o ad L,, L i, for i =, 1,. We remark ha i geeral we do o have uiquee i Propoiio 3.7. For ha purpoe we eed he followig moooiciy codiio, ee heorem 4.1 ad 4.4 below. 14

15 Aumpio 3.8 For ay µ 1, µ P() ad for Φ = F, G, we have Φ(x, µ1 ) Φ(x, µ ) µ 1 (dx) µ (dx). (3.9) hi aumpio i crucial for he uiquee of MFE, ad i he mea ime i a key codiio for he eimae i he paper, which eure furher he exiece of our weak oluio o he maer equaio. However, for F, G aifyig (3.9), i i ulikely heir mooh mollifier coruced i Secio will maiai he ame moooiciy propery. hi i oe of he mai difficulie we ecouer i hi paper. 3.3 he mai reul I he re of he paper, we will prove he followig mai reul. Uder Aumpio 3.5, 3.6, ad 3.8, V i uiformly Lipchiz coiuou i (x, µ) (uder W 1 ) ad uiformly Hölder- 1 coiuou i. Moreover, for each, V (, ) aifie he moooiciy codiio (3.9). See heorem 4.. Uder Aumpio 3.5, 3.6, ad 3.8, he abiliy propery hold for V i erm of he mollificaio of (F, G). See heorem 4.4. Uder Aumpio 3.5 ad if i mall, he we have he claical oluio heory for he maer equaio. I paricular, we eablih a poiwie repreeaio formula for µ V. See heorem 5.. Uder Aumpio 3.5, we have he uiquee of our vaihig weak oluio; ad we obai he exiece uder addiioal Aumpio 3.6 ad 3.8. See heorem 6.. Uder Aumpio 3.5, 3.6, ad 3.8, we have he exiece ad uiquee of our Sobolev oluio, provided ha µ F, µ G are coiuou. See heorem 7.3 ad Sabiliy of he PDE yem We fir udy he abiliy of u wih repec o ρ. heorem 4.1 Le Aumpio 3.5, 3.6, ad 3.8 hold. For i = 1,, le (u i, ρ i ) be a claical oluio o PDE (3.5) wih iiial codiio µ i P 1. he up u 1 (, x) u (, x) C W 1 (µ 1, µ ). (4.1) (,x), Moreover, he followig moooiciy propery hold: u 1 (, x) u (, x)ρ 1 (, x) ρ (, x)dx. (4.) 15

16 Proof Wihou lo of geeraliy, we aume =. Deoe u := u 1 u, ad imilarly for he oher oaio. Sep 1. By (3.5) we have ρ(, x) 1 xxρ(, x) x ρ 1 (, x) x u 1 (, x) ρ (, x) x u (, x) = ; u(, x) + 1 xxu(, x) 1 x u 1 (, x) x u (, x) + F (x, ρ 1 ) F (x, ρ ) = ; (4.3) ρ(, ) = µ, u(, x) = G(x, ρ 1 ) G(x, ρ ). Muliply he fir equaio by u ad he ecod equaio by ρ, iegrae over x, ad he apply he iegraio by par formula, we obai u(, x)ρ(, x)dx + ρ 1 x u 1 ρ x u x u 1 x u 1 x u ρ +F (x, ρ 1 ) F (x, ρ )ρ(, x) dx =. Noe ha he ρ 1 x u 1 ρ x u x u 1 x u 1 x u ρ = ρ1 + ρ x u. ρ 1 + ρ x u (, x)dxd = u(, x)µ(dx) G(x, ρ 1 ) G(x, ρ )ρ(, x)dx F (x, ρ 1 ) F (x, ρ )ρ(, x)dxd. Sice F, G aify Aumpio 3.8, we have ρ 1 + ρ x u (, x)dxd u(, x)µ(dx) x u W 1 (µ 1, µ ), (4.4) where he ecod iequaliy hak o (.). Moreover, he fir iequaliy of (4.4) implie (4.) a =. We ca prove (4.) a arbirary imilarly. Sep. We ex eimae W 1 (ρ 1, ρ ). Le ξ 1, ξ be uch ha L ξi = µ i ad W 1 (µ 1, µ ) = E ξ, where ξ := ξ 1 ξ. Recall (3.8) ad le X i olve he followig SDE: X i = ξ i x u i (, X i )d + B. he L X i = ρ i ad hu W 1 (ρ 1, ρ ) E X, where X := X 1 X. Noe ha X = ξ = ξ x u 1 (, X 1 ) x u (, X )d x u 1 (, X 1 ) x u 1 (, X ) + x u(, X )d. 16

17 he, by (3.8) ad (4.4), we obai E X E ξ + C E X d + ( C E X d + W 1 (µ 1, µ ) + C x u(, x) ρ (, x)dxd C E X d + W 1 (µ 1, µ ) + C x u 1 W 1 1 (µ 1, µ ). Applyig he Growall iequaliy we obai ) 1 x u(, x) ρ (, x)dxd W 1 (ρ 1, ρ ) E X C W 1 (µ 1, µ ) + x u 1 W 1 1 (µ 1, µ ). (4.5) Sep 3. Recall (4.3) for he equaio of u: u(, x) + 1 xxu(, x) 1 x u 1 (, x) + x u (, x) x u + F (x, ρ 1 ) F (x, ρ ) = ; u(, x) = G(x, ρ 1 ) G(x, ρ ). (4.6) Coider SDE: X x = x 1 he we have he adard Feyma-Kac formula: u(, x) = E G(X x, ρ 1 ) G(X x, ρ ) + x u 1 + x u (, X x )d + B. (4.7) F (X x, ρ 1 ) F (X x, ρ )d. (4.8) Deoe X x X := lim x+ε ε X x ε. he X x = 1 1 xx u 1 + xx u (, X x ) X x d. By (3.8) we ee ha X x C. Differeiae (4.8) we have x u(, x) = E x G(X x, ρ 1 ) x G(X x, ρ ) X x + x F (X x, ρ 1 ) x F (X x, ρ ) X x d. By Aumpio 3.6 ad (4.5), we have x u(, x) C up W 1 (ρ 1, ρ ) C W 1 (µ 1, µ ) + C x u 1 W 1 1 (µ 1, µ ). Similarly we ca prove he eimae a (, x) ad hu x u C W 1 (µ 1, µ ) + x u 1 W 1 1 (µ 1, µ ). 17

