Properties of the interface of the symbiotic branching model

Transcription

1 Popeies of he ineface of he symbioic banching model Jochen Blah 1 and Macel Ogiese 1 (Vesion of 4 Mach 1) Absac The symbioic banching model descibes he evoluion of wo ineacing populaions and if saed wih complemenay Heaviside funcions, he ineface whee boh populaions ae pesen emains compac. In his pape, we show ighness of he diffusively escaled soluions and hus povide a fis sep owads a scaling limi fo he ineface. The cucial esimae involves a mixed fouh momen bound which we analyse using a paicle sysem dualiy. As a coollay, we obain an esimae on he momens of he widh of an appoximae ineface. 1 Mahemaics Subjec Classificaion: Pimay 6K35, Seconday 6J8. Keywods. Symbioic banching model, muually caalyic banching, sepping sone model, popagaion of ineface, momen dualiy. 1 Inoducion 1.1 The Symbioic banching model and is ineface The symbioic banching model of Eheidge and Fleischmann EF4 is a sochasic spaial model of wo ineacing populaions, paameized by ϱ 1, 1 govening he coelaion beween he wo diving noises. Moe pecisely, i is descibed by he iniial value poblem associaed wih he sysem of sochasic paial diffeenial equaions du (x) = u (x) + γu (x)v (x) dw 1 (x), SBM(ϱ, γ) u,v : dv (x) = v (x) + (1) γu (x)v (x) dw (x), wih posiive suiable iniial condiions u (x), v (x), x R. Hee, γ > is he banching ae and W = (W 1, W ) is a pai of coelaed sandad Gaussian whie noises on R + R wih coelaion ϱ 1, 1, i.e., fo 1,, E W i 1 (A 1 )W j (A ) { ( 1 )l(a 1 A ), i = j, = () ϱ( 1 )l(a 1 A ), i j, whee l denoes he Lebesgue measue and A 1, A ae Boel ses. Soluions of his model have been consideed igoously in he famewok of he coesponding maingale poblem 1 Insiu fü Mahemaik, Technische Univesiä Belin, Saße des 17. Juni 136, 163 Belin, Gemany. 1

2 in Theoem 4 of EF4, which saes ha, unde naual condiions on he iniial condiions u ( ), v ( ), a soluion exiss fo all ϱ 1, 1. Fuhe, he maingale poblem is wellposed fo all ϱ 1, 1), which implies he song Makov popey excep in he bounday case ϱ = 1. In EF4 i has also been obseved ha fo ϱ = 1, he sysem educes o he hea equaion wih Wigh-Fishe noise discussed e.g. by Tibe in Ti95, and ha fo ϱ =, he sysem is he so-called muually caalyic model of Dawson and Pekins DP98. An impoan ool fo he analysis of he symbioic banching model is he following unifom vesion of a esul on he asympoic behaviou of he momens of SBM obained by Blah, Döing and Eheidge in BDE11, Theoem.5. They define he so-called ciical cuve of he symbioic banching model p : ( 1, 1) (1, ) by p(ϱ) = π accos( ϱ), (3) and denoe is invese by ϱ(p) = cos( π p ) (fo p > 1). This cuve sepaaes he uppe igh quadan in wo aes: below he chaaceisic cuve, whee momens emain bounded, and above he chaaceisic cuve, whee momens incease o infiniy as : Theoem 1.1 (BDE11). Suppose (u, v ) is a soluion o he symbioic banching model wih iniial condiions u = v 1. Le ϱ ( 1, 1) and γ >. Then, fo evey x R, ϱ < ϱ(p) iff E 1,1 u (x) p is bounded unifomly in all. In paicula, if ϱ < ϱ(p), hee exiss a consan C(γ, ϱ) so ha, unifomly fo all x R and, E 1,1 u (x) p C(γ, ϱ),. Remak 1.. (i) Of couse, due o symmey, he same esul holds fo he v populaion. Tha hee is a finie bound independen of x follows since he sysem is unde he (1, 1) saing condiion anslaion invaian. (ii) Moeove, fo any x 1,..., x 4 we have by he genealized Hölde inequaliy ha E 1,1 u (x 1 )u (x )v (x 3 )v (x 4 ) max i=1,...,4 Eu (x i ) and similaly if some of he v s ae eplaced by u (and vice vesa). C(γ, ϱ), Naual quesions abou such (sysems of) SPDEs ae elaed o hei longem behaviou, in paicula he speed of popagaion of waves and inefaces fo suiable iniial condiions, such as complemenay Heaviside iniial condiions, i.e. u (x) = 1 R (x) and v (x) = 1 R +(x), x R. Definiion 1.3. The ineface a ime of a soluion (u, v ) of he symbioic banching model csbm(ϱ, κ) u,v wih ϱ 1, 1 is defined as Ifc = cl { x : u (x)v (x) > }, whee cl(a) denoes he closue of he se A in R.

