Elements of Mathematical Oncology. Franco Flandoli

Transcription

1 Elemen of Mahemaical Oncology Franco Flandoli

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3 Conen Par. Sochaic Differenial Equaion, linear Parial Differenial Equaion and heir link 5 Chaper. Brownian moion and hea equaion 7. Inroducion 7. Simulaion of Brownian moion 8 3. Abou he problem of finding a deniy from a ample 4. Macrocopic limi of Brownian moion 3 5. On he weak convergence of meaure, in he random cae 7 6. Hea equaion a Fokker-Planck and Kolmogorov equaion 8 7. Two-dimenional imulaion 9 Chaper. SDE and PDE 3. Sochaic differenial equaion 3. Simulaion of SDE in dimenion one 5 3. The mehod of compacne for SDE 7 4. Link beween SDE and linear PDE 3 5. Simulaion of he macrocopic limi 37 Par. Ineracing yem of cell and nonlinear Parial Differenial Equaion 43 Chaper 3. Compacne reul 45. Compac e in he pace of meaure valued funcion 45. Compacne crieria in infinie dimenional funcion pace 48 Chaper 4. Mean field heory 53. Mean field model. Compacne of he empirical meaure 53. Paage o he limi Uniquene for he PDE and global limi of he empirical meaure 6 4. The men field SDE and propagaion of chao 6 Chaper 5. Local and inermediae ineracion 67. Simulaion of ineracing paricle 67. Local ineracion The macrocopic limi of paricle wih a ize The PDE aociaed wih local ineracion 73 3

4 4 COTETS 5. Inermediae ineracion: preparaion Rigorou reul on inermediae ineracion 8 Par 3. Growh and change of pecie in populaion of cell and nonlinear Parial Differenial Equaion 95 Chaper 6. Example of macrocopic yem in Mahemaical Oncology 97. An advanced model of invaive umor wih angiogenei 97. The Fiher-Kolmogorov-Perovkii-Pikunov model 5 3. A probabiliic repreenaion 4. Exience, uniquene, invarian region 4 5. Remark on he full invaive model wih angiogenei 9 6. Simulaion abou he full yem 8 7. Fick or Fokker-Planck? 9 8. Modelling he crowding-driven diffuion 3 9. Concluion 3 Chaper 7. The mahemaic of proliferaion a he microcopic level 33. Inroducion 33. The model Preliminarie on Iô formula for procee defined on random ime inerval Back o our cae Eimae on he maringale erm Eimae on h 4 Bibliography 47

5 Par Sochaic Differenial Equaion, linear Parial Differenial Equaion and heir link

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7 CHAPTER Brownian moion and hea equaion. Inroducion We recall ha, given a probabiliy pace Ω, F, P and a filraion F, namely wha we call a filered probabiliy pace Ω, F, F, P, a coninuou Brownian moion i a ochaic proce B wih he following properie: i i i a coninuou adaped proce ii B = a.. iii for every, he r.v. B B i Gauian, and i i independen of F. If coninuiy i no precribed, i can be proved ha here i a coninuou verion. A Brownian moion in R d i a ochaic proce B wih value in R d, B = B,..., B d uch ha i componen B i are independen real valued Brownian moion. We alo recall ha he hea equaion in R d, wih diffuion conan k >, i he Parial Differenial Equaion PDE u = k u, u = = u. Here u = u = u x denoe he oluion, a funcion u : [, T ] R d R we may replace [, T ] by [, and u denoe he iniial condiion, a funcion u : R d R. Thi ecion i devoed o he decripion of a few properie of hee objec and heir link. We reric he aenion o k = for impliciy of noaion and how he modificaion in he general cae a he end of he ecion... Hea kernel. Le p x be he funcion p x = π d/ exp x / defined for >. I i called he hea kernel. We have p x = p x. Indeed, p x = d/ π d/ π exp x / + π d/ exp x / x / = p x d + x 7

8 8. BROWIA MOTIO AD HEAT EQUATIO i p x = p x x i / i p x = p x x i / + p x / x = p x i. Mo of he reul of hi ecion depend on hi imple compuaion. However, le u now rear progreively from Brownian moion and inveigae i alo a a numerical level.. Simulaion of Brownian moion.. Simulaion of a rajecory of Brownian moion. Le B, be a real valued Brownian moion, on a filered probabiliy pace Ω, F, F, P. Le X be an F -meaurable random variable. Conider he ochaic proce X = X + B. I will be he oluion of he ochaic differenial equaion dx = db, X = = X when hi concep will be clarified. The following imple code, wrien wih he free ofware R, imulae a rajecory, in he piri of explici Euler cheme of dicreizaion of ochaic differenial equaion n i he number of ime ep; ju o inroduce one more R command, we ue an uniform diribued iniial condiion: n = ; d =.; h = qrd X = : n X[] = runif,, for in : n { X[ + ] = X[] + h rnorm } plox, ype = l, col = 3 line : n Le u re he role of he quaniy h = qrd: he random variable correponding o he ofware command X[ + ] X[] i X +d X, which i equal o X +d X = B +d B hence i i, by definiion of Brownian moion, a Gauian, d. A uch, i can be repreened in he form B +d B = dz where Z i,. Thi i why, in he code, X[ + ] X[] i eled equal o h rnorm. oice finally ha he independence of incremen of Brownian moion i refleced, in he code, by he fac ha rnorm compue independen value a each ieraion of he "for" cycle.

9 . SIMULATIOS OF BROWIA MOTIO 9.. Simulaion of everal independen Brownian moion. Wha we are going o do can be equivalenly decribed a he imulaion of everal Brownian rajecorie, bu in he piri of yem of many paricle we prefer o hink a we had everal independen Brownian moion and we imulae a ingle rajecory for each one of hem. Conider a equence B i, i =,,... of independen Brownian moion on a filered probabiliy pace Ω, F, F, P. Conider a equence X i, i =,,... of random iniial condiion, F -meaurable, independen and idenically diribued wih law having deniy ρ x wih repec o Lebegue meaure on R d. Conider he following imple differenial equaion having oluion dx i = db i, X i = = X i X i = X i + B i. Le u ee a picure wih a myriad of color: = 5; n = ; d =.; d =.5; h = qrd X = marixnrow =, ncol = n X[, ] = rnorm,, d for in : n { X[, + ] = X[, ] + h rnorm } ploc, n, c, for k in : { linex[k, ], col = k } Fir, noice he hape of he envelop, roughly like, correponding o he propery V ar [ B i ] =.

10 . BROWIA MOTIO AD HEAT EQUATIO We may look more cloely he diribuion of poin a ome ime. To have a beer reul, we increae he number of poin and avoid o produce he previou picure which i ime conuming: =; n=; d=.; d=.5; h=qrd X=marixnrow=,ncol=n X[,]=rnorm,,d for in :n- { X[,+]=X[,]+h*rnorm } nn= hix[,nn],3,false linedeniyx[,nn],bw=dx[,nn]/3 Her are wo picure correponding o he ime ep and : Thee picure arie he following queion: when he number of "paricle" end o infiniy, doe hi profile converge o a well defined limi profile? And could we obain he limi profile by a le ime conuming mehod, for inance by olving a uiable equaion? To ge a feeling of he pracical problem, ake =. We ee ha now he ofware, on an ordinary lapop require a non-negligible amoun of ime; and he profile i more and more regular. In a ga or fluid, he number of molecule i of he order of ; in a living iue affeced by a cancer, he number of umor cell may be of he order of 9 ; boh are incredibly larger han =. The profile hould be exremely regular, bu no affordable by a direc imulaion of he paricle yem. 3. Abou he problem of finding a deniy from a ample In he previou ecion, given he value X,..., X a ome ime, we ried o repreen graphically hee poin by mean of a probabiliy deniy, which reflec heir degree of concenraion.

11 3. ABOUT THE PROBLEM OF FIDIG A DESITY FROM A SAMPLE The hiogram i he fir eay way: he pace i pariioned in cell here inerval of equal lengh and he number of poin in each cell i couned. One can plo he hiogram giving he number of poin per cell or i normalizaion wih area one, o be compared wih a probabiliy deniy. In order o obain a more regular probabiliy deniy funcion, here are variou mehod. One of hem i paricularly conneced wih cerain heoreical inveigaion ha we are going o perform in hee lecure: i i he "kernel moohing" mehod. I ar from a "kernel", namely a probabiliy deniy K x. Then he kernel i recaled by he formula d i he paial dimenion K ɛ x := ɛ d K ɛ x and, given he poin, we have o perform he average ρ x := K ɛ x X i. i= oice ha K ɛ x dx = hence ρ x dx = i= K ɛ x X i dx = i= K ɛ x dx =. In oher word, he reuling funcion ρ x i a probabiliy deniy funcion. To implemen hi mehod one ha o chooe a kernel K cerain R command chooe a Gauian kernel by defaul, bu he reul i only mildly affeced by hi choice. Wha really make a difference i he choice of he recaling facor ɛ, alo called he "bandwidh". The general idea i ha for large ɛ we ge a raher fla and very mooh profile ρ x; for mall ɛ he profile ocillae. One ha o chooe an inermediae value of ɛ, which i he mo diffi cul problem of he implemenaion of he mehod a i i he choice of he cell in he hiogram. Wrong choice really give reul ha are no accepable, oo far from realiy. I i quie naural o expec ha he value of he bandwidh i relaed o he andard deviaion of poin. However, he imple choice ɛ =ddaa doe no give he be reul. Correcion by a facor improve he reul, bu he choice of he facor i no eay. There are ad hoc rule, quie incredible, like he one a he help page?bw.nrd or ofware R. Below we propoe o ue ddaa/5, a a rial, bu i i no alway he be. Concerning differen varian of implemenaion of he kernel moohing idea, ee he help of R under he name?kmooh,?bkde,?deniy and for inance he paper a hp://via.had.co.nz/paper/deniyeimaion.pdf. The reader i uggeed o ry he following exercie: a Gauian ample i generaed, he hiogram i ploed and, opionally, alo he rue deniy ha generaed he ample i ploed over he hiogram. Then, one can over-plo he deniy given by a mehod of kernel moohing, for differen value of he bandwidh. Here i an example of code:

12 . BROWIA MOTIO AD HEAT EQUATIO Z=rnorm,, hiz,5,false Z=orZ Y=dnormZ,, linez,y linedeniyz,bw=dz/5,col="red" and here are wo example wih wrong choice of he bandwidh we wrie only he la line: linedeniyz,bw=.,col="red", linedeniyz,bw=,col="red"

13 4. MACROSCOPIC LIMIT OF BROWIA MOTIOS 3 4. Macrocopic limi of Brownian moion Fir of all le u inroduce he o called empirical meaure S := δ X i. I i a random probabiliy meaure on Borel e, convex combinaion of random dela Dirac meaure. A he poiion X i of each ingle paricle we pu a poinwie ma of ize. By random probabiliy meaure we may looely ju mean a probabiliy meaure depending on ω Ω; more rigorouly we mu explain in which ene we mean he meaurabiliy in ω and he imple way i o ay ha φ x S dx mu be meaurable, for each φ C b R d. If we imagine hu meaure a a bunch of very mall poin mae, one one ide we may ge he feeling ha he global ma i more concenraed here han here, bu on he oher ide we do no ee any profile, imilar o a Gauian or oher. The only "aliude" we ee i, in a ene. To exrac a profile we may mollify he aomic meaure S by a convoluion wih a kernel: given a probabiliy deniy θ x, eing θ ɛ x := ɛ d θ ɛ x i= we perform he convoluion u x := θ ɛ S x := R d θ ɛ x y S dy.

14 4. BROWIA MOTIO AD HEAT EQUATIO The funcion u x i a regularized verion of S ame made above for K. We have θɛ S x = and i i a probabiliy deniy he proof i he θ ɛ x X i hence hi operaion coincide wih he kernel moohing mehod decribed above wih K = θ. Afer hee preliminarie, we ak he following queion: doe S converge, a, o a limi probabiliy meaure, maybe having a deniy ρ x? Under he previou aumpion, hi i rue. Dealing wih convergence of meaure, le u recall ha we call weak convergence he propery 4. lim φ x S dx = φ x ρ x dx R d R d i= when i hold for every coninuou bounded funcion φ. T. Under he aumpion pecified a he beginning of Secion.,for every e funcion φ C b R d we have propery 4. in he ene of almo ure convergence. Moreover, ρ x = p x y ρ y dy R d where p x i he deniy of a Brownian moion in R d, namely p x = π d/ exp x /. Finally, he funcion ρ x i mooh for > and aifie he Cauchy problem for he hea equaion ρ = ρ, ρ = = ρ he iniial condiion i aumed a a limi in L R d a P. Given, he r.v. X i are i.i.d. and hu he ame i rue for he r.v. φ X i, wih φ C b R d ; moreover, by he boundedne of φ, he r.v. φ X i have finie momen. Therefore, by he rong Law of Large umber, i= φ X i [ ] E φ X in he ene of almo ure convergence. Since X i = X i + Bi and ince he erm X i and Bi are independen, wih deniie ρ and p x repecively, hen alo X i ha deniy, given by he convoluion of ρ and p x, deniy we have denoed above by ρ x. Moreover E [ φ X ] = φ x ρ x dx R d and hu propery 4. i fully proved, along wih he convoluion formula for ρ x. Finally, i i by he compuaion of Secion. i eaily follow ha ρ x aifie he Cauchy problem for he hea equaion.

15 4. MACROSCOPIC LIMIT OF BROWIA MOTIOS 5 A he beginning of hi ecion we have inroduced alo he regularizaion θ ɛ S x. Do hey converge oo, o ρ x? Thi queion correpond more cloely o he problem aed a he end of Secion.. If we keep ɛ fixed, he anwer i a rivial conequence of he la heorem under he aumpion ha θ i alo bounded coninuou: θɛ S x = θɛ ρ x lim for every x R d. We do no ge ρ x, if we keep ɛ conan. More inereing i o link ɛ and, namely examin he limi lim θɛ S x for uiable equence ɛ. When lim θɛ S x = ρ x? Le u ee an example of reul. Aume θ Cc R d, θ, θ x dx =, o ue he mo common reul on convergence of mollifier we ue in paricular he fac ha θ ɛ f f uniformly on compac e if f i coninuou, bu he nex reul i rue in larger generaliy uing appropriae exenion. T. If hen, for every > and x R d, we have lim lim E [ θɛ S ɛ d =. x ρ x ] =. P. Fir, we have ρ x i no he average of θ ɛ x X i [ E θɛ S x ρ x ] = E θ ɛ x X i ρ x i= E [ ] θɛ x X i E θɛ x X i + i= E [ ] θ ɛ x X i ρ x i= [ [ Abou he fir erm, uing he independence beween he r.v. θ ɛ x X i and he propery E θɛ x X i E θɛ, we may delee he mixed erm and wrie E [ ] θɛ x X i E θɛ x X i = [ θɛ [ ] ] E x X i E θɛ x X i i= which, being θ ɛ x X i equally diribued, i equal o i= = E [ θ ɛ x X E [ θɛ x X ] ]..

16 6. BROWIA MOTIO AD HEAT EQUATIO For he ame reaon he econd erm i conrolled by [ E θɛ S x ρ x ] E [ θ ɛ x X E [ θɛ x X ] ] + E [ θ ɛ x X ] ρ x 4 E [ θ ɛ x X ] + 4 E [ θ ɛ x X ] + E [ θ ɛ x X ] ρ x. oice ha E [ θ ɛ x X ] = R d θ ɛ x y ρ y dy we have proved above ha X ha deniy ρ. Moreover, we have θ ɛ x y ρ y dy = ρ x R d lim uniformly in x on compac e, for >. Hence E [ ] θ ɛ x X ρ x converge o zero. The equence E [ ] θ ɛ x X = θ ɛ x y ρ y dy R d i, for he ame reaon, bounded, hence he erm 4 E [ ] θ ɛ x X [ converge o zero. I remain o deal wih he erm 4 E θ ɛ x X ]. We have hence ] E [θ ɛ x X = θɛ x y ρ y dy = ɛ d ɛ d R d R d ɛ d ρ ɛ d θ ɛ R d 4 [ E ] θ ɛ x X ɛ d θ ɛ x y ρ y dy x y dy = ɛ d ρ R d θ z dz 4 ρ θ z dz. R d Thi erm goe o zero if we aume ɛ d. ɛ R. The condiion lim d = ha a very imple inerpreaion. Fir, he law from which he paricle are exraced i almo compac uppor a any probabiliy law. To implify he argumen, aume i ha uppor of linear ize. If we have paricle, and we hin for impliciy ha hey are almo uniformly diribued in he uppor, he diance beween cloe neighbor i of he order /d. If he bandwidh ɛ i of hi order or maller, he average performed by he kernel i made only on a finie numebr of paricle, omeime maybe even zero paricle, o i flucuae randomly. If, on he conrary, ɛ i much bigger han /d, hen we average over a large number of paricle and a or of LL i acive.