18 hi implie ha, by (4.5) agai, x u C W 1 (µ 1, µ ), ad hece W 1 (ρ 1, ρ ) C W 1 (µ 1, µ ). he, by Aumpio 3.5 ad (4.8), u(, x) C 1 up W 1 (ρ 1, ρ ) C W 1 (µ 1, µ ). Similarly we prove he eimae a (, x) ad hece complee he proof. Clearly heorem 4.1 implie he uiquee of claical oluio for PDE (3.5). We he defie V (, x, µ) := u(, x), where u i he oluio o PDE (3.5) wih ρ(, ) = µ. (4.9) heorem 4. Le Aumpio 3.5, 3.6, ad 3.8 hold. he he V defied by (4.9) i uiformly Lipchiz coiuou i x ad i µ uder W 1, ad uiformly Hölder- 1 coiuou i. Moreover, for ay, V (, ) aifie he moooiciy codiio (3.9). Proof he Lipchiz coiuiy i (x, µ) ad he moooiciy propery follow from Propoiio 3.7 ad heorem 4.1. o ee he regulariy i, fix (, µ) ad le (u, ρ) be he claical oluio o PDE (3.5). Noe ha u(, x) = V (, x, ρ ). he, for = + δ, V (, x, µ) V (, x, µ) V (, x, µ) V (, x, ρ ) + u(, x) u(, x) C W 1 (ρ, ρ ) + C δ C W (ρ, ρ ) + C δ C δ, where he la iequaliy hak o (3.7). Remark 4.3 he above V i uiformly Lipchiz coiuou i µ uder W 1. I erm of he µ V defied i (.3), i i more aural o ue W. However, due o Remark.3 (i), we eed he roger Lipchiz coiuiy uder W 1. We ow eablih he abiliy of u wih repec o F, G. heorem 4.4 For i = 1,, aume F i, G i aify Aumpio 3.5 ad le (u i, ρ i ) be a claical oluio o PDE (3.5) wih coefficie (F i, G i ). Deoe F := F 1 F ad imilarly he oher oaio. (i) If (F 1, G 1 ) aifie Aumpio 3.8, he 1 6 u C 1 F + G. (4.1) (ii) Aume furher ha boh (F 1, G 1 ) ad (F, G ) aify Aumpio 3.6. he u C F + G + x F + x G. (4.11) 18

19 Proof (ii) We fir prove (4.11) uder Aumpio 3.6. I hi cae he mai idea i very imilar o ha of heorem 4.1. For impliciy we aume =. Sep 1. Noe ha ρ = ad F 1 (x, ρ 1 ) F (x, ρ ) = F 1 (x, ρ 1 ) F 1 (x, ρ ) + F (x, ρ ); G 1 (x, ρ 1 ) G (x, ρ ) = G 1(x, ρ 1 ) G 1(x, ρ ) + G(x, ρ ). Followig imilar argume a i heorem 4.1 Sep 1, epecially by he moooiciy codiio (3.9) for (F 1, G 1 ) ad (.), we ca eaily how ha ρ 1 + ρ x u (, x)dxd G(x, ρ )ρ(, x)dx F (x, ρ )ρ(, x)dxd C x F + x G up W 1 (ρ 1, ρ ). (4.1) Sep. We ex eimae W 1 (ρ 1, ρ ). Aume L ξ = µ ad X i olve he followig SDE: he clearly X i = ξ x u i (, X i )d + B. E X C E X d + hu, by (4.1) ad oe ha L X = ρ, W 1 (ρ 1, ρ ) E X C hi implie ha E x u(, X ) d. E x u(, X ) d C ( E C ( x F + x G up W 1 (ρ 1, ρ ) Sep 3. Similar o (4.3) we have ) 1 x u(, X ) d ) 1 up W 1 (ρ 1, ρ ) C x F + x G. (4.13) u(, x) + 1 xxu(, x) 1 x u 1 (, x) + x u (, x) x u + F 1 (x, ρ 1 ) F (x, ρ ) = ; u(, x) = G 1 (x, ρ 1 ) G (x, ρ ). For he X x i (4.7), we have u(, x) = E G 1 (X x, ρ 1 ) G (X x, ρ ) + 19 F 1 (X x, ρ 1 ) F (X x, ρ )d. (4.14)

20 he, by Aumpio 3.5 ad (4.13), u(, x) C 1 F + G + up W 1 (ρ 1, ρ ) C F + G + x F + x G. Similarly we prove he eimae a (, x) ad hece obai (4.11). (i) We ow prove (4.1) wihou aumig Aumpio 3.6. Deoe ε := F + G. Fir, by he fir lie of (4.1) we have ρ 1 + ρ x u (, x)dxd Cε. (4.15) o eimae W 1 (ρ 1, ρ ), oe ha we cao ue (3.8) aymore, we hall iead ue (.). Le L ξ = ρ ad B be a P -Browia moio. Deoe X := ξ + B, ad for i = 1,, θ i := x u i (, X ), B i := B θ i d, dp ( i := M i := exp θ dp db i 1 ) θ i d. he B i i a P i -Browia moio, ad ρ i i he P i -diribuio of X. By (3.7) we have θ i C 1, E P M i p + M i p C 1,p for all p 1, i = 1,. (4.16) Now for ay fucio ϕ a i (.), we have ϕ(x)ρ(, x)dx = E P 1 ϕ(x ) E P ϕ(x ) = E P M 1 M ϕ(x ) = E P Noe ha M 1 M ϕ(x ) ϕ(ξ) E P M 1 M B = E P 1 1 M ( E P 1 1 M M 1 ) 1 ( ) 1 E P 1 B M M 1 M 1 B C 1 (E P 1 1 M M 1 ) 1. (4.17) ( = exp θ db 1 1 ) θ d, where θ := θ 1 θ. he, for ay δ >, by (4.15) ad (4.16) we have E P 1 1 M M 1 E P 1 1 M M 1 Cδ + 1 δ EP 1 1 { θdb1 δ, θ d δ} + 1 { θdb1 >δ} + 1 { θ d>δ} 1 M Cδ + C ( 1 E P 1 δ Cδ + C 1 δ ( E P 1 M 1 θ db 1 + θ d θ db 1 + ( ) 1 θ d θ d ) ) 1 Cδ + C 1 ε. δ

21 Se δ := ε 1 6, he E P 1 M 1 C M 1 1 ε 1 3, ad hu ϕ(x)ρ(, x)dx C 1ε 1 6, hak o (4.17). herefore, we derive from (.) ha W 1 (ρ 1 (), ρ ()) Cε 1 6 = C F + G 1 6. (4.18) Fially, by (4.14) ad Aumpio 3.5 we have u(, x) C 1 F + G + up W 1 (ρ 1, ρ ) C 1 F + G 1 6. Similarly we ca prove he eimae a (, x) ad hece obai (4.1). 5 he claical oluio i mall ime duraio he aalyi i he previou ecio provide he Lipchiz coiuiy ad abiliy of he value fucio V. I hi ecio we focu o he differeiabiliy of V i erm of µ, which i he key for he claical oluio of he maer equaio. For hi purpoe, i i more coveie o ue he FBSDE yem (3.6). For mooh coefficie F ad G, 11 how ha, roughly peakig, V i mooh if eiher (F, G) aify he moooiciy codiio (3.9) or if he ime duraio i mall. For weak oluio, we hall apply hee reul o he maer equaio wih coefficie (F, G ), which are he mooh mollifier of (F, G). However, ice (F, G ) ypically do o aify (3.9), o i hi ecio we focu o he cae ha i mall. We fir udy he differeiabiliy i x, which i more or le adard. Propoiio 5.1 Aume F, G aify Aumpio 3.5. here exi ome coa δ 1 = δ 1 (L, L 1 ) > uch ha he followig hold wheever δ 1. (i) he FBSDE yem (3.6) ha a uique oluio wih Z,ξ ad Z,x,ξ bouded by C 1. (ii) he mappig ξ Y,x,ξ i law ivaria ad hu V (, x, µ) := Y,x,ξ wih L ξ = µ i well defied. Moreover, V i uiformly Lipchiz coiuou i (x, µ) P ad Hölder- 1 coiuou i, ad i bouded by C 1. (iii) x V (, x, µ) exi for < ad i bouded by C 1. Moreover, if x F, x G exi ad are coiuou, he x V i coiuou o Θ ad we have he followig repreeaio: x V (, x, µ) = x Y,x,ξ, (5.1) 1

22 where x X,x,ξ x Y,x,ξ + = 1 x Z,x,ξ = x G(X,x,ξ d, L X,ξ ) x X,x,ξ x F (X,x,ξ, L X,ξ) x X,x,ξ + Z,x,ξ x Z,x,ξ d x Z,x,ξ db. (5.) Proof (i) he exiece i due o Propoiio 3.4 ad 3.7. he uiquee (amog oluio wih bouded Z) follow from he adard coracio mappig argume (c.f. 4 Secio 8.), ice i mall here. We emphaize ha, he argume i 4 Secio 8. require uiformly Lipchiz coiuou coefficie, while (3.6) ivolve Z, which i i geeral o uiformly Lipchiz coiuou. However, ice we kow a priori ha Z i bouded by C 1 ad by rericig o he oluio wih bouded Z, we may acually view he coefficie a uiformly Lipchiz coiuou ad hu all he argume remai valid. (ii) I i clear ha ξ Y,x,ξ i law ivaria. Sice F ad G are bouded by L ad Z,x,ξ i bouded by C 1, he by akig expecaio i he la equaio of (3.6) we ee ha V (, x, µ) = Y,x,ξ i bouded by C 1. he regulariy of V follow follow he adard eimae for he FBSDE yem (agai wih uiformly Lipchiz coiuou coefficie). (iii) Noe ha V (, x, µ) = u(, X,x,ξ ) = u(, x), where (u, ρ) i he claical oluio o he PDE (3.5) wih iiial codiio ρ = µ. he x V (, x, µ) = x u(, x) i bouded by C 1. Noe ha, give L X,ξ, he la wo equaio i (3.6) i a adard FBSDE. Whe x F, x G exi ad are coiuou, differeiae hem formally i x we obai FBSDE (5.). By 4 heorem we ee ha FBSDE (5.) i wellpoed. I follow from adard argume i FBSDE lieraure ha (5.1) hold, which implie furher he coiuiy of x V. he followig reul maily follow from 11. However, he poiwie repreeaio formula (5.3) i ew, o our be kowledge. heorem 5. Aume F, G C 1,1 ( P) aify Aumpio 3.5. he here exi a coa δ 1 uch ha he followig hold wheever δ 1. (i) µ V exi ad i coiuou o Θ. Moreover, we have he followig repreeaio formula: by omiig (, ξ) i he upercrip whe here i o cofuio, µ V (, x, µ, x ) = E Ẽ µ G(X x x x, L X, X ) X + µ G(X x, L X, X x, ) X x, M x + M x µ F (X x, L X, x x X ) X + µ F (X x, L X, X x, ) X x, d, (5.3)