17 5. O THE WEAK COVERGECE OF MEASURES, I THE RADOM CASE 7 5. On he weak convergence of meaure, in he random cae Above we have proved ha, for every φ C b R d, we have a.. lim φ x S dx = φ x ρ x dx. R d R d The even of zero probabiliy where convergence doe no hold may depend on φ. Hence, a priori, we canno ay ha, for P -a.e. ω Ω, he equence of meaure { S ω } converge weakly, ince hi propery preciely mean: here exi an even Ω Ω wih P Ω = uch ha for all ω Ω and all φ C b R d we have lim φ x S ω dx = φ x ρ x dx. R d R d Unil now, we have a e Ω φ for each φ wih a imilar propery and heir inerecion i no under conrol. The reul i rue. S C. There exi an even Ω Ω wih P Ω = uch ha, for all ω Ω, ω dx converge weakly o ρ x dx. P. The idea i o prove he aerion fir for a dene counable e of e funcion and hen exend o all e funcion by an eimae. However, le u ee he deail. Le {φ α } α be a dene equence in C c R d he e of compac uppor coninuou funcion, deniy meaured wih repec o he uniform convergence noice ha C b R d, on he conrary, would no be eparable. Since, for each α, we have 4. wih φ = φ α, being a counable numebr of properie we may ay ha here exi an even Ω Ω wih P Ω = uch ha, for all ω Ω and every α, we have 5. lim φ α x S ω dx = R d φ α x ρ x dx. R d For every φ C c R d and α we have φ x S ω dx φ x ρ x dx R d R d φ α x φ x S ω dx R d + φ α x S ω dx φ α x ρ x dx R d R d + φ α x φ x ρ x dx R d φ α φ S R ω dx + ρ x dx d R d + φ α x S ω dx φ α x ρ x dx. R d R d

18 8. BROWIA MOTIO AD HEAT EQUATIO Recall ha R S d ω dx =, R ρ d x dx =. Given φ C c R d, givern ε >, le α be uch ha φ α φ ε. Then φ x S ω dx φ x ρ x dx R d R d ε + φ α x S ω dx φ α x ρ x dx. R d R d Therefore, recalling 5., for every ω Ω we have lim up φ x S ω dx φ x ρ x dx ε. R d R d Since ε > i arbirary and he limup doe no depend upon ε, we deduce ha he limup i equal o zero. Finally, recall ha, if a equence of probabiliy meaure converge o a probabiliy meaure hi i eenial, and here i i rue over all e funcion of C c R d, he i converge weakly namely over all e funcion of C b R d. The proof i complee. R. A imilar reul i rue if we replace a.. convergence wih convergence in probabiliy. 6. Hea equaion a Fokker-Planck and Kolmogorov equaion The hea equaion i relaed o Bownian moion in hree way: a macrocopic limi he heorem above, a a Fokker-Planck equaion; a a Kolmogorov equaion. Le u ee here he la wo inerpreaion. 6.. Hea equaion a Fokker-Planck equaion. We hall ee below in Secion 4. he general meaning of Fokker-Planck equaion. The hea equaion i a paricular cae of i. In Secion 4. we deal wih meaure-valued oluion. Here, due o he impliciy of he paricular cae, we may deal wih regular oluion of he Fokker-Planck equaion. Conider he equaion dx = db, X = = X where X ha deniy ρ x. The reul i: he deniy ρ x of X i oluion of he Cauchy problem for he hea equaion ρ = ρ, ρ = = ρ. We have already proved hi reul in Theorem. Thi ecion doe no preen a new reul bu only ini on he link beween SDE and PDE which ae under appropriae aumpion ha he law of he oluion o an SDE aifie in a uiable ene a PDE. 6.. Probabiliic repreenaion formula. Finally, le u ee ha, for he hea equaion u = u, u = = u one ha he following probabiliic repreenaion formula, in erm of a Brownian moion B : u x = E [u x + B ].

19 7. TWO-DIMESIOAL SIMULATIOS 9 Indeed, E [u x + B ] = = u x + y p y dy p x y u y dy and we know ha hi expreion give a oluion of he hea equaion. Thi link i a paricular cae of he link beween SDE and Kolmogorov equaion, udied below in Secion Two-dimenional imulaion We complee he fir Chaper by a few imulaion of Brownian moion in dimenion, which will come back in ubequen chaper. 7.. Brownian moion in D. The nex code i a way o imulae a rajecory of a D Brownian moion: n=; d=.; h=qrd X=:n; Y=:n X[]=; Y[]= for in :n- { X[+]=X[]+h*rnorm Y[+]=Y[]+h*rnorm } plox,y,ype="l", col=3; ablineh=; abline, The following wo varian how a movie, in wo differen form: n=; d=.; h=qrd X=:n; Y=:n

20 . BROWIA MOTIO AD HEAT EQUATIO X[]=; Y[]= ploc-,,c-, ablineh= abline, for in :n- { X[+]=X[]+h*rnorm Y[+]=Y[]+h*rnorm linex[:],y[:],ype="l", col=3 }... n=; d=.; h=qrd X=:n; Y=:n X[]=; Y[]= for in :n- { X[+]=X[]+h*rnorm Y[+]=Y[]+h*rnorm ploc-,,c-, ablineh= abline, linex[:],y[:],ype="l", col=3 } 7.. Several Brownian moion in D. Wih everal Brownian moion, we canno plo anymore he full rajecore, he picure would be oo full. We may, for inance, plo he final poiion: n=; =; d=.; h=qrd X=marixnrow=,ncol=n; Y=X X[,]=rnorm,,; Y[,]=rnorm,, for in :n- { X[,+]=X[,]+h*rnorm Y[,+]=Y[,]+h*rnorm } ploc-,,c-,; linex[,n],y[,n],ype="p", col= ablineh=; abline,

21 7. TWO-DIMESIOAL SIMULATIOS Thi i a imple example of paricle yem. Wih a grea degree of abracion. we could hink of i a a e of cancer cell, embedded ino a iue or in viro. We would like o ee he moion of he paricle. We ue a number of rick: i we plo only a few ime, by mean of he command T=5, if%%t==, ec.; ii we clean he previou poin wih he command polygonc-,,,-,c-,-,,,col="whie", border=a. Moreover, he mehod omeime work poorly, i depend on he value of T compared o he complexiy of he pecific code. The example below ha been uned o work. n=; =; d=.; h=qrd X=marixnrow=,ncol=n; Y=X X[,]=rnorm,,; Y[,]=rnorm,, T=5 ploc-,,c-,,ype="n" for in :n- { X[,+]=X[,]+h*rnorm Y[,+]=Y[,]+h*rnorm if%%t== { polygonc-,,,-,c-,-,,,col="whie", border=a linex[,+],y[,+],ype="p", col= ablineh= abline, } }

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23 CHAPTER SDE and PDE. Sochaic differenial equaion.. Definiion. We call ochaic differenial equaion SDE an equaion of he form. dx = b, X d + σ, X db, X = = X where B i a d-dimenional Brownian moion on a filered probabiliy pace Ω, F, F, P, X i F -meaurable, b : [, T ] R d R d and σ : [, T ] R d R d d have ome regulariy pecified cae by cae and he oluion X i a d-dimenional coninuou adaped proce. The meaning of he equaion i he ideniy. X = X + b, X d + σ, X db where we have o aume condiion on b and σ which guaranee ha b, X i inegrable, σ, X i quare inegrable, wih probabiliy one; σ, X db i an Iô inegral, and more preciely i i i coninuou verion in ; and he ideniy ha o hold uniformly in, wih probabiliy one. The generalizaion o differen dimenion of B and X i obviou; we ake he ame dimenion o have le noaion. Even if le would be uffi cien wih more argumen, le u aume ha b and σ are a lea coninuou, o ha he above menioned condiion of inegrabiliy of b, X and σ, X are fulfilled. In mo cae, if X = x i deerminiic, when we prove ha a oluion exi, we can alo prove ha i i adaped no only o he filraion F bu alo o F B, he filraion aociaed o he Brownian moion; more preciely, i compleion. Thi i ju naural, becaue he inpu of he equaion i only he Brownian moion. However, i i o naural if implicily we hink o have a uiable uniquene. Oherwie, in principle, i i diffi cul o exclude ha one can conruc, maybe in ome arificial way, a oluion which i no B-adaped. Indeed i happen ha here are relevan example of ochaic equaion where oluion exi which are no B-adaped. Thi i he origin of he following definiion. D rong oluion. We have rong exience for equaion. if, given any filered probabiliy pace Ω, F, F, P wih a Brownian moion B, given any deerminiic iniial condiion X = x, here i a coninuou F -adaped proce X aifying. in paricular, we may chooe F = F B and have a oluion adaped o B. A rong oluion i a oluion adaped o F B. 3

24 4. SDES AD PDES D weak oluion. Given a deerminiic iniial condiion X = x, a weak oluion i he family compoed of a filered probabiliy pace Ω, F, F, P wih a Brownian moion B and a coninuou F -adaped proce X aifying.. In he definiion of weak oluion, he filered probabiliy pace and he Brownian moion are no pecified a priori, hey are par of he oluion; hence we are no allowed o chooe F = F B. When X i random, F -meaurable, he concep of weak oluion i formally in rouble becaue he pace where X ha o be defined i no precribed a priori. The concep of rong oluion can be adaped for inance replacing F B wih F B F, or ju aying ha, if X i a oluion on a precribed pace Ω, F, F, P where X and B are defined, hen i i a rong oluion. If we wan o adap he definiion of weak oluion o he cae of random iniial condiion, we have o precribe only he law of X and pu in he oluion he exience of X wih he given law. Le u come o uniquene. Similarly o exience, here are wo concep. D 3 pahwie uniquene. We ay ha pahwie uniquene hold for equaion. if, given any filered probabiliy pace Ω, F, F, P wih a Brownian moion B, given any deerminiic iniial condiion X = x, if X and X are wo coninuou F -adaped proce which fulfill., hen hey are indiinguihable. D 4 uniquene in law. We ay ha here i uniquene in law for equaion. if, given wo weak oluion on any pair of pace, heir law coincide... Srong oluion. The mo claical heorem abou rong oluion and pahwie uniquene hold, a in he deerminiic cae, under Lipchiz aumpion on he coeffi cien. Aume here are wo conan L and C uch ha b, x b, x L x x σ, x σ, x L x x b, x C + x σ, x C + x for all value of and x. The econd condiion on b and σ i wrien here for ake of generaliy, bu i we aume, a aid above, ha b and σ are coninuou, i follow from he fir condiion he uniform in ime Lipchiz propery. T 3. Under he previou aumpion on b and σ, here i rong exience and pahwie uniquene for equaion.. If, for ome p, E [ X p ] <, hen [ E ] up X p <. [,T ] P....

25 . SIMULATIO OF SDE I DIMESIO OE 5.3. Weak oluion. Le u ee only a paricular example of reul abou weak oluion. Aume ha σ i conan and non-degenerae; for impliciy, aume i equal o he ideniy, namely conider he SDE wih addiive noie dx = b, X d + db. Moreover, aume b only meaurable and bounded or coninuou and bounded if we prefer o mainain he general aumpion of coninuiy. The key feaure of hee aumpion are: he noie i nondegenerae hence more rericive han above for rong oluion bu b i very weak, much weaker han he uual Lipchiz cae. Under uch aumpion on b, if we do no have he noie db in he equaion, i i eay o make example wihou exience or wihou uniquene. T 4. Under hee aumpion, for every x R d, here exi a weak oluion and i i unique in law.. Simulaion of SDE in dimenion one.. Linear example. Conider he equaion, wih λ >, dx = λx d + σdb, X = x. I Euler dicreizaion, on inerval of conan ampliude, ha he heoreical form X n+ X n = λx n d + σ d B n+ B n d d := n+ n where he r.v. Z n = B n+ B n d are andard Gauian and independen. The algorihmic form can be he following one; fir we conruc a funcion, he drif, hen we wrie he main par of he code: drif=funcionx -x n=; d=.; h=qrd; ig= X=:n; X[]= for in :n- {X[+]=X[]+ d*drifx[] + h*ig*rnorm} plox,ype="l", col=; ablineh=; abline,

26 6. SDES AD PDES E. Try wih oher iniial condiion and oher value of λ and σ... onlinear example. Conider he equaion called "wo-well-poenial" dx = X X 3 d + σdb, X = x. I Euler dicreizaion i X n+ X n = X n X 3 n d + σ dzn d := n+ n, Z n = B n+ B n d. The code i: drif=funcionx x-x^3 n=; d=.; h=qrd; ig=.5 X=:n; X[]= for in :n- {X[+]=X[]+ d*drifx[] + h*ig*rnorm} plox,ype="l", col=4; ablineh=; abline,

27 3. THE METHOD OF COMPACTESS FOR SDES 7 E. Try wih oher iniial condiion and oher value of σ and n..3. Imporan exercie. In boh cae of he wo example above, plo an iogram and a fied non-parameric coninuou deniy of he diribuion a ime for inance for =,, 5,. 3. The mehod of compacne for SDE 3.. Compacne in C [, T ] ; R d. Recall ha a e i called relaively compac if i cloure i compac. Every ube of a compac e i relaively compac. Recall claical Acoli-Arzelà heorem: a family of funcion F C [, T ] ; R d i relaively compac in he uniform opology if i for every [, T ], he e {f ; f F } i bounded ii for every ε > here i δ > uch ha f f ε for every f F and every, [, T ] wih δ. Recall he definiion of he Hölder eminorm, for f : [, T ] R d, f f [f] C α = up and obviouly of he upremum norm f = up [,T ] f. Simple uffi cien condiion for i and ii are i here i M > uch ha f M for all f F ii for ome α,, here i R > uch ha [f] C α R for all f F.

28 8. SDES AD PDES Hence he e K M,R = { f C [, T ] ; R d } ; f M, [f] C α R are relaively compac in C [, T ] ; R d. The Sobolev pace W α,p, T ; R d, wih α, and p >, i defined a he e of all f L p, T ; R d uch ha T T f f p [f] W α,p := +αp dd <. We endow W α,p, T ; R d wih he norm f W α,p = f L p + [f] W α,p. I i known ha and W α,p, T ; R d C ε [, T ] ; R d if α ε p > [f] C ε C ε,α,p f W α,p. Uing hee new pace, imple uffi cien condiion for i and ii are i and ii for ome α, and p > wih αp >, here i R > uch ha [f] W α,p R for all f F. Indeed, if ii hold, here exi ε > uch ha α ε p >, hence uch ha [f] C ε C ε,α,p f W α,p; moreover, f L p T /p f T /p M and herefore [f] C ε C ε,α,p f W α,p C ε,α,p T /p M + R for all f F, which implie he validiy of ii. Therefore he e { K M,R = f C [, T ] ; R d } ; f M, [f] W α,p R are relaively compac in C [, T ] ; R d, if αp >. 3.. Applicaion o SDE. Conider he SDE dx = b, X d + σ, X db, X = = x wih bounded coninuou coeffi cien. We may dream of a generalizaion of Peano heorem, namely ju exience of a oluion. Le b n, σ n be a equence of coninuou funcion, each one uniformly Lipchiz in x wih conan ha may depend on n: equibounded: b n, x b n, y L n x y σ n, x σ n, y L n x y b n + σ n C here C > i independen of n and uch ha b n b, σ n σ, uniformly on compac e of [, T ] R d. Le {X n } be he oluion of he equaion Le Q n be heir law on C [, T ] ; R d. dx n = b n, X n d + σ n, X n db, X n = = x.