23 where refer o idepede copie a i (.4), ad X x, := EX,ξ F B σ(ξ1 {ξ x} ), Z x, := EZ,ξ F B σ(ξ1 {ξ x} ); Γ x = x G(X x, L X ) + x F (Xr x, L Xr ) + Γ x r + Zr x γr x dr γr x db r ; M x = 1 + M x r Γ x r + Z x r db r ; (5.4) ad ( X x, Y x, Z x, X x, Y x, Z x ) olve he followig liear McKea-Vlaov ype FBSDE: X x = 1 Z x d; Y x = x G(X x, L X ) X x + X x, +Ẽ µ G(X x, L X, X x ) X x + + µ G(X x, L X, Y x, x, x, X ) X + = x G(X x,, L X ) X x, = Z x, d, x F (X x, L X ) X x + Z x Z x d + +Ẽ µ G(X x,, L X, X x ) X x + + µ G(X x, L X, Z x, db. x, x, X ) X + µ F (X x, L X, X x ) X x d µ F (X x, L X, x, X ) x F (X x, L X ) X x, Z x db x, X d P(ξ = x); (5.5) + Z x, µ F (X x,, L X, X x ) X x d µ F (X x, L X, x, X ) Z x, d x, X d P(ξ x) I paricular, whe x i o a aom of ξ, amely P(ξ = x) =, he X x = X, Z x = Z, ad he FBSDE (5.5) become X x = 1 Z x d; Y x = x G(X x, L X ) X x + Y x, X x, = x G(X, L X ) X x, + +Ẽ µ G(X, L X, X x ) X x + + µ G(X, L X, X ) X x, + = Z x, d, x F (X x, L X ) X x + Z x Z x d x F (X, L X ) X x, µ F (X, L X, X x ) X x d µ F (X, L X, X ) Z x db ; + Z Z x, d (5.6) x, X d Z x, db. (ii) If F ad G are mooh eough i all variable, he V i mooh eough i all variable ad i paricular i a claical oluio o he maer equaio (3.4). 3

24 Proof We fir emphaize ha Aumpio 3.5 implie ha F ad G are Lipchiz coiuou i µ uder W a well. he proof for (ii) i leghy bu quie raighforward, by combiig he argume i (i) ad 11, we hu omi i. We hall prove (i) i five ep. o implify he preeaio, we aume F =. he preece of F doe o caue ay difficuly. Moreover, for oaioal impliciy, we aume = ad omi i X,ξ ec. Sep 1. We fir how ha he FBSDE (5.4) ad (5.5) are wellpoed for δ 1. Ideed, a we kow ha Z ad Z x are bouded by C 1, he o i Z x,. hu all he (radom) coefficie of he liear FBSDE (5.5) are bouded ad he (5.5) i wellpoed. o ee (5.4), we oe ha he erm Γ x r γr x i o Lipchiz coiuou. However, deoe ˆΓ x := M x Γ x ad ˆγ x := M x γ x + Γ x Γ x + Z x, oe ca verify raighforwardly ha (M x, ˆΓ x, ˆγ x ) aifie he followig liear FBSDE wih bouded coefficie (aumig F = for impliciy): M x = 1 + Zr x Mr x + ˆΓ x r db r, ˆΓx = x G(X x, L X )M x ˆγ r x db r. (5.7) he (5.7) i wellpoed for δ 1, hece o i (5.4). Sep. For ay ξ, η L (F ) wih L ξ = µ, followig adard argume we have lim E ε up Xξ+εη X ξ ε µ X ξ,η =, (5.8) where ( µ X ξ,η, µ Y ξ,η, µ Z ξ,η ) aifie he liear McKea-Vlaov FBSDE: µ X ξ,η = η µ Z ξ,η d, µ Y ξ,η = x G(X ξ, L X ξ ) µ X ξ,η + Z ξ µ Z ξ,η d + Ẽ µ G(X ξ, L X ξ, X ξ ) µ µ Z ξ,η db, ξ,η X (5.9) Nex, by (5.8), oe ca how ha lim E ε up Y x,ξ+εη Y x,ξ ε µ Y x,ξ,η =, (5.1) where ( µ X x,ξ,η, µ Y x,ξ,η, µ Z x,ξ,η ) aifie he liear (adard) FBSDE: µ X x,ξ,η µ Y x,ξ,η + = µ Z x,ξ,η d = x G(X x,ξ, L X ξ ) µ X x,ξ,η Z x,ξ µ Z x,ξ,η d + Ẽ µ G(X x,ξ, L X ξ, X ξ ) µ µ Z x,ξ,η db. ξ,η X (5.11) 4