29 3. THE METHOD OF COMPACTESS FOR SDES 9 L. The family {Q n } i igh in C [, T ] ; R d. P. Sep. For pedagogical reaon, we ar wih a parially inuffi cien proof, o clarify he role of cerain argumen. Recalling he relaively compac e K M,R above, i i uffi cien o prove ha, given ɛ >, here are M, R > uch ha 3. P X n K c M,R < ɛ for all n. Condiion 3. mean A uffi cien condiion i and Sep. The fir one i eay: which implie P X n > M or [X n ] C α > R < ɛ. P X n > M < ɛ/ and P [X n ] C α > R < ɛ/. P X n > M M E [ T up X n x + b n, X n d + up [,T ] C + up σ n, X n db E [ up X n [,T ] [,T ] ] C + E C + E [ [ up [,T ] up [,T ] up X n [,T ] [,T ] ] σ n, X n db ] [ T C + E σ n, X n d C. up σ n, X n db σ n, X n db ] / Hence, given ɛ >, we may chooe M > uch ha P X n > M < ɛ/. Sep 3. The econd one, X n X n P > R < ɛ/ however i more diffi cul ince i involve a double upremum in ime and maringale inequaliie do no help. A way o prove hi propery i o ue a quaniaive Kolmogorov crierion; anoher i a varian of hee ighne argumen baed on opping ime, developed by Aldou. ] /

30 3. SDES AD PDES However we have een above anoher cla of compac e baed on Sobolev pace. The advanage of W α,p, T ; R d wih repec o C ε [, T ] ; R d i ha he opology i enirely defined by inegral, which merge wih expecaion beer han a upremum. Le u ue hem in he nex ep. Sep 4. Recalling now he relaively compac e K M,R above, uing he ame argumen a in ep and he reul of ep, we are lef o prove ha here exi α, and p > wih αp >, wih he following propery: given ɛ >, here i R > uch ha for every n. We have ow, for, P [X n ] W α,p > R < ɛ/ P [X n ] W α,p > R R E [ T = C R T T T X n X n p +αp dd ] E [ X n X n p ] +αp dd. X n X n = b n r, Xr n dr + σ n r, Xr n db r p X n X n p C b n r, Xr n p dr + C σ n r, Xr n db r C p p + C σ n r, Xr n db r and, by he o called Burkholder-Davi-Gundy inequaliy, [ p] [ E σ n r, Xr n db r CE Therefore The inegral T T P [X n ] C ε > R C R = C R C p/. T T T T p/ ] σ n r, Xr n dr p/ +αp dd +α p dd. dd i finie if α < + α ; we need o ue p >, becaue of he conrain p αp >. Thu we may find R > uch ha he previou expreion i maller han ɛ/, a required. R 3. By he ame compuaion, one can how ha B W α,p, T ; R d a.., for every α < and p >, hence B Cε [, T ] ; R d a.., for every ε <, a we already know. I alo provide a quaniaive conrol on P [B] C ε > R, uually no aed when Kolmogorov regulariy heorem i given.

31 3. THE METHOD OF COMPACTESS FOR SDES 3 R 4. The fracional Sobolev opology above provide one of he imple proof of Kolmogorov regulariy heorem; obviouly one ha o accep he Sobolev embedding heorem, which in a ene incorporae argumen imilar o hoe of dyadic pariion in he claical proof of Kolmogorov heorem. R 5. From he previou proof we may deduce he following well-known crierium ee he book of Billingley: if i hold and here are p, β, C > uch ha E [ X n X n p ] C +β for all n, hen he equence {Q n } i igh in C [, T ] ; R d. We have proved ha {Q n } i igh in C [, T ] ; R d. Hence here are ubequence which converge weakly. Le u ake one of hem and, ju for impliciy of noaion, le u denoe i by {Q n }. Thu we are auming ha {Q n } converge weakly o ome probabiliy meaure Q on Borel e of C [, T ] ; R d. Our aim i o prove he exience of a oluion of he SDE. Aume for a econd ha he previou fac imply he exience of a coninuou proce X uch ha X n converge o X a.., in he uniform opology. Thi aerion i obviouly fale: convergence in law doe no imply a.. convergence. However, i i rue in a more involved way, uing a Skorohod repreenaion heorem. The deail are no rivial, for inance becaue he new procee X n, on a new probabiliy pace, given by uch heorem, do no aify he SDE, a priori; one can prove ha hey aify i in a uiable weak ene. So, le u mi hee deail, alo becaue hey will no ener he dicuion in he cae of he macrocopic limi, our final inere. And aume alhough no rue ha he original equence X n converge o X a.., in he uniform opology on compac e. I follow ha b n r, X n r dr σ n r, X n r db r b r, X r dr σ r, X r db r in probabiliy, by he P -a.. uniform convergence of b n r, X n r rep. σ n r, X n r o b r, X r rep. σ r, X r and he equiboundedne of b n r, X n r rep. σ n r, X n r. I follow ha X aifie he SDE The zero-noie example. Concerning hi la iue, he convergence, here i an inereing finie dimenional example which anicipae wha happen in he cae of macrocopic limi. I i he cae of he equence of equaion dx ε = b, X ε d + εdb, X = = x when b i bounded coninuou. Under hi aumpion here i exience and uniquene in law, bu alo in he rong ene, for every ε >. We claim ha he family {Q ε } of he law of {X ε } i igh in C [, T ] ; R d and each limi meaure Q of he family ha he following propery: Q C x =

32 3. SDES AD PDES where C x C [, T ] ; R d i he e of all oluion of he deerminiic equaion dx = b, X, X = = x. d The proof of ighne of {Q ε } i idenical o he one given above, we leave i a an exercie. Le {Q εn } be a weakly converging ubequence and Q be i limi. Conider he funcional Ψ : C [, T ] ; R d R defined a Ψ f = up f x b, f d. [,T ] I ha he propery ha Ψ f = if and only if f C x. The funcional Ψ i coninuou on C := C [, T ] ; R d, wih he uniform opology here we ue ha b i coninuou. Recall ha by Pormaneau heorem one ha Q A lim inf Q εn A for every open e A C. Hence, for every δ >, Q f C : Ψ f > δ lim inf Q εn f C : Ψ f > δ. If we prove ha hi lim inf i zero, hen Q f C : Ψ f > δ = for every ɛ >, hence Q f C : Ψ f = =, which prove Q C x = we leave a an exercie o prove ha C x i a cloed e, hence Borel. We have Q εn f C : Ψ f > δ = P Ψ X εn > δ = P up x = P = P Xεn [,T ] up [,T ] ε n B > δ b, X εn up B > δ/ε n ε n [,T ] δ E 4. Link beween SDE and linear PDE [ d > δ up B [,T ] 4.. Fokker-Planck equaion. Along wih he ochaic equaion. defined by he coeffi - cien b and σ, we conider alo he following parabolic PDE on [, T ] R d : p 4. = i j a ij p div pb, p = = p called Fokker-Planck equaion. Here a = σσ T. Alhough in many cae i ha regular oluion, in order o minimize he heory i i convenien o inroduce he concep of meaure-valued oluion µ ; moreover we reric o he cae of probabiliy meaure. To give a meaning o cerain inegral below, we aume beide oher aumpion depending on he reul b, σ bounded coninuou ].

33 4. LIKS BETWEE SDES AD LIEAR PDES 33 bu i will be clear ha more cumberome reul can be done by lile addiional concepual effor. We looely wrie µ 4. = d i j a ij µ div µ b, µ = = µ i,j= bu we mean he following concep. By µ, φ we mean R φ x µ d dx. By C b [, T ] R d we denoe he pace of bounded coninuou funcion ϕ : [, T ] R d R and by C, b [, T ] R d he pace of funcion ϕ uch ha ϕ, φ, iφ, i j φ C b [, T ] R d. D 5. A meaure-valued oluion of he Fokker-Planck equaion 4. i a family of Borel probabiliy meaure µ [,T ] on R d uch ha µ, ϕ,. i meaurable for all ϕ C b [, T ] R d and µ, φ, µ, φ, = µ, φ + d a ij i j φ + b φ, d for every φ C, b [, T ] R d. T 5 exience. The law µ of X i a meaure-valued oluion of he he Fokker-Planck equaion 4.. P. Le φ be of cla C, b dφ, X = φ, X d + φ, X dx + i,j= [, T ] R d. By Iô formula for φ, X, we have d i j φ, X a ij, X d i,j= = φ, X d + φ, X b, X d + φ, X σ, X db + d i j φ, X a ij, X d. We have E T φ, X σ, X d < we ue here ha σ and φ are bounded, hence E φ, X σ, X dw = and hu all erm are finie by he boundedne aumpion E [φ, X ] E [φ, X ] = E + i,j= φ, X d + E φ, X b, X d d E i j φ, X a ij, X d. i,j= Since E [φ, X ] = R φ, x µ d dx and imilarly for he oher erm we ge he weak formulaion of equaion 4.. The preliminary propery ha µ, ϕ,. = E [ϕ, X ] i meaurable for all ϕ C b [, T ] R d i eay. R 6. Under uiable aumpion, like he imple cae when a ij i he ideniy marix, if µ ha a deniy p hen alo µ ha a deniy p,, ofen wih ome regulariy gained by he parabolic rucure, and hu he Fokker-Planck equaion in he diff erenial form 4. hold.

34 34. SDES AD PDES T 6 uniquene. Aume ha he backward parabolic equaion called backward Kolmogorov equaion u + d a ij i j u + b u = i,j= u =T = ψ on [, T ] R d ha, for every T [, T ] and ψ Cc R d, a lea one oluion u of cla C, b [, T ] R d. Then he Fokker-Planck equaion 4. ha a mo one meaure-valued oluion. P. If µ i a meaure-valued oluion of he Fokker-Planck equaion and u i a C, b [, T ] R d oluion of he Kolmogorov equaion, hen from he ideniy which define meaure-valued oluion and he ideniy of Kolmogorov equaion µ T, ψ µ, u,. Then, if µ i, i =,, are wo meaure-valued oluion of he Fokker-Planck equaion wih he ame iniial condiion µ, we have µ T, ψ = µ T, ψ. Thi ideniy hold for every ψ Cc R d, hence µ T = µ T. The ime T [, T ] i arbirary, hence µ = µ. Obviouly he weak apec of he previou uniquene reul i he aumpion, no explici in erm of he coeffi cien. The reaon i ha here are wo main cae when uch implici aumpion i aified. One i he cae when b and σ are very regular; he oher i when σ i non-degenerae. A an example of he econd cae, le u menion he fundamenal cae of hea equaion. E. Conider he cae b =, σ = Id. The forward Kolmogorov equaion v = v = = ψ d i v i,= on [, T ] R d ha he explici oluion v, x = p x y ψ y dy which i infiniely differeniable wih all bounded derivaive. The funcion u, x = v T, x i hen an explici and regular oluion of he backward Kolmogorov equaion above. In hi cae, herefore, he Fokker-Planck equaion ha a unique meaure-valued oluion.

35 4. LIKS BETWEE SDES AD LIEAR PDES Backward Kolmogorov equaion. Along wih he ochaic equaion. defined by he coeffi cien b and σ, we conider alo he following backward parabolic PDE on [, T ] R d, called backward Kolmogorov equaion: u + d a ij i j u + b u =, u =T = ψ. i,j= To expre in full generaliy he relaion wih he SDE we have o inroduce he SDE on he ime inerval [, T ], wih any [, T ]: X = x + b, X d + σ, X db, [, T ]. Obviouly, on [, T ], we have he ame reul a on [, T ], in paricular rong exience and pahwie uniquene under Lipchiz aumpion. Aume hee condiion and denoe he unique oluion, defined on ome filered probabiliy pace, by X,x. The relaion wih he backward Kolmogorov equaion i [ ] u, x = E ψ. Thi relaion hold under differen aumpion and for oluion u wih differen degree of regulariy. Le u ar wih he mo elemenary reul. P. If u i a oluion of he backward Kolmogorov equaion of cla C, [ ] [, T ] R d, wih bounded u and σ, hen u, x = E ψ X,x T. P. Given [, T ], we apply Iô formula o u, X,x on [, T ]. The compuaion i he ame done in he proof of Theorem 5. Since u olve he backward Kolmogorov equaion, we ge T ψ T, X,x T = u, x + u, X,x σ, X,x db where we have ued he ideniy X,x = x. Uing he boundedne aumpion, u, X,x σ, X,x i of cla M, hence [ T ] E u, X,x σ, X,x db =. [ ] The relaion u, x = E ψ follow. X,x T R 7. Le u re he dualiy of he previou problem. The law µ of X ac on e funcion ψ by ψdµ which i equal o E [ψ X ]; hence here i a form of dualiy beween [ he law ] µ and he expreion E [ψ X ]. The PDE aified by he law µ and he PDE aified by E ψ are in dualiy, a rigorouly claimed by Theorem 6. I i a general fac of linear problem ha exience for a problem give uniquene for he dual one: hink o he fac ha full range of a marix implie kernel zero for he ranpoe marix; hence Theorem 6 i naural. X,x T X,x T

36 36. SDES AD PDES 4.3. Macrocopic limi. We may reformulae Theorem 5 a a macrocopic limi of a yem of non-ineracing paricle. Le B i, i, be a equence of independen Brownian moion in R d, defined on a filered probabiliy pace Ω, F, F, P. Le X i be independen Rd -r.v., F -meaurable, wih he ame law µ. Le b, σ a in Theorem 5; more preciely, le u aume hey are bounded coninuou and aify he Lipchiz condiion of he claical heorem of exience and uniquene one may ak le, inc ehere we only need uniquene in law. Conider he equence of SDE in R d dx i = b, X i d + σ, X i db i, X i = = X i. One can prove ha he procee X i have he ame law; denoe he marginal a ime of X i by µ ; we already know ha µ i a meaure-valued oluion of he Fokker-Planck equaion. The procee X i are independen, ince each X i i adaped o he correponding Brownian moion B i and iniial condiion X i, which are independen. One can provide rigorou proof of hee fac, which however are exremely inuiive. Conider, for each, he empirical meaure S := i= δ X i. Conider alo mollifier θ ɛ x = ɛ d θ ɛ x, wih θ Cc R d, θ, θ x dx =. T 7. For every [, T ], a.. we have weak convergence of S o he oluion µ of he PDE 4.. Moreover, if µ ha a deniy ρ, wih ρ C b R d for ome [, T ], hen for every x R d we have lim θɛ S x = ρ x in mean quare L ɛ Ω-limi, a oon a lim d =. P. The proof of he fir claim i he ame done above for he Brownian moion: given φ C b R d, he r.v. φ X i are bounded i.i.d., hence by he rong Law of Large umber we have, a.., lim S, φ = E [ φ X ] = µ, φ. Then one can find a full probabiliy even Ω Ω where he ame convergence hold for every φ C b R d, by he argumen of Corollary. The proof of he econd claim i alo he ame a he one of Theorem, where he exience of he deniy ρ i ued, along wih i coninuiy and finiene of ρ. For he equel of hee lecure i i very imporan o exrac he following meage from hi reul. If a family of cell X i are ubjec o a drif b, X i which move hem in a cerain direcion, a he PDE level he deniy ρ x of hoe cell will be ubjec o he ranpor erm div ρb. And converely, if we read uch a erm in he PDE for a deniy, he inerpreaion i a drif which ac on he paricle of ha deniy. The mo common cae of vecor field b i a gradien field of a poenial U b = U in which cae we ay ha paricle move along he gradien of U. The ranpor erm ha he form div ρ U ha we hall mee everal ime below in cancer model.

37 5. SIMULATIOS OF THE MACROSCOPIC LIMIT 37 Similarly, bu hi i in a ene more obviou, if he cell X i are ubjec alo o a random moion decribed by a Brownian moion db, i hen he deniy i ubjec o he diffuion erm ρ. Thi relaion i generalized o he cae of a non homogeneou, non ioropic random moion of he form σ, X i db i and he correponding diffuion erm ij i j a ij ρ in he PDE; however, he homogeneou ioropic cae db i i he rule, in abence of pecial phenomena. Finally, 5. Simulaion of he macrocopic limi 5.. Simulaion of imple PDE. Fir of all, le u ee a imple code o imulae he hea equaion ρ = k ρ ρ = = ρ on [, T ] Rd in dimenion d =. The mehod i he finie difference one, wih explici Euler in ime. There i a famou and eenial rule o know: he ime ep d and pace ep dx mu be relaed by he condiion omeime called Von eumann abiliy condiion k d dx. A he boundary, we have impoed no-flux condiion. T= ; x=5; dx=.; K=; T= L=dx*x; d=dx^/4*k x.vir=x+ u = marixnrow=x.vir, ncol=t X=eq,L,dx u[,]=dnormx,l/,5 M=maxdnormX,L/,5 ploc,l,c-.,.,ype="n" for in :T- { for i in :x { u[i,+] = u[i,] + K * d * u[i+,]-*u[i,]+u[i-,] / dx^ } u[,+]=u[,+]; u[x.vir,+]=u[x,+] if%%t== { polygonc,l,l,,c-.,-.,.,.,col="whie", border=a linex,u[,+] ablineh=; ablineh=m } }

38 38. SDES AD PDES Running he imulaion for a hor while, he effec of he boundary ar o appear and i i no o nice. However, if we e equal o zero a he boundary inead of he no-flux condiion, lowly he ma diappear, which i even wore in a ene. We ugge o change d=dx^/4*k ino d=dx^/*k and d=dx^/*k o ee he role of he abiliy condiion. ow we add a drif o he hea equaion, in he Fokker-Plank form ρ = k ρ div ρb ρ = = ρ on [, T ] Rd and we chooe a drif b which concenrae ma around a poin: b x = C x x e C x x. T= ; x=5; dx=.; K=; T= L=dx*x; d=dx^/4*k x.vir=x+ u = marixnrow=x.vir, ncol=t X=eq,L,dx b = -X-L/-*exp-.*abX-L/-^ u[,]=dnormx,l/,5 M=maxdnormX,L/,5 ploc,l,c-.,.,ype="n" for in :T- { for i in :x { u[i,+] = u[i,] + K * d * u[i+,]-*u[i,]+u[i-,] / dx^ - d*b[i+]*u[i+,]-b[i]*u[i,]/dx }

39 5. SIMULATIOS OF THE MACROSCOPIC LIMIT 39 u[,+]=u[,+]; u[x.vir,+]=u[x,+] if%%t== { polygonc,l,l,,c-.,-.,.,.,col="whie", border=a linex,u[,+] ablineh=; ablineh=m } } If we wan an inerpreaion, i can be he cae of cell which move from heir original poiion and gaher around a blood veel. 5.. Paricle imulaion and comparion wih he PDE. We fir run he previou imulaion of he hea equaion black line ogeher wih he imulaion of Brownian paricle aring wih he ame iniial deniy a he hea equaion; we repreen he profile of paricle by mean of he "deniy" R-command. T= ; x=5; dx=.; K=/; T=; = L=dx*x; d=dx^/4*k; h=qrd x.vir=x+ u = marixnrow=x.vir, ncol=t; XMB=marixnrow=,ncol=T X=eq,L,dx u[,]=dnormx,l/,5 XMB[,]=rnorm,L/,5 M=maxdnormX,L/,5 ploc,l,c-.,.,ype="n" for in :T- { XMB[,+]=XMB[,]+h*rnorm for i in :x { u[i,+] = u[i,] + K * d * u[i+,]-*u[i,]+u[i-,] / dx^ }

40 4. SDES AD PDES u[,+]=u[,+]; u[x.vir,+]=u[x,+] if%%t== { polygonc,l,l,,c-.,-.,.,.,col="whie", border=a linex,u[,+] linedeniyxmb[,+],bw=dxmb[,+]/5,col= ablineh=; ablineh=m } } Up o minor difference, he evoluion of he profile i he ame: Le u now dicu he econd example above. The deniy i hrinked in a mall region, hence he variaion in number of poin per pace become larger. If one rie wih eh ame parameer above, he reul i mode. The following parameer give on he conrary quie good reul: T= ; x=5; dx=.; K=/; T=; = L=dx*x; d=dx^/4*k; h=qrd drif=funcionx -x-l/-*exp-.*abx-l/-^ x.vir=x+ u = marixnrow=x.vir, ncol=t; XMB=marixnrow=,ncol=T X=eq,L,dx u[,]=dnormx,l/,5 XMB[,]=rnorm,L/,5 M=maxdnormX,L/,5 ploc,l,c-.,.,ype="n" for in :T- { XMB[,+]=XMB[,]+d*drifXMB[,]+h*rnorm for i in :x { u[i,+] = u[i,] + K * d * u[i+,]-*u[i,]+u[i-,] / dx^ - d*drifx[i+]*u[i+,]- drifx[i]*u[i,]/dx } u[,+]=u[,+]; u[x.vir,+]=u[x,+] if%%t== { polygonc,l,l,,c-.,-.,.,.,col="whie", border=a linex,u[,+]

41 5. SIMULATIOS OF THE MACROSCOPIC LIMIT 4 linedeniyxmb[,+],bw=dxmb[,+]/,col= ablineh=; ablineh=m } } oice he change in bw=dxmb[,+]/ wih repec o bw=dxmb[,+]/5, neceary o appreciae maller cale rucure. Thi of coure caue wider flucuaion in he profile unle he number of paricle i very high.

42

43 Par Ineracing yem of cell and nonlinear Parial Differenial Equaion

44

45 CHAPTER 3 Compacne reul. Compac e in he pace of meaure valued funcion On Pr R d, he e of all probabiliy meaure on Borel e of R d, we may inroduce everal meric. There i a general kind of meric given by δ µ, ν = i µ, φ i ν, φ i + µ, φ i ν, φ i i= or δ µ, ν = i= i µ, φ i ν, φ i where {φ i } i a uiable equence of bounded coninuou funcion. Anoher one, bu on Pr R d, he e of probabiliy meaure µ wih finie fir momen R x µ dx <, i he Waerein meric d { } W µ, ν = up µ, φ ν, φ : [φ] Lip φ φ where [φ] Lip i he Lipchiz eminorm [φ] Lip = up in fac hi i no he uual definiion bu a heorem, named Kanorovich-Rubinein characerizaion. In boh cae we have a complee eparable meric pace and he convergence i equivalen o he weak convergence of probabiliy meaure. A a conequence of he general verion of Acoli-Arzelà heorem in meric pace, we have he following fac. P. Le E = Pr R d wih he meric d = δ above or E = Pr R d wih he Waerein meric d = W. A family of meaure-valued funcion F C [, T ] ; E i relaively compac in C [, T ] ; E if i for every [, T ], for every ɛ >, here exi r ɛ, > uch ha µ B, r ɛ, > ɛ for every µ F ii for every ɛ >, here exi δ > uch ha d µ, µ < ɛ for every µ F and, [, T ] uch ha < δ. To prove he econd condiion in example, in he cae of he Waerein meric, we may ue he following fac. L. If, for ome α, and p wih αp >, and a conan C > one ha T T W µ, µ p +αp dd C 45

46 46 3. COMPACTESS RESULTS for every µ F, hen ii hold, wih E = Pr R d. P. Clearly, ii hold if here exi θ, C > uch ha up W µ,µ θ C, for every µ F. Chooe θ > uch ha α θ p >. Then, for ome conan C >, and herefore I follow ha µ, φ µ, φ C θ [ µ, φ ] W α,p { } W µ, µ C θ up [ µ, φ ] W α,p : [φ] Lip. W µ, µ T up θ C up [φ] L ip C = C T T T T T µ, φ µ, φ p +αp dd up [φ]l ip µ, φ µ, φ p +αp dd W µ, µ p +αp dd. To prove he econd condiion in example, for he meric δ, i i ueful o have he following crierion. L 3. Aume ha, for every φ C b R d, one ha he following propery: for every ɛ >, here exi δ > uch ha µ, φ µ, φ < ɛ for every µ F and, [, T ] uch ha < δ. Then condiion ii above hold, wih E = R d. In paricular, a uffi cien condiion i ha here exi α, wih he following propery: for every φ C b R d here i a conan C φ > uch ha [ µ, φ ] C α C φ for every µ F. Anoher uffi cien condiion i ha here exi α, and p wih αp > wih he following propery: for every φ C b R d here i a conan C φ > uch ha for every µ F. hence [ µ, φ ] W α,p C φ P. Given ɛ >, le k be uch ha k ɛ/. oice ha i µ, φ i µ, φ i + µ, φ i µ, φ i i k ɛ/ i=k+ d µ, µ k i= i=k+ i µ, φ i µ, φ i + µ, φ i µ, φ i + ɛ/.

47 . COMPACT SETS I THE SPACE OF MEASURE VALUED FUCTIOS 47 Since k i finie, here exi δ > uch ha µ, φ i µ, φ i < ɛ/ for every µ F and, [, T ] uch ha < δ and i =,..., k. Then d µ, µ ɛ k i + ɛ < ɛ for every µ F and, [, T ] uch ha < δ. Thi i propery ii. i= R 8. The previou reul i omewha in he ame piri a Aubin-Lion lemma: he regulariy in ime neceary for ighne i uffi cien when he pace regulariy i meaured in a very weak ene. Compacne in ime and in pace are deal wih eparaely. L 4. Le [, T ] be given. If here exi a conan C > uch ha R d x µ dx C for every µ F, hen condiion i i fulfilled, for ha value of. P. µ B, r c r hence, given ɛ >, we may chooe r > C ɛ. x µ dx C R d r R 9. We may replace R d x µ dx in he hypohei of he lemma by R d g x µ dx wih any increaing funcion g uch ha lim r + g r = +. C. Le α, and p wih αp >, be given. For any M, R > he e K M,R C [, T ] ; Pr R d defined a { T } T W µ, µ p K M,R = µ : up x µ dx M, [,T ] R d +αp dd R i relaively compac in C [, T ] ; Pr R d. C 3. Le α, and p wih αp >, and g be an increaing funcion uch ha lim r + g r = +. For every M > and every funcion φ C φ from C b R d o,, he e K M,C C [, T ] ; Pr R d defined a K M,C = { µ : up [,T ] i relaively compac in C [, T ] ; Pr R d. R d g x µ dx M, [ µ, φ ] W α,p C φ for every φ C b R d}

48 48 3. COMPACTESS RESULTS. Compacne crieria in infinie dimenional funcion pace.. Claical Aubin-Lion lemma. If E, d i a meric pace, here i a naural exenion o he pace C [, T ] ; E: a family of funcion F C [, T ] ; E i relaively compac if i for every [, T ], he e {f ; f F } i relaively compac ii for every ε > here i δ > uch ha d f, f ε for every f F and every, [, T ] wih δ. There are generalizaion o L p -pace. One i named Kolmogorov-Riez, no dicued here. We menion he following one. L 5 Aubin-Lion. Le E E E be hree Banach pace, wih compac embedding E E bounded e of E are relaively compac in E and coninuou embedding E E. Le p, q. Then L p, T ; E W,q, T ; E i compacly embedded ino L p, T ; E. Thi heorem i very powerful for applicaion o PDE... An example. Le D be a bounded regular domain in R d ; conider he PDE u = u + b u u + c u u = = u, u D =. where b, c : L D R are coninuou, b bounded and c wih linear growh. We wan o prove he exience of a oluion of cla u C [, T ] ; L D L, T ; W, D. Le u develop he o called a priori eimae: if a oluion u exi and i uffi cienly regular, hen d u, x dx = u u + b u u + c u dx. d Uing alo he boundary condiion, one ha u udx = u, x dx u b u u + c u dx C b u u dx + C c u + u dx u dx + C b u dx + C c + u dx + C c D and hu we have u, x dx + u, x dxd = u x dx + C u, x dxd.

49 . COMPACTESS CRITERIA I IFIITE DIMESIOAL FUCTIO SPACES 49 I follow, by Gronwall lemma and hen again by he ame inequaliy, T u n, x dx + u n, x dxd C up [,T ] wih a conan C > independen of n. Thi prove in paricular ha he equence {u n } i bounded in L, T ; W, D, which i one half of he aumpion of Aubin-Lion lemma. The econd half i omewha eaier: from he differenial equaion and he fac ha he linear operaor i bounded from W, D o he negaive order Sobolev pace W,, i follow ha { } u n i bounded in L, T ; W,. The we apply Aubin-Lion lemma wih p, q =, E = W, D, E = L D, E = W, and ue in eenial way he compacne of he embedding W, D L D Rellich heorem. We deduce ha {u n } i relaively compac in L, T ; L D in he rong opology hi i he imporan informaion in order o pa o he limi in nonlinear erm. The equence i alo weakly relaively compac in L, T ; W, D and weak ar relaively compac in C [, T ] ; L D. Then here exi a ubequence which converge in all hee hree opologie o a funcion u which i hen of cla L, T ; W, D and C [, T ] ; L D. Thank o he rong convergence in L, T ; L D one can pa o he limi and prove ha u i a oluion hi ep depend on he equence of approximaing problem and hu we have o mi he deail..3. Fracional verion of Aubin-Lion lemma and a zero-noie example. Aubin-Lion lemma i no uiable for ochaic problem ince procee uually are no differeniable. A we have een in he cae of SDE, a crierion baed on fracional Sobolev pace i more uiable. The definiion of L 6. Le E E E be hree Banach pace, E, E reflexive, wih compac embedding E E and coninuou embedding E E. Le p, q >, α,. Then i compacly embedded ino L p, T ; E W α,q, T ; E L p, T ; E. Le u ee he expeced cheme of i applicabiliy o he following zero-noie problem: in a bounded regular domain D R d we conider he SPDE du ε = u ε + b u ε u ε + c u ε d + εdb u ε = = u, u ε D =. wih b and c a in he example. above. Here, for ake of impliciy, aume ha B i a onedimenional Brownian moion, bu hi iue can be generalized a lo. The rigorou analyi of uch a problem require everal deail which are no imporan for our main aim of paricle yem and would deeriorae he underanding of he main poin we wan o emphaize. Thu he following dicuion i inenionally approximae from he viewpoin of rigor, including wha we call "proof" below. Fir, le u accep he following claim, for horne of expoiion: for every ε, here exi a lea one oluion u ε wih pah of cla Y = C [, T ] ; L D L, T ; W, D. Conider he

50 5 3. COMPACTESS RESULTS law Q ε of u ε, a a probabiliy meaure on Borel ube of Y. Denoe by C u he e of all oluion of cla Y of he deerminiic PDE of example.. We wan o prove ha he family {Q ε } i igh in L, T ; L D and all i limi poin Q are uppored on C u. L 7. {Q ε } i igh in L, T ; L D. P. We wan o apply he fracional Aubin-Lion lemma wih p = q =, E = W, D, E = L D, E = W, a in he deerminiic cae. Given δ > we have o find wo conan M, R > uch ha T. P u ε, x + u ε, x dxd > M < δ T T. P u ε W, d + for ome α,. By Iô formula Hence T u ε u ε W, +α > R d u ε, x = u ε u ε + b u ε u ε + c u ε d = + u ε, x dx + u x dx + + u ε εdb + ε d. u ε dxεdb + ε D u ε, x dxd u ε b u ε u ε + c u ε dxd a an example, le u menion ha concerning rigor, he reul of hi formula i correc bu he proof require addiional argumen, for inance becaue we have reaed u ε, x a an Iô proce, a fac which i rue only when u ε i a real valued proce, while here i i only a diribuion. A above in he deerminiic cae u x dx + u dxd + C u dxd + C + u ε dxεdb which implie u ε, x dx + u ε, x dxd + u x dx + C u ε dxεdb. < δ u dxd + C

51 . COMPACTESS CRITERIA I IFIITE DIMESIOAL FUCTIO SPACES 5 If we know ha u ε dx i of cla M hi depend on he properie of u ε ha have been proved abou i exience, an iue ha we have mied, we ake expecaion and prove [ ] [ ] E u ε, x dx + E u ε, x dxd u x dx + C E u dx d + C [ ] which provide fir a uniform-in-ε bound on E u ε, x dx by Gronwall lemma, [.3 up [,T ] E hen he bound.4 E ] u ε, x dx C u ε, x dxd C by he inequaliy ielf. If know only ha u ε dx i of cla Λ, we have o apply a opping ime of he form { } τ = inf > : u ε, x dx > o ha he proce u ε τ, x dx i now of cla M ; hen we apply he ame compuaion an remove τ wih Faou lemma a he end, in he final inequaliie.3-.4, by aking he limi a. Thi i he fir half of our effor, becaue, pliing. in wo par for impliciy of expoiion, we have T P u ε, x dxd > M M T E u ε, x dxd C M T and hu for M large hi probabiliy i arbirarily mall. The proof for P u ε, x dxd > M i imilar, uing.3. Concerning he econd half, namely., he par concerning T uε W, d follow from he previou compuaion becaue u ε W, C uε L. For he oher par we have T T u ε u ε W P, +α > R T T R E u ε u ε W, +α ] = T T E [ u ε u ε W, R +α ] o we have o eimae E [ u ε u ε W,. From he equaion aified by u ε we have u ε u ε = u ε + b u ε u ε + c u ε dr + ε B B

52 5 3. COMPACTESS RESULTS u ε u ε W, u ε + b u ε u ε + c u ε W, dr + ε B B / / u ε + b u ε u ε + c u ε W, dr + Cε B B u ε u ε W, u ε + b u ε u ε + c u ε W, dr + Cε B B ] E [ u ε u ε W, T I i no diffi cul o prove ha ] E [ u ε + b u ε u ε + c u ε W, hence, uing.4, we ge ] E [ u ε + b u ε u ε + c u ε W, dr + Cε. ] E [ u ε u ε W, C ] CE [ u ε W, + C and herefore ] T T E [ u ε u ε W, R +α C R if we chooe α < /. Thu we may chooe R large o make he probabiliy above mall.