25 I paricular, (5.1) implie, hu, by he defiiio of µ V, lim V (, x, L ξ+εη) V (, x, L ξ ) ε ε µ Y x,ξ,η =. E µ V (, x, µ, ξ)η = µ Y x,ξ,η. (5.1) Moreover, recall (5.4), oe ca verify raighforwardly ha d M x,ξ µ Y x,ξ,η Γ x,ξ µ X x,ξ,η = db, ad all he procee have he deired iegrabiliy. he µ Y x,ξ,η = M x,ξ µ Y x,ξ,η Γ x,ξ µx x,ξ,η = E = E Ẽ M x,ξ µg(x x,ξ, L X ξ, X ξ ) µ M x,ξ X ξ,η µy x,ξ,η Γ x,ξ µx x,ξ,η. (5.13) Sep 3. I hi ep we prove (5.3) i he cae ha ξ i dicree: p i = P(ξ = x i ), i = 1,,. We fir oe ha, by oherwie akig codiioal expecaio o (5.9), codiioal o F B σ(ξ), we may aume wihou lo of geeraliy ha η i σ(ξ)-meaurable, wih η = η i o {ξ = x i }, ad hu η = η i1 {ξ=xi }. Sice (5.11) i liear, we have µ Y x,ξ,η = η i µ Y x,ξ,1 {ξ=x i }. he (5.1) implie η i µ Y x,ξ,1 {ξ=x i } = By he arbirarie of η i, hi implie ha Fix i. For W = X, Y, Z, deoe E µ V (, x, µ, x i )η i pi. µ V (, x, µ, x i ) = 1 p i µ Y x,ξ,1 {ξ=x i }. (5.14) W i := EW ξ F B, ξ = x i = W x i,ξ, W i := E µ W ξ,1 {ξ=x i } F B, ξ = x i; (5.15) W i, := EW ξ F B σ(ξ1 {ξ x i }), W i, := 1 p i p i E µ W ξ,1 {ξ=x i } F B σ(ξ1 {ξ x i }). Noe ha Ẽ µ G(X ξ, L X ξ = Ẽ µ G(X ξ, L X ξ, X ξ ) µ X ξ,1 {ξ=x i }, X i ) X i + µ G(X ξ, L X ξ, X i, ) X i, p i. (5.16) 5

26 ake codiioal expecaio o (5.9), codiioal o F B {ξ = x i} ad F B σ(ξ1 {ξ x}), repecively, we have X i = 1 Z i d, X i, = Z i, d, Y i = x G(X i, L X ξ ) X i + Z Z i d i ZdB i +Ẽ µ G(X i, L X ξ, X i ) X i + µ G(X i i, i,, L X ξ, X ) X p i ; Y i, +Ẽ = x G(X i, µ G(X i,, L X ξ ) X i, +, L X ξ Z i, Z i, d, X i ) X i + µ G(X i,, L X ξ, i, X Z i, ) X i, db (1 p i ). hi i FBSDE (5.5) i he pree cae. Moreover, by (5.14), (5.13), ad (5.16) we have = E Ẽ µ V (, x, µ, x i ) = 1 E p Ẽ i M x,ξ µg(x x,ξ, L X ξ M x,ξ µg(x x,ξ, L X ξ, X i ) X i + M x,ξ which prove (5.3) o he uppor of ξ (whe F = )., X ξ ) µ X ξ,1 {ξ=x i } µg(x x,ξ, L X ξ, X i, ) X i,, (5.17) Sep 4. We ow prove (5.3) i he cae ha ξ ha coiuou diribuio, agai aumig F = ad = for impliciy. For each 1, le x i := i for i =,...,, ξ := x i1 xi 1,x i )(ξ), ad p i := P(ξ = x i ). I i clear ha ξ ξ 1. he by (5.1) ad he abiliy of FBSDE oe ca eaily how ha, for ay η L (F ), E µ V (, x, µ, ξ)η = µ Y x,ξ,η = lim µy x,ξ,η = lim E µ V (, x, L ξ, ξ )η. (5.18) Fix x, ad le a(µ, x ) deoe he righ ide of (5.3). he Sep 3 implie ha µ V (, x, L ξ, ξ ) = a(l ξ, ξ ). (5.19) Le (X,i, X,i, X,i, ) ec correpod o (ξ, x i ). Sice ξ ha coiuou diribuio, we have lim max 1 i p i =. Now for ay x, le i (x ) be he i uch ha x x i(x ) 1, x i(x )). By he abiliy of FBSDE, oe ca how ha (M x,ξ, X x,ξ, X ξ, X,i(x ), X,i (x ), X,i (x ),, X,i (x ), ) coverge o (M x,ξ, X x,ξ, X ξ, Xx, X x, X, X x, ) uiformly, uiformly i x, where ( X x, X x, ) i defied by (5.6). he, by he repreeaio (5.17), we ee ha lim a(l ξ, x i (x ) ) = a(µ, x ). ogeher wih (5.18) ad (5.19), hi implie ha E µ V (, x, µ, ξ)η = E a(µ, ξ)η. 6

27 hi prove (5.3) whe ξ i coiuou. Moreover, by he abiliy of FBSDE (5.6), we ee ha here exi a modulu of coiuiy fucio ρ (idepede of x), uch ha a(µ 1, x 1) a(µ, x ) ρ ( x 1 x + W (µ 1, µ ) ), (5.) a log a µ 1, µ are coiuou. hi implie he uiform coiuiy of µ V i (µ, x ) whe µ i coiuou, uiformly i x (ad ). he coiuiy of µ V wih repec o (, x) alo follow from adard argume for FBSDE. Sep 5. Fially we prove he geeral cae. For ay µ P, here exi µ P uch ha each µ i coiuou ad lim W (µ, µ) =. By he uiform regulariy (5.) ad applyig he Arzelá-Acoli heorem, poibly alog a ubequece, we have lim a(µ, x ) = b(µ, x ), where b i alo uiformly coiuou ad he covergece i uiform, uiformly i x (ad ). he followig he ame argume a for (5.18) we have µ V (, x, µ, x ) = b(µ, x ) ad hu µ V (, x, ) i uiformly coiuou o P. he coiuiy i (, x) ca be proved imilarly by coiderig he uiform limi of µ V (, x, µ, x ). I remai o ideify b(µ, x ) wih a(µ, x ), whe µ i o coiuou. For hi purpoe we fix x ad prove he equaliy i wo cae. Fir, aume P(ξ = x ) =, le µ be coiuou wih lim W (µ, µ) =. eaily how ha lim a(µ, x ) = a(µ, x ). b(µ, x ) = a(µ, x ). By he abiliy of FBSDE (5.6), oe ca he by he defiiio of b we ee ha Nex, aume P(ξ = x ) >. Le x i, ξ, ec. be he dicree oe a i Sep 4. We emphaize ha, for fixed x, we ill have lim p i (x ) = P(ξ = x ), bu he covergece may o be uiform i x. he agai by he abiliy of FBSDE (5.5) we derive from (5.17) ha lim a(l ξ, x i (x ) ) = a(µ, x ). By Sep 3 we have Ea(L ξ, ξ )η = E µ V (, x, L ξ, ξ )η = Eb(L ξ, ξ )η. hi implie ha a(l ξ, ) = b(l ξ, ) o he uppor of ξ. I paricular, ice lim p i (x ) = P(ξ = x ) >, we have a(l ξ, x i (x ) ) = b(l ξ, x i (x )) for large eough. he by he uiform coiuiy of b we have a(µ, x ) = lim a(l ξ, x i (x ) ) = lim b(l ξ, x i (x ) ) = b(µ, x ). hi complee he proof. Remark 5.3 (i) While heorem 5. i eablihed for mall, he repreeaio formula hold rue for arbirary, provided ha all he ivolved FBSDE are wellpoed. hi i he cae whe F, G alo aify Aumpio 3.6 ad 3.8, ee heorem 7.3 Sep 1 below. 7