53 CHAPTER 4 Mean field heory. Mean field model. Compacne of he empirical meaure Conider he ineracing paricle yem dx i, = K j= X i, X j, d + σdb i wih i =,...,, K : R d R d bounded Lipchiz, σ >. We aume o have a filered probabiliy pace Ω, F, F, P and independen Brownian moion B i in R d. inerac wih every oher paricle X j, uch ha he diance i in he uppor of K. Even if K i compac uppor, in he limi when we mu R. Each paricle X i, X i, X j, expec ha he number of paricle X j, ineracing wih X i, goe o infiniy ince, a we hall ee, he empirical meaure will converge o a probabiliy meaure, o paricle remain relaively concenraed. For hi reaon, hi model i alo called "long range". I i no ricly correc if we wan o decribe local ineracion, like membrane ineracion beween cell. On he iniial [ condiion ] X i,, we aume hey are F -meaurable and: X i, a up i, E < b here i µ Pr R d uch ha S, φ µ, φ in probabiliy, for every φ C b R d. Thi i rue in paricular when X i, = X i, where { X i } i i a equence of i.i.d. F -meaurable r.v. wih law µ Pr R d. Denoe by S he empirical meaure and by Q he law of S on C [, T ] ; Pr R d. T 8. Under aumpion a, { Q } i relaively compac on C [, T ] ; Pr R d. P. Sep. Given ɛ >, we have o find a compac e K ɛ E uch ha Q K ɛ > ɛ for every. We look for a e of he form K M,R a decribed in Corollary. We have Q KM,R c = P S KM,R c T T P x S W S dx > M + P, S p R d +αp dd > R. up [,T ] 53

54 54 4. MEA FIELD THEORY We eparaely prove ha boh probabiliie are maller han ɛ/, for every. In boh cae we apply Chebihev inequaliy. We have x S dx = X i, R d hence P up [,T ] R d x S dx > M [ ] X i, where he propery E up [,T ] T T W S P, S p +αp dd > R Thu we have o prove ha i= M E [ up [,T ] x S R d [ = M E up [,T ] i= [ = E up M i= X i, X i, [,T ] dx ] ] ] C M C will be checked below in Sep. Similarly, we have T T R E W S, S p +αp dd = T T E [ W S, S p ] R +αp dd. [ E W S, S p ] C +β for ome p and β >. We hall prove hi below in Sep 3. If o, chooe α > o mall ha αp < β and ge T T W S P, S p +αp dd > R C R. ow, given ɛ > above, we may find M, R > uch ha Q K M,R c < ɛ. Sep. We imply have X i, X i + X i j= K X i, + K T + σ B i hence, recalling ha E [ X i ] = x µ dx <, [ ] [ E up X i, C + σe [,T ] X j, d + σ B i up [,T ] B ] C.

55 . MEA FIELD MODEL. COMPACTESS OF THE EMPIRICAL MEASURE 55 Sep3. In order o eimae W S, S we fir eimae S, φ S, φ wih Lipφ. We have S, φ S, φ = φ X i, φ X i, i= φ i= X i, i= X i, X i, φ X i,. Hence W S, S X i, i= X i,. Therefore reaing i= a an inegraion w.r.. a probabiliy meaure on he naural number,...,, hence uing Hölder inequaliy W S, S p X i, i= X i, p [ E W S, S [ X i, Thu we have o eimae E [ X, E X, X i, p ] i= [ X i, E X i, p ]. p ]. The yem i reverible, hence we could reduce o p ], bu i i uneenial, ju concepually inereing a paradigm i ha he quaniaive analyi of paricle i = i uffi cien. From he SDE we have X i, X i, = j= K X i, r Xr j, dr + σ B i B i hence X i, X i, K Xr i, j= K + σ B i B i Xr j, dr + σ B i B i

56 56 4. MEA FIELD THEORY which implie [ X i, E X i, p ] [ C p + CE B i B i p] C p + C p/ C p/ renaming each ime he conan. Chooing p > and eing β = p >, he proof i now complee becaue we have proved [ E W S, S p ] C +β. I may be noiced ha, from he proof of hi heorem and he reul of he previou ecion, we may exrac a general crierion of compacne of a family of empirical meaure S, independenly of he precie equaion aified by he paricle: P 3. If, for ome p, β, C > [ E x S R d up [,T ] [ Q W S, S dx ] C p ] C +β for every, hen he famyly of law Q of S i relaively compac in Pr C [, T ] ; Pr R d. The fac ha Q are law of S i no eenial. A general crierion i: P 4. Se C = C [, T ] ; Pr R d. Aume ha G Pr C i a family of probabiliy meaure on C. If, for ome p, β, C > x µ dx dq µ C R d C up [,T ] for every Q G, hen G i relaively compac. C W µ, µ p dq µ C +β Here he noaion µ and for he generic elemen of C [, T ] ; Pr R d.. Paage o he limi Le u now examine he limi poin of { Q }, he family of law of he empirical proce. The aim i o prove ha, in he limi, we ge oluion of he nonlinear Fokker-Planck equaion. µ = σ µ div µ K µ

57 . PASSAGE TO THE LIMIT 57 wih iniial condiion µ. However, hee oluion are, a lea a priori, only probabiliy meaure, depending on ime and a priori alo on randomne. Thu we ue he concep of meaure-valued oluion, compleely imilar o he one given in a previou ecion in he linear cae. R. Le u re he non-linear and non-local characer of hi equaion: i i quadraic, due o he produc µ K µ, and i i non-local, due o he convoluion K µ. R. Recall ha imple quadraic equaion like u = u do no have global oluion, for cerain iniial condiion. Hence he exience of global oluion, included in he heorem of paage o he limi, i a non-rivial reul; he a-priori bound on oluion given by he fac ha hey are meaure of ma one, conribue. R 3. on-localiy i more eviden in he cae when µ ha a deniy u, which hen aifie in weak ene u = σ u div u K u. The change of u x a poin x involve he value of K u x in fac of value in a neighbor, due o he differenial operaion div, which i baed on u y for all y R d or a lea in he ranlaion a x of he uppor of K. Le u fir give he inuiion abou he reul. Preliminarily, noice ha K x X j, j= = K x y S dy =: K S x hence he SDE of he ineracing paricle can be rewrien a dx i, = K S X i, d + σdb. i Le φ x be a e funcion of call Cb. [, T ] R d. By Iô formula we have dφ X i, = φ + φ X i, X i, d + φ X i, σdb i + σ φ K S X i, d. X i, d

58 58 4. MEA FIELD THEORY Therefore, being S, φ = i= φ where = Therefore we have proved: d S, φ = X i, i=, φ φ i= φ i= i= X i, X i, X i, σ φ S, φ d + S, φ K S M φ, = i= d K S σdb i X i, L 8. The empirical meaure S d S, φ = S, φ d + S, φ K S d d + dm φ, φ X i, σdb i. aifie he ideniy X i, d + σ d + dm φ, S, φ d + σ S, φ d. R 4. We canno wrie M φ, in erm of S, a lea direcly. However, hi i rue for he quadraic variaion: [ M φ,, M φ, ] = σ i= φ X i, σ d = S, φ d. By a repreenaion heorem for ochaic inegral, here exi an auxiliary Brownian moion β uch ha M φ, = σ S, φ dβ. In hi omewha arificial way we may conider he ideniy aified by S a a cloed equaion. From he ideniy of he lemma, if we aume rue for a econd ha S weakly converge o a limi meaure µ, we ee a lea inuiively ha µ aifie he nonlinear Fokker-Planck equaion.. To be rigorou, however, we have o deal wih he weak convergence of Q, he law of S.

59 . PASSAGE TO THE LIMIT 59 T 9. If Q i a weak limi poin of a ubequence of { Q }, hen Q i uppored on he meaure-valued oluion of he nonlinear Fokker-Planck equaion wih iniial condiion µ. P. For every φ C. b d S, φ = S, φ Ψ φ µ := up [,T ] µ = σ µ div µ K µ [, T ] R d, conider he funcional d + S, φ K S µ, φ µ, φ d + dm φ, + σ S, φ d. µ, φ + σ φ + φ K µ d. I i coninuou on C [, T ] ; Pr R d. Hence, if Q k i a ubequence which weakly converge o Q, by Pormaneau heorem we have Q Ψ φ µ > δ lim inf k Q k Ψ φ µ > δ. Bu Q k Ψ φ µ > δ = P Ψ φ S k > δ = P up S k, φ µ, φ S k, φ + σ φ + φ K S k d > δ. [,T ] From he ideniy of Lemma 9 we ge Q k Ψ φ µ > δ = P P P up [,T ] S k µ, φ + up S k M φ, k [,T ] S k µ, φ > δ/ + P µ, φ > δ up M φ, k [,T ] M φ, k > δ > δ/. The fir erm goe o zero, a k, by he aumpion on S. For he econd erm we have [ ] P up M φ, k > δ/ 4δ E up M φ, k [,T ] [,T ] CE [ M φ, k T ] becaue M φ, k i a maringale. ow, i i well known ha [ ] E φ X i, k db i φ X i, k db j =

60 6 4. MEA FIELD THEORY if i j becaue B i and B j are independen. Hence ] [ = σ k E E [ M φ, k T k i= By he iomery formula for Iô inegral, hi i equal o which i bounded by = σ k k i= ] φ X i, k db i. φ X i, k d C σ k φ T a k. Therefore Q Ψ φ µ > δ =. Since hi hold rue for every δ >, we deduce Q Ψ φ µ = =. Hence, for every φ Cb. [, T ] R d, Q i uppored on he e of meaure-valued funcion µ uch ha µ, φ = µ, φ + µ, φ + σ φ + φ K µ d. Thi implie ha Q i uppored on he e of meaure-valued oluion of he nonlinear Fokker-Planck equaion, by an argumen of counable deniy of funcion. 3. Uniquene for he PDE and global limi of he empirical meaure T. The nonlinear., wih iniial condiion µ Pr R d, ha one and only one meaure-valued oluion. { P. Exience ha been proved above: he uppor of any limi meaure Q of he family Q } i non empy and i i made of uch oluion. The proof of uniquene i poponed. C 4. The family { Q } of law of he empirical procee S weakly converge o δ µ where µ C [, T ] ; Pr R d i he unique oluion of equaion.. P. From he compacne of { Q } we know ha from every ubequence we may exrac a furher ubequence which converge o ome limi meaure Q, uppored on meaure-valued oluion of.. By he previou uniquene heorem, Q = δ µ where µ C [, T ] ; Pr R d i he unique oluion of equaion.. Since hi hold rue for all limi poin Q, and he weak convergence in Pr C [, T ] ; Pr R d i a meric convergence, we deduce ha he full equence { Q } converge, o δ µ.

61 4. THE ME FIELD SDE AD PROPAGATIO OF CHAOS 6 4. The men field SDE and propagaion of chao Anoher approach following e.g. [4] o mean field heory i poenially le general han he compacne one bu i give more informaion. I ar wih a preliminary analyi of he limi SDE 4. dx = b X d + σdw 4. b x = K x y µ dy 4.3 µ = Law of X wih X a given F -meaurable r.v.; aume for impliciy E [ X ] <. One can prove ha: T. There i a unique rong oluion X, µ of problem P. We give only he cheme of he proof he deail are no diffi cul. Denoe by Pr R d he pace of probabiliy meaure on R d wih finie fir momen, endowed wih he -Waerein meric W. Conider he pace C [, T ] ; Pr R d of coninuou familie µ [,T ], wih µ Pr R d, endowed wih he meric d µ, ν = up W µ, ν. [,T ] Fir, one conider he map Γ in C [, T ] ; Pr R d, T o be choen, defined a Γµ = Law Y µ dy µ = b µ Y µ d + σdw, Y µ = X b µ x = K x y µ dy. I i a conracion for mall T. Le u how only hi compuaion. By definiion/characerizaion of -Waerein meric, we have By he equaion, becaue Y µ Y ν W Γµ, Γν E [ Y µ L K b µ Y µ b ν Y ν d b µ Y µ b µ Y ν d + Y µ Y ν ]. b µ Y ν b ν Y ν d Y ν d + L K W µ, ν d b µ x bµ x K x y K x y µ dy L K x x b µ x bν x = L K L K K x y µ ν dy L KW µ, ν

62 6 4. MEA FIELD THEORY by he oher definiion/characerizaion of -Waerein meric. Thu, by Gronwall lemma, We conclude Y µ Y ν e L K L K W µ, ν d. T W Γµ, Γν e L K L K W µ, ν d. Then Γ i a conracion in C [, T ] ; Pr R d for T mall enough, namely uch ha e L K L K T <. Thi give u local exience and uniquene of µ, fixed poin of Γ, from which one deduce exience and uniquene of a rong oluion on [, T ] by a claical reul on SDE and idenificaion of µ a he law of hi oluion, due o he definiion of Γ. The argumen can hen be repeaed on inerval of equal lengh, becaue he choice of T depend only on L K. D 6. A meaure-valued oluion µ [,T ] of equaion?? wih iniial condiion µ Pr R d i a family in C [, T ] ; Pr R d uch ha?? hold for every φ Cc R d. T. Given µ Pr R d, here exi a unique meaure-valued oluion µ [,T ] of equaion?? wih iniial condiion µ Pr R d. P. We give only he cheme of he proof. Exience readily follow from he previou heorem: i i uffi cien o apply he reul on Fokker-Planck equaion given in Chaper. The proof of uniquene i more involved. Fir hi i he more echnical ep one can prove uniquene of meaure-valued oluion µ [,T ] of he linear Fokker-Planck equaion µ = σ µ div µ b wih iniial condiion µ Pr R d. The definiion i σ µ, φ µ, φ = b µ, φ d + µ, φ d for all φ Cc R d. The aumpion on b here i ha i i a given uniformly-in-ime Lipchiz coninuou vecor field b : [, T ] R d R d. One proof i by dualiy: one fir prove ha, given an arbirary pair T [, T ] and g Cc R d, he backward Kolmogorov equaion on [, T ] u + σ u + b u = u T = g ha a uffi cienly regular oluion o apply he rule of calculu required in he nex argumen. Second, one exend he definiion of meaure-valued oluion from every φ Cc R d o every ime-dependen e funcion u in he call of regulariy of oluion of he previou Kolmogorov equaion. A hi poin we know ha µ, u µ, u = µ, u σ d + b µ, u d + µ, u d

63 4. THE ME FIELD SDE AD PROPAGATIO OF CHAOS 63 for every T. By he equaion aified by u we ge µ, u = µ, u and in paricular µ, g = µ, u. Since g Cc R d i arbirary, hi idenifie µ. Hence he linear equaion ha a unique oluion. ow, le ν be a meaure-valued oluion of equaion?? wih iniial condiion µ Pr R d. I i oluion of ν = σ ν div ν b where b = K ν. I i unique by he argumen ju explained; bu alo he marginal of he law ν of a proce Y olving dy = b Y d + σdb i a oluion. Hence Y, ν i anoher oluion of problem , hu i i equal o X, µ ; namely ν i equal o µ. ow, for every i, conider dx i = v X i, d + σdw i v x, = K x y µ dy µ = Law of X i. The nex heorem i he key ep in he proof of he macrocopic limi bu i i alo a very inereing reul in ielf: i decribe he aympoic in dynamic of each paricle X i. I immediaely ha an inereing corollary, called propagaion of chao. T 3. We have P. We have X i X i = = + + E [ ] up X i X i [,T ] C. K X i X j d j j j K v X i, d K X i X j K X i X j d j K X i X j K X i X j d X i X j v X i, d

64 64 4. MEA FIELD THEORY hence By exchangeabiliy [ ] [ ] X i E X i X i L K E X i d + L K + E K [ ] [ ] X i E X i X i L K E X i d + which implie, by Gronwall lemma, [ ] X i E X i e L KT j X i X j v E K j j E K j [ ] X j E X j d X i, d. X i X j v X i X j v X i, d. X i, d Thi i a grea eimae: i conrol he difference beween he oluion of a coupled yem and he oluion of he independen one, by an expreion which depend only on he independen one. We have E K j X i X j v X i, = E E K j x X j v x, x=x i hence E K j x X j E K v x, E K j V ar [K x X ] j X i X j v [ ] X i E X i x X j X i, e L KT CT. v x, / C C / Before we complee he main program abou he convergence of S, we may deduce from he previou heorem he o called propery of propagaion of chao: he independence of iniial condiion propagae o an approximae independence of poiion a any ime, in he limi a. And,

65 4. THE ME FIELD SDE AD PROPAGATIO OF CHAOS 65 he previou heorem alo ell u, in he limi each independen paricle aifie an SDE he equaion for X i in ineracion wih a mean field he deniy of X i ielf, no more wih ingle paricle. C 5. For every given k and every ϕ,..., ϕ k C b R d Lip R d, we have [ k lim E ] k k ϕ i X i = E [ϕ i X ] = ϕ i, µ. i= P. I i uffi cien o explain he proof for k =. We have i= i= E [ ] ϕ X ϕ X E [ϕ X ] E [ϕ X ] ] = E [ϕ X ϕ X ϕ X ϕ X and in abolue value i i bounded above by [ ϕ ] E X ϕ X ϕ X ϕ X [ ϕ ] [ ϕ E X ϕ X ϕ ] + ϕ E X ϕ X [ ] X ϕ Lip ϕ Lip E X which converge o zero a. R 5. The proof alo how ha 4.4 E [ ] ϕ X ϕ X E [ϕ X ] E [ϕ X ] C ϕ Lip ϕ Lip /. We complee he ecion wih he following reul, which olve he problem of he macrocopic limi, ince we already know ha µ i he oluion of he PDE: T 4. If µ denoe he law of X, we have [ E S, φ µ, φ ] C for all φ C b R d Lip R d, for a uiable conan C >. P. We give only an idea of proof. One ha E [ S, φ µ, φ ] = E [ S, φ ] + µ, φ E [ S, φ ] µ, φ. By exchangeabiliy, [ S E, φ ] = [ E φ X i ] φ X j = [ E φ X ] + E [ φ X ] φ X ij = [φ E X ] + E [ φ X ] φ X µ, φ = E [φ X ] E [ S, φ ] µ, φ = E [ φ X i ] E [φ X ] = E [ φ X ] E [φ X ]. i i i j

66 66 4. MEA FIELD THEORY Hence [ S E, φ ] = [ E φ X ] + E [ φ X ] φ X + E [φ X ] E [ φ X ] E [φ X ] = [φ E X ] E [ φ X ] φ X + E [ φ X ] φ X E [φ X ] + E [φ X ] E [ φ X ] E [φ X ]. The erm in he fir line i bounded by φ /. The erm in he econd line i bounded by C φ Lip / ee he remark above and imilarly he erm in he hird line.

67 CHAPTER 5 Local and inermediae ineracion. Simulaion of ineracing paricle We imulae he yem of ineracing paricle under he ineracion kernel K α,r x = g α,r x x x where R > i a cu-off parameer which allow u o appreciae he difference beween long or hor range ineracion and α i a poibly mulidimenional ineniy parameer which again help o ee differen cae. The funcion g α,r will be choen wih ome crierion. The kernel K α,r, applied o he difference beween paricle poiion K α,r X i, X j, X i, = g α,r X j, X i, X i, X j, X j, ha he direcion of he line beween he paricle, hence produce a diplacemen of X i, in hi X i, direcion; and i ha ineniy g α,r X j,. We may diinguih beween repulive and aracing kernel: if g α,r x > i i repulive, becaue he vecor X i, X j, move X i, g α,r x < away from X j,. If on he conrary i i aracing. Uually, if we hink o a repulive mechanim, we wan ha i i ronger when he paricle are cloer. Hence we ake g α,r poiive and decreaing. A imple example, which ake ino accoun he range-parameer R, i g α,r x = α x R x R wih α >. To imulae uch yem we may inroduce a few funcion norma=funcionx,y qrx^+y^ H=funcionr ignr+/ g=funcionr,r,alp alp*h-r/r*-r/r 67

68 68 5. LOCAL AD ITERMEDIATE ITERACTIOS and run he code, for differen value of α and R: =; n=5; d=.; ig=; h=qrd; T=; L=5; L = alp=; R= X=marix,,n; Y=X; X[,]=rnorm,,L; Y[,]=rnorm,,L ploc-l,l, c-l,l, ype="n" for in :n-{ for i in :{ DX= X[i,]-X[,]; DY= Y[i,]-Y[,] Kx=gnormaDX,DY,R,alp*DX/normaDX,DY+. Ky=gnormaDX,DY,R,alp*DY/normaDX,DY+. X[i,+]=X[i,] + d*meankx + h*ig*rnorm Y[i,+]=Y[i,] + d* meanky + h*ig*rnorm } if%%t== {polygonc-l,l,l,-l,c-l,-l,l,l, col="whie", border=a Cir=:*+.5 ymbolx[,+],y[,+], circle = Cir, add=true,inche=false } } Here for inance are wo ample for alp=; R=, a hor diance of ime: I i inereing alo o run he code for negaive value of α... Dependence of g α,r on R. In he example above, here i a funcion g α r uch ha g α,r r = g α r/r. In he example i i g α r = α r r. We hall alway aume uch kind of dependence below, alo in he heoreical argumen.

69 3. THE MACROSCOPIC LIMIT OF PARTICLES WITH A SIZE 69. Local ineracion Aume now we wan o imulae a family of cancer cell which are ubjec only o he following mechanim: a random moion and he volume conrain. By he laer we mean ha wo cell canno overlap hey have o occupy differen porion of pace. In realiy, cell have a very high degree of deformaion, hence if we idealize hem a ball we have o accep ome degree of overlapping of he ball, bu no oo much. A imple way o include in our dynamic he volume conrain i o inroduce a repulive kernel K α,r which ha a very hor range R, of he order of cell ize. In he imulaion above, ince we have aken cell of radiu.5 recall Cir=:*+.5, and K α,r ac on he cener X i,, X j, of he cell, i mu ac only when he cener have a diance beween and when he diance beween he cener i larger han, he cell do no ouch and hu no volume conrain applie. Thu we have o ake R =. Moreover, ince he volume conrain i really mandaory, we chooe α very large. Here are wo picure of he imulaion when alp=; R=: 3. The macrocopic limi of paricle wih a ize 3.. Recaling he radiu of cell. Le u now examine he macrocopic limi. If we idealize paricle a poin, he concep of macrocopic limi i he one dicued unil now in he lecure. However, when we aume ha paricle have a ize, ome addiional argumen apply. Aume he paricle are cell. Idealize hem a ball of radiu R cell. In doing he macrocopic limi we increae he number of cell. Compare hen he cae of and cell. Do we ake he ame radiu or do we recale he radiu? Thi dilemma correpond preciely o wo differen viewpoin, ha we now explain. We may decide ha we do no recale he radiu: he radiu i alway R cell, ay /, independenly of he increaing number of cell. Thi viion correpond o an oberver han live a he level of cell, ha ee he cell a "large" individual. A mall umor ay cell occupie hen a mall porion of pace, a larger umor ay cell occupie a porion of cell which i much larger ay ime he previou one. Thi i he microcopic viion. To make a comparion wih differen kind of mahemaical model he o called cellular auomaa, hi viion i capured very well by conidering he laice Z d and aume ha each node of he laice can be eiher empy or occupied by a cell.

70 7 5. LOCAL AD ITERMEDIATE ITERACTIOS The macrocopic viion, on he conrary, correpond o our viion of a iue, where he preence of, or 9 cell ju correpond o a differen degree of moohne, granulariy of he objec we oberve. The cell occupy a porion D of he macrocopic pace independen of he number of cell; we ee hem more or le a a coninuum, wih a differen degree of moohne depending on heir number. If we adop hi viion, he radiu R cell mu be recaled depending on he number of cell, oherwie cell hould necearily overlap. The correc rule come from he limi cae of cell of maximal deniy and uniformly diribued: a cube [, ] d i compleely filled by cubic cell occupying d [ he pace ai, a i + ] n. The cube [, ] d i covered by n d of uch mall cube. So, if he number of i= cell i, heir diameer mu be a number n uch ha nd =. We deduce n = /d, n = /d. Summarizing, he radiu of a cell mu be of he order R cell /d. Thi rule implie ha, a he macrocopic level, we hould expec a deniy funcion ρ, x which doe no exceed he hrehold, up o excepional even which hould rapidly relax o. 3.. Conac ineracion when he radiu i recaled. Aume now we conider ineracing cell in he cae when he ineracion i of conac ype, like he volume conrain or oher which occur a he membrane of he cell, like adheion, ha we hall dicu laer on. The radiu of ineracion i herefore of he order of he radiu or diameer of he cell. Thu we ake R = /d in he definiion of he funcion g α,r r. Moreover, auming he form g α,r r = g α r/r, we reach he following fac: he ineracion kernel, for conac ineracion and for he purpoe of a macrocopic decripion, hould be of he form x Kα x g α /d x x Thi precripion however i no compleely correc or, more preciely, in he ymbol we could ill have a facor depending on. Le u ee why. When we have wrien above ha he ineracion kernel i K α,r x, we had in mind he mean field model dx i, = j= K α,r X i, X j, d + σdb. i amely, we had in mind o average he conribuion of all paricle X j,. Such viion i naural when inerac wih all paricle, or a lea wih a non-negligible proporion. On he conrary, in he X i, cae of conac ineracion, he cell X i, inerac only wih a very mall and finie number of cell, hoe in conac wih ielf. If we average heir conribuion by mean of he facor, in he limi a he ineracion become evanecen and only he noie remain. Therefore we have o impoe

71 a differen caling: dx i, 3. THE MACROSCOPIC LIMIT OF PARTICLES WITH A SIZE 7 = α j= g /d X i, X j, X i, X i, X j, X j, d + σdb. i Here we have dropped he dependence of g from an ineniy parameer and collec he ineniy wih i dependence on he recaling in he prefacor α. Which α? Ju o confirm he feeling, in he previou model paricle X i, inerac only wih very few paricle X j,, hoe a diance of he order /d from X i, ; he huge um j= reduce o he um of very few erm. The ineniy of each ingle erm i however α no The ineniy facor α : ime recaling. The choice of α can be made in wo differen way. One, no "hone", i o conjecure he limi, macrocopic, PDE and ee which α inuiively i required o converge. The oher i o inveigae more cloely he microcopic picure. The facor α i a or of arefac due o he deire of a macrocopic decripion. If we go back o he microcopic viion, he one where he paricle have a fixed radiu, he ineracion kernel hould no depend on he number of paricle. In ha viion paricle are decribed by heir poiion Y i, ubjec o he dynamic 3. dy i, = j= Here, for horne of noaion, we have e b Y i, b y = j= Y j, d + σdw i. g y y y wih g having a relaively mall uppor, if we have in mind conac ineracion; ay uppor of diameer, and paricle have alo diameer. The iniial condiion Y i, canno be independen of, oherwie we ar wih a huge number of paricle one over he oher. To have ome preadne of paricle alo a ime = we ake, for inance, Y i, wih Gauian law of andard deviaion which i proporional o he radiu of he occupied pace equal o /d o he occupied volume i of he order. We expec ha, during he dynamic, he paricle remain a a diance of ha order from he origin. ow le u recale he poiion Y i, in uch a way ha we ee all cell in a region of diameer of order. Thu we inroduce /d Y i,. However, if we do o, he equaion aified by he new poiion /d Y i, conain he recaled Brownian moion /d W i. Such random moion i oo mall. To underand hi, aume we have no ineracion he heory hould cover hi paricular cae. If we ake he empirical meaure aociaed o /d W i, hen he macrocopic PDE doe no conain he Laplacian recall ha σw i produced σ u; repeaing ha

72 7 5. LOCAL AD ITERMEDIATE ITERACTIOS proof one can ee ha an infinieimal diffuion coeffi cien lead o u. So, here i omehing wrong in he imple idea ha he recaling i /d Y i,. Wha i wrong i ha alo ime ha o be recaled. Inuiively, hink o he cae wihou ineracion and wih a dicreized form of he Brownian diplacemen. A he microcopic level, each ime uni every paricle ha a diplacemen of order. If we ju recale /d W i, each ime uni he paricle have diplacemen of order /d. Eenially hey do no move, for very large. The dynamic i oo low o be viible. There i a dynamic, bu he recaling /d W i, ha frozen i. If we wan o coninue o ee i, we need o accelerae, o recale alo ime. Recall ha /d W i ha he law of a Brownian moion. Hence he ime recaling mu be /d /d. Therefore we conider he new poiion X i, := /d Y i, /d. The paricle X a ime = i he ame a he paricle Y a ime /d : we accelerae he movie of paricle Y. P 5. If Y i, aify 3., hen X i, dx i, = /d j= b /d X i, aify X j, d + σdb i, wih he iniial condiion X i, := /d Y i,, where he independen Brownian moion B i, are given by B i, := /d W i. /d P. Y i, Y i, /d = Y i, + = Y i, + j= /d r= /d = Y i, + j= j= b Y i, b Y i, Y j, d + σw i Y j, d + σw i /d b Y i, Y j, /d dr + σw i r /d r /d /d X i, = /d Y i, + /d = X i, + j= j= /d b /d Xr i, b Y i, Y j, /d dr + σ /d W i r /d r /d /d Xr j, dr + σb i,.

73 4. THE PDE ASSOCIATED WITH LOCAL ITERACTIO 73 C 6. The correc recaled dynamic o inveigae he macrocopic limi ha he form dx i, = /d j= g /d X i, X j, X i, X i, X j, X j, d + σdb. i 3.4. Simulaion. The correc code for a conac ineracion of repulive form i herefore: =3; n=5; d=.; ig=; h=qrd; T=; L=; L = ; d= alp=*^/d; R=^-/d X=marix,,n; Y=X; X[,]=runif,-L/,L/; Y[,]=runif,-L/,L/ ploc-l,l, c-l,l, ype="n" for in :n-{ for i in :{ DX= X[i,]-X[,]; DY= Y[i,]-Y[,] Kx=gnormaDX,DY,R,alp*DX/normaDX,DY+. Ky=gnormaDX,DY,R,alp*DY/normaDX,DY+. X[i,+]=X[i,] + d*meankx + h*ig*rnorm Y[i,+]=Y[i,] + d* meanky + h*ig*rnorm } if%%t== {polygonc-l,l,l,-l,c-l,-l,l,l, col="whie", border=a Cir=:*+.5*^-/d ymbolx[,+],y[,+], circle = Cir, add=true,inche=false } } Here we have recaled α, R and he ize of he circle according o he heory. Moreover, we have aken a much beer ime ep o avoid ha overlap beween paricle could caue oo high arificial diplacemen. Finally, we have replaced he Gauian iniial condiion wih a uniform one in order o impoe, a ime zero, an iniial condiion very cloe o deniy. We ugge o run he code for =,, 3 and ee ha he image i very imilar, ju le and le granular. 4. The PDE aociaed wih local ineracion Thi ecion i more diffi cul han previou one, hence we kech i conen. Fir, we reformulae he model wih conac ineracion in a form which i uiable for he inveigaion of i macrocopic limi, 4.. Then, ince he macrocopic limi problem i very diffi cul and innovaive, we change he problem and freeze he caling parameer in he ineracion poenial, 4..

74 74 5. LOCAL AD ITERMEDIATE ITERACTIOS Wih hi rick we may apply Mean Field heory and deduce a PDE, paramerized by he freezed parameer, ay M. Then we end M o infiniy, o find a new PDE, called Porou Media equaion, which in a ene correpond o infiniely many paricle and infiniely recaled poenial, 4.3. However, we have no done he limi ogeher, wih M =, a i hould be. We have fir aken, hen M. I i he ame? We do no know he anwer. We preen a few imulaion ha indicae ha he wo reul are a lea numerically very cloe, for cerain clae of problem, 4.4. Encouraged by he numerical reul, we decribe in deail a parial reul in he direcion of howing ha he wo limi are he ame. I hold rue for a modified verion of he recaling, called inermediae or moderae ineracion heory, Reformulaion of he SDE wih local ineracion. Given he funcion g r, r, of he previou lecure, le G r, r, be a primiive of g r and le V x = G x. E. g r = r r, r, G r = r / r. We conider V a a "poenial". Wriing V for V, we have V x = g x x x. Hence we may wrie equaion dx i, = /d j= g /d X i, X j, X i, X i, X j, X j, d + σdb i in he form Seing furher we have dx i, = /d j= V /d X i, V x = V /d x V x = /d V /d x X j, d + σdb. i hence we may wrie he previou SDE a dx i, = j= V Wih hee new noaion, recall he reul: X i, X j, d + σdb. i

75 4. THE PDE ASSOCIATED WITH LOCAL ITERACTIO 75 C 7. The correc recaled dynamic o inveigae he macrocopic limi of a yem wih local ineracion V i 4. dx i, = V X i, X j, d + σdb i. j= 4.. The Mean Field equaion wih a concenraed poenial. Le u decuple he number of paricle from he caling of he poenial and conider he equaion 4. dx i,,m = V M X i,,m X j,,m d + σdb i j= where now we have wo parameer: i he number of paricle, M i a parameer which define he recaling V M x = MV M /d x. Obviouly he rue conac-ineracion problem i o ake M = and end. Bu we already know he reul of ending if we keep M fixed i i he Mean Field heory: he empirical meaure S,M = i= δ X i,,m weakly converge o he meaure-valued oluion µ of he Mean Field equaion µ = σ µ + div µ V M µ. Aume we need i below ha µ ha a deniy u x. More preciely, boh µ and u x depend on M; o le u wrie u M x. Thi funcion i a weak oluion of he PDE u M 4.3 = σ um + div u M V M u M. We ak wo queion: doe u M converge o he oluion u of ome PDE? I hi PDE he correc one for he limi of he empirical meaure S = S,? We may anwer he fir queion, bu he econd one i open Taking he limi in he Mean Field PDE. Le u fir inveigae he limi, a M, of oluion u M of equaion 4.3. We do no wan o be rigorou here i i poible o ae a rigorou reul and hu aume ha u M converge, in a uiable ene, o a funcion u. We wan o find he equaion aified by u. We have, for every e funcion φ C, u M, φ = u M, φ + σ u M, φ d V M u M, u M φ d hence, under relaively weak convergence properie, he fir hree erm converge, hence we have u, φ = u, φ + σ u, φ d lim M In addiion, if we prove ha V M um v in a uiable ene, we ge u, φ = u, φ + σ u, φ d V M u M, u M φ d. v, u φ d.