28 (ii) By uig he liearized yem of PDE (3.5), 8 (Corollary 3.9) provided a poiwie repreeaio formula for he gradie δv δµ (, x, µ, x ). Noe ha µ V (, x, µ, x ) = x δv δµ (, x, µ, x ), o 8 implie a repreeaio formula for µ V (, x, µ, x ) a well, by ivolvig a forward backward PDE yem whoe iiial value i he derivaive of he Dirac meaure. Our repreeaio formula (5.3) ivolve rog oluio of FBSDE ad hold uder weaker echical codiio. We oe ha, ulike he coecio bewee (3.5) ad (3.6), he forward PDE i 8 doe o repree he deiy of he forward SDE i (5.5), o he coecio bewee (5.3) ad heir repreeaio formula i o clear o u. Recall Remark.3-(ii) ha our mollifier doe o have uiform covergece for µ U. A a applicaio of our repreeaio formula, we have he followig poiwie covergece reul for µ V. We fir remark ha, for Φ = F, G C 1,1 ( P), le Φ be a mooh mollifier of Φ where he mollificaio i µ i a i Secio ad he mollificaio i x i adard. he i i clear ha (Φ, x Φ ) (Φ, x Φ) uiformly, ad µ Φ coverge o µ Φ i he ee of heorem. (iii). I he re of he paper, we hall alway ue mollifier i hi way. Corollary 5.4 Aume (F, G) C 1,1 ( P) aify Aumpio 3.5 ad δ 1 for he δ 1 i heorem 5.. Le (F, G ) be he mooh mollifier of (F, G) ad V be he claical oluio o he maer equaio (3.4). he, for ay <, Proof lim up (x,µ,x ) P ( µ V µ V )(, x, µ, x ) =. (5.1) Wihou lo of geeraliy we aume =. Fix (x, µ, x ). Beide C 1, a below we ue aoher geeric coa c, which deped o L ad L 1 i Aumpio 3.5, he uiform coiuiy of µ F, µ G, ad, bu o o (x, µ, x ), uch ha lim c =. For he oaio i he proof of heorem 5., we add ubcrip or upercrip for he correpodig erm defied hrough (F, G ). Fir, ice (F, G ) coverge o (F, G) uiformly, by (3.6) oe ca eaily how ha Z C 1, E up X X + Z Z d c, ad imilar eimae hold for (X,x, Z,x ) ad (X,x,, Z,x, ). Nex, followig he ame argume for he eimae of ρ(, x) i (3.7), oe ca how ha ρ 1 (, ˆx ) + ρ (, ˆx ) C 1, where ρ 1 (, ˆx ) ad ρ (, ˆx ) are he deiie of X,x ad X,x,, repecively. hi, 8

29 ogeher wih heorem. (iii), implie ha Similarly, we have µ E G (ˆx, L X, X,x ) µ G(ˆx, L X, X,x ) µ G (ˆx, L X, ˆx ) µ G(ˆx, L X, ˆx ) ρ 1 (, ˆx )dˆx C 1 µ G (ˆx, L X, ˆx ) µ G(ˆx, L X, ˆx ) dˆx c. µ E G µ G(ˆx, L X,, X,x ) c ; µ E F µ F (ˆx, L X, X,x ) + µ F µ F (ˆx, L X, X,x, ) c. he, recall ha ( x F, x G ) ( x F, x G) uiformly, by (5.5) we ca how ha E up E up X,x + X,x, C, X,x X x + X,x, X x, c,, where c, i ome geeric coa which may deped o bu o o (x, µ, x ) uch ha lim c, =. he, by (5.3) ad he uiform coiuiy of ( µ F, µ G), we have (agai omiig he F erm for impliciy), E µ V (, x, µ, x ) E µ V (, x, µ, x ) = E Ẽ M,x µg (X,x, L X,x,x, X ) X M x µ G(X x x x, L X, X ) X +M,x µg (X,x, L X, X,x, ) X,x, M x µ G(X x, L X, X x, ) X x, E Ẽ M,x,x X M x x X µ G (X,x, L X,x, X ) + M x x X µ G (X,x, L X,x, X ) µ G(X,x, L X,x, X ) + M x x X µ G(X,x, L X,x, X ) µ G(X x x, L X, X ) + M,x X,x, M x X x, µ G (X,x, L X, X,x, ) c,, + M x X x, + M x X x, which implie he corollary immediaely. µ G (X,x, L X, X,x, ) µ G(X,x, L X, X,x, ) µ G(X,x, L X, X,x, ) µ G(X x, L X, X x, ) 9