76 76 5. LOCAL AD ITERMEDIATE ITERACTIOS Thi i he weak form of he PDE u = σ u + div uv. We have only o idenify v. We have, for every e funcion θ C, V M u M, θ = u M, V M θ. Moreover, V M θ x = V M y x θ y dy = V M y x θ y dy. Aume now ha V M i proporional o a equence of mollifier: hi i rue by he formula V M x = MV M /d x if V i proporional o a mooh probabiliy deniy V x = σ f x where σ > and f i a pdf. In hi cae, ince θ i mooh and compac uppor, V M y x θ y dy σ θ x, hence V M θ x σ θ x and finally V M u M, θ σ u, θ = σ u, θ. We have found v = σ u and herefore he limi PDE u 4.4 = σ u + σ u. Thi equaion i known in he lieraure a Porou Media equaion. R 6. oice ha he equaion doe no depend on he deail of he ineracion poenial V, only on σ = V x dx. Thi look range, from he viewpoin of he paricle yem. The reul gueed o far i correc from he viewpoin of he PDE: i i a heorem ha he oluion u M of he Mean Field equaion converge o he oluion u of he Porou Media equaion Open problem and a few imulaion. Le u repea he open problem. We have a few objec: he paricle yem wih conac ineracion 4. and i empirical meaure S ; he paricle yem wih mean field ineracion V M 4. and i empirical meaure S,M ; he oluion u M of he mean field PDE 4.3; he oluion u of he porou media PDE 4.4. We know ha in uiable ene. Do we have alo lim S,M = u M lim M um = u lim S = u?

77 4. THE PDE ASSOCIATED WITH LOCAL ITERACTIO 77 The anwer i no known. In hi ecion we give ome evidence of he fac ha he anwer, alhough heoreically uncerain, numerically i very cloe o be poiive, in he cae of repulive ineracion wih inegrable poenial V. The aim of he following imulaion i o ee, a he paricle level, wha happen if we conider paricle wih poenial V M x, for differen choice of M. We conider he cae, in d =, V x = x x x x and we recale i a above: V M x = M M /d V M /d x = M V Mx. The following funcion define par of V wih a generic recaling, where in he equel we hall ake R = M, α = M. norma=funcionx abx H=funcionr ignr+/ g=funcionr,r,alp alp*h-r/r*-r/r The code i now M=; M= =; loc=; n=; d=.; ig=; h=qrd; T=; L=5; L = e= marix/,, alp=m^; R=/M alp=m^; R=/M X=marix,,n; X=marix,,n; X[,]=rnorm,,L; X[,]=X[,]; ploc-l,l, c,.6, ype="n" for in :n-{ DX.= marixx[,],,; DX=DX.-DX. DX.= marixx[,],,; DX=DX.-DX. Kx=gnormaDX,R,alp*DX/normaDX+. Kx=gnormaDX,R,alp*DX/normaDX+. X[,+]=X[,] + d*kx%*%e + h*ig*rnorm X[,+]=X[,] + d*kx%*%e + h*ig*rnorm if%%t== {polygonc-l,l,l,-l,c,,.6,.6, col="whie", border=a linedeniyx[,+],bw=dx[,+]/4,col="green" linedeniyx[,+],bw=dx[,+]/4,col="red" }

78 78 5. LOCAL AD ITERMEDIATE ITERACTIOS } R 7. The previou code, hank o he help of paricipan, ha been improved wih repec o previou one, by aking a marix-approach o he compuaion of ineracion beween paricle. Thi peed-up coniderably he compuaion. R 8. Taking value like -3 inead of 4-5 in bw=dx[,+]/4 mooh-ou he profile bu one hen upec ha fane i due o moohing and no o diffuion. The reul eem o be ha M doe no maer: he diffuion i eenially he ame. The reul i imilar for large deniie: M=; M= =; loc=; n=; d=.; ig=; h=qrd; T=; L=5; L =. e= marix/,, alp=m^; R=/M alp=m^; R=/M X=marix,,n; X=marix,,n; X[,]=rnorm,,L; X[,]=X[,]; ploc-l,l, c,, ype="n" for in :n-{ DX.= marixx[,],,; DX=DX.-DX. DX.= marixx[,],,; DX=DX.-DX. Kx=gnormaDX,R,alp*DX/normaDX+. Kx=gnormaDX,R,alp*DX/normaDX+. X[,+]=X[,] + d*kx%*%e + h*ig*rnorm X[,+]=X[,] + d*kx%*%e + h*ig*rnorm if%%t== {polygonc-l,l,l,-l,c,,,, col="whie", border=a linedeniyx[,+],bw=dx[,+]/4,col="green" linedeniyx[,+],bw=dx[,+]/4,col="red" } } Replacing he poenial wih he following one doe no change he picure: V x = x x x x norma=funcionx abx

79 5. ITERMEDIATE ITERACTIO: PREPARATIO 79 H=funcionr ignr+/ g=funcionr,r,alp *alp*h-r/r*-r/r^ Obviouly we hould go o M =, bu i eem ha nohing change. The concluion eem o be ha aking he limi fir and hen M give a very imilar reul a impoing M = and aking. Some geomeric inuiion, no decribed here, eem o confirm hi reul for inermediae value of he deniy. When he deniy i very large, he cae M = could have a ronger diffuion, bu hi i no biologically relevan. When he deniy i very mall he diffuion of he Mean Field equaion hould prevail, bu he Brownian diffuion in ha cae become dominan. Overall, we hink ha he Porou Media equaion i a very good model for conac repulive ineracion, when V i inegrable. E 3. Compare he imulaion of he Porou Media PDE 4.4 wih he paricle yem wih rue conac ineracion 4.. For hi reaon, le u examine in deail he following cae. 5. Inermediae ineracion: preparaion 5.. Inermediae regime. The open problem above, namely wheher he limi of he conacineracion cae i equal o aking fir and hen M, ha a full rigorou and poiive oluion if we recale he ineracion poenial in a weaker way. Karl Oelchäger idenified hi inermediae problem, beween mean field and conac ineracion, called inermediae or moderae regime, where i i poible o prove ha S converge o he oluion of he porou media equaion. I i he cae when V x = β V β/d x wih β, β, for a uiable β <. For β = we have mean field ineracion; for β = we have conac ineracion. The regime β, β i hu more local han mean field bu no o realiic a conac ineracion. The rericion o he inermediae regime will be ued o prove ighne. Bu alo he problem of paage o he limi i very diffi cul, a explained below. In hi cae, i i no he rericion on β o play a role bu an aumpion on he rucure of V a he convoluion of wo kernel, aumpion again idenified by Oelchäger. Le u dicu fir he iue of convergence. 5.. Taking he limi. A in a previou ecion we have: L 9. The empirical meaure S d S, φ = S, φ d S, φ V S aifie he ideniy d + dm φ, + σ S, φ d

80 8 5. LOCAL AD ITERMEDIATE ITERACTIOS for all φ Cb. [, T ] R d, where M φ, = i= φ X i, σdb i. In order o underand he diffi culy o ake he limi in hi ideniy, aume for a econd ha S weakly converge o ome meaure µ. We can eaily pa o he limi in he erm S, φ and S, φ ; he maringale erm M φ, goe o zero in mean quare he proof i eay; we hall ee below imilar compuaion. Bu he limi of he nonlinear erm S, φ V S i much more diffi cul. Recall, from general fac of analyi, ha weak convergence of funcion i no uffi cien o pa o he limi in nonlinear erm: he equence f n x = in nx, x [, ] converge weakly o zero, bu fn doe no. Hence he ole propery of weak convergence of he meaure S canno be uffi cien. Anoher general fac we know from analyi i ha if S weakly converge o µ and f i a equence of funcion which converge uniformly o f, hen f, S = f, µ Bu uniform convergence i eenial. lim E 3. If υ n = δ xn, υ = δ x, x n x, hen υ n υ. If f n converge uniformly o f, we ee ha f n, υ n = f n x n converge o f x. Bu weaker convergence do no imply he ame reul: here i no reaon why f n x n hould converge o f x if we have only poinwie convergence, or L p convergence, and o on. Can we ay ha V S converge uniformly o ome limi? Thi i a very diffi cul queion. The only eay hing we can ay i ha i weakly converge: V S, θ σ θ x µ dx = σ θ x u x dx in he cae when µ ha a differeniable deniy u x. Indeed V S, θ = S, V θ σ θ x µ dx = σ θ x u x dx. Implicily we have underood ha, in hi problem, ooner or laer we have o prove ha µ ha a deniy. One way i o inveigae he convergence of he mollified empirical meaure h x = W S x. Afer hee preliminary remark on he diffi culy of aking he limi in he non-linear erm, le u decribe he brillian rick devied by Karl Oelchläger, under an addiional aumpion. Aume ha ake σ = here for impliciy of expoiion V = W W where W are claical mollifier and, from now on, we wrie W x = W x.

81 5. ITERMEDIATE ITERACTIO: PREPARATIO 8 Then we have S, φ V S = φ S, W W S = W φ S, W S where we have ued he propery W f, g = f, W g eay o prove. ow, aume we can prove ha he quaniy W φ S, W S φ W S, W S i mall for large namely ha commuing W wih he poinwie produc φ i irrelevan, in he limi. If o, we have o deal wih φ W S, W S and here we ju need weak L -convergence of W S and rong L -convergence of W S. Recall indeed ha if f n f in L R d and g n g in L R d, hen f n, g n f, g f n, g n f, g n + f, g n f, g f n f g n + f, g n f, g and now he fir erm goe o zero by rong convergence of f n o f and boundedne of g n ; he econd erm goe o zero by weak convergence of g n o g. In he nex ecion we decribe a modified model where i i poible o prove uch properie and hen o pa o he limi rigorouly Bound on paricle poiion. Our ulimae aim i o prove ighne of he law Q of he empirical meaure S. [In fac, a he end, we follow a differen approach, namely we prove ighne of mollified empirical meaure, bu i i convenien for pedagogical reaon o argue abou ighne of he law Q of S ] To reach hi reul, he fir and more imporan eimae, following he compuaion made for he Mean Field cae, would be o prove ha [ ] E up X i, C. Bu now X i, X i, + [,T ] j= V X i, X j, d + σ B i and V i unbounded wih repec o, o we canno repea he eimae of he mean field cae. oice ha V X i X j d = u, X i where j u, x = V S x = V x X j j.

82 8 5. LOCAL AD ITERMEDIATE ITERACTIOS Hence he ineracing paricle yem can be wrien in he form = u X i d + σdb i. Thu X i, dx i, X i, + u If we prove uiable eimae on u, we have a ool o prove E X i d + σ B i. [ X i, up [,T ] ] C. The nex ecion how ha here i hope o prove an eimae on u. To compare he empirical funcion u ued here wih he funcion u of he nex ecion, noice ha u x = V S x, if i converge, i hould converge o he oluion u of he Porou Media equaion, by he general conjecure underlying hi ecion Energy eimae on he Porou Media equaion. We fir decribe an a priori eimae for he Porou Media equaion ince, we hink, i i concepually he idea behind he nex compuaion on S. For impliciy of noaion, conider he equaion u = σ u + div u u. For hi equaion, he andard energy eimae read d u dx = u u σ d dx = u u + div u u dx = σ u dx u u dx hence σ u dx + u dx + u u dx d = u dx where we re he remarkable fac ha u div u u dx = u u u dx = u u dx. Thi provide eimae on u. Recall ha a he end of he previou ubecion we idenified a a poible ool exacly he conrol of u. Thu we hall ry now o repea hee energy compuaion on u. wih 6. Rigorou reul on inermediae ineracion In hi ecion we prove he following reul. We conider he SDE dx i, = V X i, X j, d + σdb i j= V x = β V β/d x

83 6. RIGOROUS RESULTS O ITERMEDIATE ITERACTIO 83 wih β, β, for a uiable β < aed by he nex heorem. Recall ha we wrie W x = W x. T 5. Aume ha V = σ W W where W i a mooh compac uppor probabiliy deniy funcion and σ i a poiive conan. Se V x = β V β/d x, W x = β W β/d x o ha V = σ W W and e w x := W S x. Aume β d d +. Finally, aume ha he iniial condiion X i, aify E w x [ X i, dx C, E ] C R d and S, φ u, φ in probabiliy, for every φ Cc, wih u L R d, u. Then he family of law Q of he funcion w x are igh on he pace Y decribed below, which include L loc [, T ] R d. They have he unique limi Q = δu, and u i he unique weak oluion of he Porou Media equaion u = σ u + σ u wih iniial condiion u. The proof of hi heorem require everal ep ha we divide in he nex ubecion. 6.. Energy eimae on he mollified empirical meaure. Recall Lemma 9: d S, φ = S, φ d S, φ V S d + dm φ, + σ S, φ d. and he definiion In hi ecion we prove he following wo reul. w u x = W S x = W x X j j x = V S x. L. For he mollified empirical meaure w x, under he aumpion V = σ W W, we have Rd w x dx + σ x dx d + x S dx d = w x dx + R d R d w R d w x dm dx + R d σ V S R d i= W x X i, d dx.

84 84 5. LOCAL AD ITERMEDIATE ITERACTIOS C 8. Under he furher aumpion β Rd E w x dx + σ E w x dx + C. R d R d w d d+, here i a conan C > uch ha x dx d + E u x S dx d R d Le u prove he lemma. Replace φ y by W x y, wih x given a a parameer, and hink S, φ a an inegraion in he y variable, o ha S, φ = S, W x = w x and o on for he oher erm: S, W x = W S x = w x S, W x V S = y W x y V S y S dy = div W x y V S y S dy = div W V S S. We ge where dw x = div W V S M x = i= S x d + dm x + σ w x d W x X i, σdb i. Le u apply Iô formula o w x, wih x reaed again a a parameer. We have d w x = w x dw, x + d [ w x ] = w x div W V S S x d + σ w x w x, x d + w x dm + d [ M x ] and d [ M x ] = σ W x X i, d. i=

85 6. RIGOROUS RESULTS O ITERMEDIATE ITERACTIO 85 Think o he previou ideniy a inegraed in ime, a i i rigorouly. Then we inegrae in dx, namely we compue he differenial of he "energy" R w d x dx: w x dx = w x dx R d R d + w x div W V S S x d dx R d + σ w x w x d dx R d + w x dm dx R d σ + W x X i, d dx. R d i= Le u exchange inegraion he funcion are inegrable under minor aumpion on W. We have w x w x d dx R d = = w x w x dx d R d w x dx d R d ha we hall wrie on he lef-hand-ide of he ideniy above. Moreover, and hi i he main poin of he compuaion, w x div W V S S x d dx R d = w x div W V S S x dx d R d = w x W V S S x dx d R d and, by a rule already ued above, = W w x V S R d x S dx d. From he aumpion V = σ W W i follow V = σ W W we check i below and hu W w x = W W S = W W S = V S = V S.

86 86 5. LOCAL AD ITERMEDIATE ITERACTIOS we have ued he properie f g h = f g h and f g = f g, eay o prove. We deduce σ R d = w x div W V S R d V S Le u prove ha V = σ W W : W W x = σ x S dx d. W x + y W y dy = σ β = σ β W y β/d x = β V β/d x = V x. S x d dx β W β/d y x W β/d y dy W y dy = σ β W W β/d x The lemma i prove. Le u now prove he corollary. We have o prove ha he la erm in he ideniy of he lemma i bounded by a conan, in expeced value he maringale erm w, x dm ha zero average. The required bound will follow from he following reul. oice ha we hall perform ju ideniie and even wihou expeced value, hence he reul, in erm of range of β, i opimal. L. If β d d+, here i a conan C > uch ha, for all in [, T ], R d σ i= W x X i, r dr dx C. P. We have R d σ = σ = σ i= i= i= W x X i, r dr dx R d W x X i, r dx dr W x dx dr R d becaue, by he change of variable x x Xr i, in he inegral in dx, we have R d W x X i, r dx = R d W x dx.