30 6 he vaihig weak oluio We are ow ready o defie he weak oluio. Defiiio 6.1 We ay V C (Θ) wih V (, ) = G i a vaihig weak oluio of maer equaio (3.4) if, for ay < δ ad ay {V } 1 C 1,, (Θ) uch ha LV i uiformly Lipchiz coiuou i (x, µ) P, uiformly i ad, ad lim up (V V )(, x, µ) + (x,µ) P he here exi δ (, δ uch ha lim heorem 6. Le Aumpio 3.5 hold. up LV (, x, µ) (,x,µ) δ, P =, (6.1) up (V V )(, x, µ) =. (,x,µ) δ, P (i) he maer equaio (3.4) ha a mo oe vaihig weak oluio V uch ha, for ay, he mappig µ V (, x, µ) i uiformly Lipchiz coiuou, uiformly i x. (ii) Aume furher ha Aumpio 3.6 ad 3.8 hold. he he fucio V defied by (4.9) i he uique vaihig weak oluio of he maer equaio (3.4), ad i paricular i aifie he properie i heorem 4.. Proof (i) Aume here are wo vaihig weak oluio V ad ˆV. Deoe := up{, : here exi (x, µ) uch ha ˆV (, x, µ) V (, x, µ)} >. By he coiuiy of V ad ˆV we ee ha V (, ) = ˆV (, ). Now le F ad G be mooh mollifier of F ad V (, ), repecively. By heorem 5., here exi δ 1 (, uch ha he followig maer equaio (6.) ha a claical oluio V o δ 1, wih ufficie regulariy: V + 1 xxv 1 xv + F + MV =, V (, ) = G. (6.) Noe ha LV = F F. By heorem. (i) ad (ii) we ee ha {V } 1 aifie (6.1) for boh V ad ˆV o δ 1,. he by he vaihig weak oluio propery here exi δ (, δ 1 uch ha lim up (,x,µ) δ, P (V V )(, x, µ) + (V ˆV )(, x, µ) =. hi implie ha ˆV (, ) = V (, ) for δ,, coradicig wih he defiiio of. herefore, we mu have = ad hece ˆV = V. (ii) o verify he weak oluio propery of V, we fix < δ ad a deired {V } 1 a i Defiiio 6.1. Deoe F := F LV ad G (x, µ) := V (, x, µ). he V i a claical oluio o he maer equaio (6.) o δ,. Noe ha (F, G ) 3

31 aify Aumpio 3.5 uiformly o δ,, excep ha F deped o a well, ad (F, V (, )) aifie Aumpio 3.8. However, i i raighforward o exed heorem 4.4 (i) o hi cae ad we obai up (V V )(, x, µ) (,x,µ) δ, P C 1 up (F F )(, x, µ) + G (x, µ) V (, x, µ) 1 6 (,x,µ) δ, P = C 1 up LV (, x, µ) + V (, x, µ) V (, x, µ) 1 6. (,x,µ) δ, P hi implie ha V i a vaihig weak oluio. he followig reul i a immediae coequece of heorem 4.4. heorem 6.3 Aume {(F, G )} aify Aumpio 3.5, 3.6, ad 3.8, ad he Lipchiz coa i Aumpio 3.5 i uiform. Le V be he uique vaihig weak oluio o maer equaio (3.4) wih coefficie (F, G ). If lim F F + G G =, he lim V V =. 7 he Sobolev oluio o moivae he defiiio of Sobolev oluio, aume V i a claical oluio of maer equaio (3.4), ad ϕ C 1,1 (, ; R). Muliply (3.4) by ϕ, iegrae over x, ad apply he iegraio by par formula we have = LV (, x, µ)ϕ(, x)dx = V (, x, µ)ϕ(, x)dx V (, x, µ) ϕ(, x)dx 1 x V (, x, µ) x ϕ(, x)dx + 1 xv (, x, µ) + F (x, µ) Ẽ µv (, x, µ, ξ) x V (, ξ, µ) ϕ(, x)dx + 1 Ẽ x µ V (, x, µ, ξ)ϕ(, x)dx. I order o reduce he order of he differeiabiliy i he la erm above, we reric o hoe µ wih a mooh deiy ρ C 1 (). he he la erm above become 1 x µ V (, x, µ, x)ρ( x)ϕ(, x)d xdx = 1 µ V (, x, µ, x)ρ ( x)ϕ(, x)d xdx. Deoe P := {µ P : µ ha a deiy fucio ρ C 1 ()}. (7.1) We fir have he followig imple lemma whoe proof i obviou ad hu i omied. 31

32 Lemma 7.1 he e P i dee i P. Coequely, V C (Θ) i uiquely deermied by i value i, P. We ow defie Defiiio 7. We ay V C,1,1 (Θ) i a Sobolev oluio o he maer equaio (3.4) if V (, ) = G, ad for ay ϕ C 1,1 (, ; R) ad µ P wih deiy ρ, we have V (, x, µ)ϕ(, x)dx = V (, x, µ) ϕ(, x)dx + 1 x V (, x, µ) x ϕ(, x)dx 1 + xv (, x, µ) F (x, µ) ϕ(, x)dx (7.) + µ V (, x, µ, x) 1 ρ ( x) + x V (, x, µ)ρ( x) ϕ(, x)d xdx. We fir verify ha our vaihig weak oluio i alo a Sobolev oluio, provided ha i i i C,1,1 (Θ). heorem 7.3 Le F, G C 1,1 ( P 1 ) aify Aumpio 3.5, 3.6, ad 3.8. he he vaihig weak oluio V of maer equaio (3.4) i alo a Sobolev oluio. Proof We proceed i wo ep. Sep 1. Fir, by heorem 4., V i uiformly Lipchiz coiuou i (x, µ) wih cerai Lipchiz coa ˆL 1 L 1, which deped o L 1 ad L. By Propoiio 5.1 ad heorem 5. we ee ha V C,1,1 ( δ 1 P). We ca e δ 1 uch ha i i deermied by ˆL 1, iead of L 1. he V ( δ 1, ) C 1,1 ( P) ad i alo aifie he moooiciy codiio (3.9). Now apply Propoiio 5.1 ad heorem 5. agai o maer equaio (3.4), bu o ierval δ 1, δ 1 wih ermial codiio V ( δ 1, ), we ee ha V C,1,1 ( δ 1 P). Repea he argume we obai V C,1,1 (Θ). Sep. o verify (7.), oe ha i i local i (ad µ). By oherwie applyig he argume i Sep 1 agai, wihou lo of geeraliy we aume δ 1. Fix ome deired ϕ ad µ, ρ a i Defiiio 7.. Le (F, G ) be he mooh mollifier of (F, G) ad V be he correpodig claical oluio. he (7.) hold for (V, F ), ad hu, for ay, by iegraig i over, we obai G (x, µ)ϕ(, x)dx = + + V (, x, µ)ϕ(, x)dx V (, x, µ) ϕ(, x)dxd xv (, x, µ) F (x, µ) ϕ(, x)dxd x V (, x, µ) x ϕ(, x)dxd µ V (, x, µ, x) 1 ρ ( x) + x V (, x, µ)ρ( x) ϕ(, x)d xdxd. 3