87 Hence we coninue he ideniie a = σ = σ β β/d = σ β β/d 6. RIGOROUS RESULTS O ITERMEDIATE ITERACTIO 87 Rd W x dx = σ β R d β W x β/d dx R d W x dx. R d β β/d W x β/d dx The aumpion β d d+ mean β + β d, hence hi quaniy i bounded. 6.. Toward he ighne of he law of S. The aemen of Theorem 5 claim he ighne of he law Q of he funcion w x, no of he law Q of he empirical meaure S on C [, T ] ; Pr R d. To reach hi reul we need in addiion he ric inequaliy β < d d+. We do no give he full proof ince i i no par of our main reul. However, he idea of proof i very inereing, compared wih he mean field cae, hence we digre on i. Recall he proof of ighne made for he mean field problem. We have fir o prove ha E [ up [,T ] X i, i= ] C. From X i, X i, + u, X i d + σ B i we have X i, i= = X i, i= X i, i= X i, i= X i, i= + i= u, X i, d + σ + u, x S R d + u, x S R d + C T u, x S R d i= dx d + σ B i i= / dx d + σ B i i= B i / dx d + σ B i i=

88 88 5. LOCAL AD ITERMEDIATE ITERACTIOS hence E [ up [,T ] + σe E i= [ up [,T ] X i, i= [ X i, ] [ ] T + C T E u, x S R d ]. i= B i / ] dx d The fir erm i bounded by aumpion. The econd one i bounded by he eimae above. The hird one i eaily bounded a we have done oher ime. The fir eimae on he empirical meaure i proved. Then, imilarly o he mean field cae, we have o eimae W S, S X i X i i= i= u r, X i, r = u, x Sr R d / dr + σ B i B i i= dx dr + σ R d u, x S r Therefore [ E W S, S ] [ E u, x Sr R d C. B i B i i= / dx dr + σ ] dx dr + σ B i B i. i= i= [ E B i B i ] Thi eimae i no uffi cien o apply our claical heorem, ince we need C +ɛ. We omi he proof of hi improvemen ince i i quie echnical, being aified wih he idea of he full proof. The echnical argumen re-ar from he ideniy of Lemma??, eimae in he average higher power of he erm on he lef-hand-ide bu maringale inequaliie are needed here, and hen goe bu being able o handle a power higher han. I i back o he previou eimae on W S, S here ha we need he lighly ronger aumpion β < d d+ wih repec o Corollary Tighne of he mollified empirical meaure. In hi ecion we prove: L. Under he aumpion of Theorem 5, he family of law Q i igh, a decribed in Theorem 5.

89 6. RIGOROUS RESULTS O ITERMEDIATE ITERACTIO 89 P. We ue he fracional Aubin-Lion lemma. Fir, le u clarify he pace. Recall he noaion: if E E E are hree Banach pace wih coninuou dene embedding, E, E reflexive, wih E compacly embedded ino E, given p, q, and α,, he pace L q, T ; E W α,p, T ; E i compacly embedded ino L q, T ; E. We ue hi lemma wih E = L D, E = W, D and E = W γ, R d, γ large enough o be choen below, where D i a regular bounded domain, and wih p = q =, α, mall enough. The lemma ae ha L, T ; W, D W α,, T ; W γ, R d i compacly embedded ino L, T ; L D. ow, conider he pace Y := L, T ; W, R d W α,, T ; W γ, R d. Uing he Fréche opology on L loc [, T ] R d defined a T d f, g = n f, x dxd B,n n= one ha ha Y i compacly embedded ino L loc [, T ] R d he proof i elemenary, uing he fac ha if a e i compac in L, T ; L B, n for every n hen i i compac in L loc [, T ] R d wih hi opology. Denoing by L w, T ; W, R d he pace L, T ; W, R d endowed wih he weak opology, we have ha Y i compacly embedded ino 6. Y := L w, T ; W, R d L loc [, T ] R d. I i imporan o conider boh pace and no only L loc given by Aubin-Lion lemma. Indeed, a we have already een above in he informal ecion, he convergence will require rong convergence in L and weak convergence of derivaive. We deduce he following fac. L 3. Given M, R >, he e } K M,R = {f : [, T ] R d R : f L,T ;W, R d M, f W α,,t ;W γ, R d R i relaively compac in Y. Le u prove ha Q i igh on Y. Given ɛ > we have o find a compac e K ɛ Y uch ha Q Kɛ c ɛ. We hall chooe M, R depending on ɛ uch ha Q KM,R c ɛ. We have Q KM,R c = P w w / K M,R P w L,T ;W, R d > M +P W α,,t ;W γ, R d > R. We have w P L,T ;W, R d > M T M E w d C W, M by he fir wo erm in he eimae of Corollary 8. Thi i le han ɛ for large M.

90 9 5. LOCAL AD ITERMEDIATE ITERACTIOS Similarly we have P w W α,,t ;W γ, R d > R T R E If we prove ha, for a uiable choice of γ, E [ w w T W γ, ] C hen we may chooe α mall and have ha P w W α,,t ;W γ, R d > R C R ɛ for large R. Recall ha w w W γ, +α d. where Hence w w x w x = σ M + x = i= w r x dr + M x M x div W V S r S r x dr W x X i, σdb i. w C w W γ, r dr W γ, + C M M W γ, + C div W V Sr T C w r + C M M + C T L W, dr W u r S r W γ+, dr S r W γ, dr having ued he fac ha γ, f W γ, C f W, C f W,, f W γ, C 3 f L, div g W γ, C 4 g W γ+,. Le γ be uch ha γ > d. Then W γ, C b. Therefore W u r Sr W γ+, = up W u r S r, φ. φ W γ,

91 We have hence We have deduced w W u r S r w I follow, from Corollary 8, Finally 6. RIGOROUS RESULTS O ITERMEDIATE ITERACTIO 9, φ = u r Sr, W φ W x y φ x dx u r y Sr dy R d R d W u r S r T C W γ, E [ w w φ u R r y Sr dy d C φ W γ, u r y Sr dy R d C φ W γ, u r y Sr dy R d / C u W γ+, r y Sr dy. R d + C M M w r L. [ E M x M x ] dx = σ R d ow, from Lemma, we deduce T dr + u W, r x Sr dx dr R d ] [ M C + CE W γ, M = σ = σ R d E i= i= R d E R d E i= [ ]. L W x X i, r db i r dx W x X i, r db i r ] dx [ ] W x X i, r dr dx. The proof of ighne i complee. E [ M M L ] C.

92 9 5. LOCAL AD ITERMEDIATE ITERACTIOS 6.4. Paage o he limi. D 7. Given u L R d, we ay ha u Y i a weak oluion of he Porou Media equaion u = σ u + σ u wih iniial condiion u if, for every φ Cc [, T ] R d, one ha T T φ + σ φ u dxd + R d The aim of hi ecion i o prove: R d σ φ u dxd + u, φ =. L 4. Every limi probabiliy meaure Q of he family Q i uppored on he e of weak oluion of he Porou Media equaion. P. Le Q k Q. To implify noaion, le u drop he noaion of he ubequence and aume for impliciy ha Q Q. Given φ Cc [, T ] R d, conider he funcional on Y T T Ψ φ u = φ + σ σ φ u dxd + φ u dxd + u, φ. R d I i coninuou on Y and one ha he propery ha u i a weak oluion of he Porou Media equaion if and only if Ψ φ u = for every φ Cc [, T ] R d. Le u prove ha Q u : Ψ φ u = =. Thi will imply he reul by a imple deniy argumen. I i uffi cien o prove convergence o zero of Q u : Ψ φ u > δ for every δ > we apply he uual argumen baed on Pormaneau heorem. Hence we have o prove ha P Ψ φ w > δ converge o zero. The reader i uggeed o wrie all he compuaion and realize ha he diffi cul poin i only he convergence o zero of P σ T w w T, φ d S V S, φ d > δ/. We have S V S, φ = φ S, V S = φ S, W W S = W φ S, W S R d = φ W S, W S + R where R = φ, w w + R = W φ S φ W S, W S.

93 Hence P σ Bu hence T w w, φ d T W φ S R 6. RIGOROUS RESULTS O ITERMEDIATE ITERACTIO 93 S V S, φ d > δ/ = P σ T P T R φ W S W x y φ y φ x S dy D φ W x y x y S dy D φ ɛ W x y S dy W φ S P σ T w w T, φ d P D φ T ɛ w D φ ɛ δ [ T E = D φ ɛ w x φ W S, w D φ ɛ w w S V S L d > δ/ σ ] d Cɛ L. R d > δ/ d > δ/ σ. L, φ d > δ/ The end of he proof of Theorem 5 i baed on he uual argumen of uniquene of he weak oluion of he PDE, which however we omi.

94

95 Par 3 Growh and change of pecie in populaion of cell and nonlinear Parial Differenial Equaion

96

97 CHAPTER 6 Example of macrocopic yem in Mahemaical Oncology. An advanced model of invaive umor wih angiogenei Le u decribe a model inroduced by [3]. Thi model i made of 7 coupled PDE-ODE. Le u immediaely emphaize ha hi i no "he model" of cancer growh. I i one model, wih i own degree of ophiicaion, and decribing a umor in a paricular phae. I i a model of mechanical ype, dealing wih apec like random moion, moion along gradien, proliferaion, change of ype... ormoxic, hypoxic and apopoic cell. Cancer cell are mainly characerized by heir endency o duplicae. However, when hey receive an inuffi cien amoun of oxygen, or when hey do no have pace enough, heir proliferaion i inhibied. The model of [3] pli he caegory of cancer cell in hree clae: normoxic cell: healhy, proliferaing umor cell, wih normal oxygen upply hypoxic cell: quiecen umor cell, wih poor oxygen upply 3 apopoic cell: deah or programmed o deah umor cell The figure below how chemaically he obviou fac ha in a hree-dimenional umor ma he hypoxic cell are hoe inide, wih an even maller core of apopoic cell.... PDE for normoxic cell. The model precribe he following PDE for normoxic cell: 97

98 98 6. EXAMPLES OF MACROSCOPIC SYSTEMS I MATHEMATICAL OCOLOGY = k } {{} background diffuion div σ }{{} crowding-driven diffuion χ div m }{{} ranpor along ECM gradien α H o oh }{{} normoxic hypoxic, x = normoxic cell deniy + c V max V }{{} proliferaion + α H o>oh H }{{} hypoxic normoxic Fir, a background diffuion i admied; and no adheion conrain i impoed; hi i he invaive phae. The value of he conan k i, however, exremely mall, he diffuion i exremely low. Second, a crowding-driven diffuion i inroduced. Thi i a very inereing erm, which will occupy our effor quie ofen, bu no now, i i oo early. Ju a a fir idea, hi erm enforce he diffuion when he deniy of cell i larger. Third, proliferaion. If we neglec pace, he ime evoluion of a complee proliferaion i given by he differenial equaion x = λx, where λ i he proliferaion rae. One can conider a model wih ime-dependen rae. In he PDE above he ime-dependen rae i c V max V: i decreae o zero when he oal deniy of cell plu ECM V = + H + A + E + m = oal deniy of cell plu ECM approache he hrehold V max. Then, ranpor: normoxic cell, beyond he random moion decribed by k, have a endency o move along he gradien of m, he ECM deniy. Finally, ome normoxic cell become hypoxic when o o H, namely he oxygen i oo low and ome hypoxic cell are reored o he normoxic ae when o>o H.... ODE for hypoxic cell. Hypoxic cell do no move and do no proliferae. Their number increae when ome normoxic cell deeriorae o he hypoxic ae for o o H. And decreae eiher when hey are reored o he normoxic ae for o>o H or when hey degenerae o apopoic cell for o o A : dh d = α H o oh α H o oh H }{{}}{{} normoxic hypoxic hypoxic normoxic α H A o oa H H, x = hypoxic cell deniy }{{} hypoxic apopoic..3. ODE for apopoic cell. Thee cell are programmed o deah. I mean ha hey diolve in a regulaed way, no by necroi and cauing infecion. The number of apopoic cell can only increae, due o he hypoxic cell ha deeriorae:

99 . A ADVACED MODEL OF IVASIVE TUMOR WITH AGIOGEESIS 99 da d = α H A o oa H }{{} hypoxic apopoic A, x = apopoic cell deniy..4. ODE for ECM. In hi model, i i aumed ha he Exracellular Marix can only deeriorae, due o he invaion of normoxic cell: dm d = βm }{{} degradaion by normoxic cell m, x = ExraCellular Marix..5. Crowding-driven diff uion. In he equaion for normoxic cell we have he erm div σ which we have called crowding-driven diffuion. The precripion for σ in [3] i up o conan σ = max,. In oher word, when he deniy of normoxic cell pae he hrehold, an addiional diffuion ar. Thi erm i very inriguing and ypical of [3]. I i imilar o up o conan σ = which correpond o alo called porou media diffuion... The endohelial cacade. Hypoxic cell need more oxygen o urvive. Thu hey iniiae a cacade of cellular ineracion. The reul i angiogenei: new vacularizaion i developed o upply he umor microveel branching from main veel in he direcion of he umor. A meenger from hypoxic cell i en o endohelial cell: i i called VEGF Vacular Endohelial Growh Facor. Here are wo picure aken from he web ee [3] for he econd one, which include ome of he complicae molecular deail:

100 6. EXAMPLES OF MACROSCOPIC SYSTEMS I MATHEMATICAL OCOLOGY See alo he movie a he web ie [9], [3].... PDE for he VEGF concenraion. VEGF i a deniy of objec a molecular level. Thee proein have alway a diffuion oppoie o cell, which diffue only under pecial circumance: g = k } 4 g {{} diffuion + α H g H }{{} producion by hypoxic cell and he conan k 4 i much bigger han k. g, x = VEGF concenraion α g E Eg }{{} upake by endohelial cell

101 . A ADVACED MODEL OF IVASIVE TUMOR WITH AGIOGEESIS Moreover, he concenraion of VEGF i produced by he hypoxic cell, hence increae due o heir preence, and VEGF i aborbed by endohelial cell hoe forming he boundary of blood veel.... PDE for endohelial ramificaion. A he picure above how, endohelial cell do no diffue a iolaed individual bu hey propagae a microveel. However, keeping rack of heir opological rucure i diffi cul and perhap no o imporan in he opinion of he auhor of hi model, hence he concep of deniy of endohelial cell, or maybe more preciely deniy of endohelial ramificaion, i inroduced. The PDE i: E = k } E {{} diffuion E, x = deniy of endohelial ramificaion χ div E g }{{} ranpor along VEGF gradien + c Eg V max V }{{} proliferaion under VEGF preence where random moion i conidered however, a for normoxic cell, he value of he conan k i exremely mall endohelial cell move along he VEGF gradien, in order o reach he area occupied by hypoxic cell in order o build new veel, hey need o proliferae; he rae of proliferaion i c g V max V, namely i i proporional o VEGF concenraion, and i inhibied by he ame volume conrain of normoxic proliferaion...3. PDE for oxygen concenraion. Oxygen i alo molecular-level hence i diffue, wih k 3 i much bigger han k alo bigger han k 4 o = k } 3 o {{} diffuion + c 3 E o max o }{{} producion by endohelial cell o, x = oxygen concenraion α o,h,e + H + E o }{{} upake by all living cell γo }{{} oxygen decay and oxygen concenraion increae proporionally o he deniy of endohelial ramificaion, bu only up o o max i decreae due o aborpion by variou cell no he apopoic one i decay lowly...4. Summary of variable. I may be ueful o ummarize he li of variable:, x = deniy of normoxic cell H, x = deniy of hypoxic cell A, x = deniy of apopoic cell E, x = deniy of endohelial cell or deniy of vaculaure o, x = oxygen concenraion g, x = angiogenic growh facor VEGF concenraion m, x = ECM ExraCellular Marix

102 6. EXAMPLES OF MACROSCOPIC SYSTEMS I MATHEMATICAL OCOLOGY..5. Summary of conan. Diffi culie. The nex li of conan i given moly o emphaize a main diffi culy wih hi kind of complex model: ome of hee conan are poorly known and parameer fi i excluded by he impoibiliy of real experimen. k = background random moiliy coeffi cien of normoxic cell k = random moiliy coeffi cien of endohelial cell k 3 = diffuion coeffi cien of oxygen k 4 = diffuion coeffi cien of angiogenic facor χ = ranpor coeffi cien of normoxic cell along ECM gradien χ = ranpor coeffi cien of endohelial cell along VEGF gradien V cr = hrehold for crowding-driven diffuion V max = limi o oal volume of cell and ECM c = proliferaion rae of normoxic cell c = proliferaion rae of endohelial cell c 3 = producion rae of oxygen α H = decay rae from normoxic o hypoxic cell α H = reoraion rae from hypoxic o normoxic cell α H A = decay rae from hypoxic o apopoic cell α H g = producion rae of VEGF from hypoxic cell α o,h,e = upake rae of oxygen from all living cell α g E = upake rae of VEGF from endohelial cell o max = maximum oxygen concenraion o H = oxygen hrehold for raniion normoxic hypoxic o A = oxygen hrehold for raniion hypoxic apopoic β = rae of ECM degradaion γ = oxygen decay rae To re he diffi culie, le u alo menion he fac ha a iue i a highly complex environmen, poibly highly heerogeneou. The fir picure of Chaper one, like for inance he nex one, are example of he geomerical complexiy of a iue. Thi complexiy i no conidered by he model above, a lea i hi idealized form.

103 . A ADVACED MODEL OF IVASIVE TUMOR WITH AGIOGEESIS Simulaion. In Chaper 3 we hall devoe ome ime o dicu numerical imulaion of he full yem and a number of i reducion. However, ju o give a fir impreion, le u ee ome of he picure repored by he paper [3]. Blue: deniy of normoxic cell; ligh blue green: exracellular marix Doed black: deniy of hypoxic cell; red: deniy of endohelial ramificaion.