33 Sed. Noe ha (F, G, V ) coverge o (F, G, V ) uiformly, ad we ca eaily how ha x V coverge o x V uiformly. Moreover, by Corollary 5.4 we ee ha µ V µ V wheever <. he, by he domiaed covergece heorem we have G(x, µ)ϕ(, x)dx V (, x, µ)ϕ(, x)dx = + + V (, x, µ) ϕ(, x)dxd xv (, x, µ) F (x, µ) ϕ(, x)dxd x V (, x, µ) x ϕ(, x)dxd µ V (, x, µ, x) 1 ρ ( x) + x V (, x, µ)ρ( x) ϕ(, x)d xdxd. hi implie (7.), hece V i a Sobolev oluio of he maer equaio (3.4). o prove he uiquee of Sobolev oluio, we fir eed a lemma. Lemma 7.4 Le U C 1 (P ), µ P wih deiy ρ, ad ψ C (Θ) be uiformly Lipchiz coiuou i (x, µ). he d d U(ρ ) = µ U(ρ, x) 1 xρ(, x) + ψ(, x, ρ )ρ(, x)dx, (7.3) where ρ i he claical oluio o he followig PDE: Proof ρ(, x) 1 xxρ(, x) div ( ρ(, x)ψ(, x, ρ ) ) =, ρ(, x) = ρ (x) (7.4) Fir i follow from adard argume ha he PDE (7.4) ha a uique claical oluio. Le U C (P ) be he mooh mollifier of U. he U (ρ ) U (ρ ) = = 1 x µ U (ρ, x)ρ(, x)dx µ U (ρ, x)ψ(, x, ρ )ρ(, x) dxd µ U (ρ, x) 1 xρ(, x) + ψ(, x, ρ )ρ(, x)dxd. Sed, by heorem., i paricular par (iii), we have U(ρ ) U(ρ ) = hi implie (7.3) immediaely. µ U(ρ, x) 1 xρ(, x) + ψ(, x, ρ )ρ x (, x)dxd. heorem 7.5 Uder Aumpio 3.5 ad 3.8, he maer equaio (3.4) admi a mo oe Sobolev oluio. 33

34 Proof We follow he mai argume i heorem 4.4. Aume we have wo Sobolev oluio V 1 ad V o (3.4). Fix µ P wih deiy ρ. For i = 1,, le V i (, x, µ) be a adard mooh mollifier of V i i x, ad ρ i he claical oluio o PDE: ρ i (, x) 1 xxρ i (, x) div(ρ i (, x) x V i (, x, ρ i ())) =, ρ i (, x) = ρ (x). Deoe u i (, x) := V i (, x, ρ i ()), v i (, x) := u i (, x) V i (, x, ρ i ()). We emphaize ha V i i a adard mollifier i x, o i µ, he lim vi + x v i =. (7.5) We fir claim ha u i i a Sobolev oluio o PDE: u i (, x) + 1 xxu i (, x) 1 xu i (, x) + F (x, ρ i ()) a i (, x) =, where a i (, x) := µ V i (, x, ρ i (), x) x v(, i x)ρ i (, x)d x. (7.6) Ideed, for ay ϕ C 1,1 (, ), by omiig he upercrip i for oaioal impliciy: d u (, x)ϕ(, x)dx d 1 = lim V ( + h, x, ρ ( + h))ϕ( + h, x) V (, x, ρ ())ϕ(, x) dx h h 1 = lim V ( + h, x, ρ ( + h)) ϕ( + h, x) ϕ(, x) dx h h 1 + lim V ( + h, x, ρ ( + h)) V ( + h, x, ρ ()) ϕ(, x)dx h h 1 + lim V ( + h, x, ρ ()) V (, x, ρ ()) ϕ(, x)dx. h h Apply Lemma 7.4 o he ecod erm ad (7.) o he hird erm, we have d u (, x)ϕ(, x)dx d = u (, x) ϕ(, x)dx µ V (, x, ρ (), x) 1 xρ (, x) + x V (, x, ρ ())ρ (, x)ϕ(, x)d xdx + 1 x u (, x) x ϕ(, x)dx + 1 xu (, x) F (x, ρ ())ϕ(, x)dx + µ V (, x, ρ (), x) 1 xρ (, x) + x u (, x)ρ (, x)ϕ(, x)d xdx = u (, x) ϕ(, x)dx + µ V (, x, ρ (), x) x v (, x)ρ (, x)ϕ(, x)d xdx + 1 x u (, x) x ϕ(, x)dx + 1 xu (, x) F (x, ρ ())ϕ(, x)dx. 34