3 The main quesion addessed in EF4 is whehe fo he above iniial condiions he socalled compac ineface popey holds, ha is, whehe he ineface is compac a each ime almos suely. This is answeed affimaively in hei Theoem 6, ogehe wih he asseion ha he ineface popagaes wih a mos linea speed, i.e. hee exiss a consan c = c(γ) such ha fo each ϱ 1, 1, hee is a (almos-suely) finie andom ime T such ha, almos suely, fo all T T, Ifc ct, ct. (4) T Howeve, due o he scaling popey of he symbioic banching model, see Lemma 8 of EF4, which saes ha if (u, v ) is a soluion o SBM(ϱ, γ) u,v, hen ( u K (x), v K (x) ) := ( u K ( Kx), v K ( Kx) ), x R, K >, is a soluion o csbm(ϱ, K γ) u K,v K wih suiably ansfomed iniial saes (u K, vk ), one migh expec ha he flucuaions of he posiion of he ineface should be of ode 1/. Indeed, wih he help of he momen esimaes of Theoem 1.1, i is possible o senghen (4) fo a (ahe small) paamee ange, see BDE11: Theoem 1.4 (BDE11). Suppose (u, v ) is a soluion of SBM(ϱ, γ) 1R,1 R + wih ϱ < ϱ(35). Then hee is a consan C(γ, ϱ) > and a finie andom-ime T such ha almos suely Ifc C T log(t ), C T log(t ), fo all T > T. T The esicion o ϱ < ϱ(35) seems aificial and comes fom he echnique of he poof. Though ϱ(35).9958 is ahe close o 1 he esul is sill ineesing since i shows ha sub-linea speed of popagaion is no esiced o siuaions in which soluions ae unifomly bounded as fo insance fo ϱ = 1. The poof is based on he mehod of Tibe fom Ti95 fo he hea equaion wih Wigh-Fishe noise employed wih impoved bounds on he momens of he symbioic banching model based on he ciical cuve, cicumvening he lack of unifom boundedness of he populaion sizes. In he ligh of he scaling popey, one migh hope ha fo a ahe lage paamee se, and possibly all ϱ, a diffusive ime-space escaling could lead o a igh sequence of sochasic pocesses. Indeed, his pogamme has been caied ou fo he discee space vesion of (1) he symbioic banching model. Fo muually caalyic model ϱ =, Klenke and Mynik consuc in a seies of papes KM1, KM11a, KM11b, a non-ivial limiing pocess fo γ and sudy hei long-em popeies. This limi is called he infinie ae muually caalyic banching pocess. Moeove, Klenke and Oele KO1 give a Toe ype appoximaion. Regading he ineface, in Coollay 1. of KO1 he auhos conjecue ha, unde suiable assumpions, a non-ivial ineface fo he limiing pocess exiss, which would in un pedic a squae-oo ode fo he flucuaions of he ineface. Recenly, his pogamme has been exended by Döing and Mynik o he case ϱ in DM11a, DM11b. These obsevaions and he above conjecue ae he saing poin of ou invesigaion. 3

4 1. Main esuls and open poblems We fis need o inoduce some suiable noaion. Fo a pai of (coninuous) funcions (u, v), we define R(u, v) := sup{x : u(x) > }, L(u, v) = inf{x : v(x) > }. (5) Fo he soluion (u, v ) of he symbioic banching model wih complemenay Heaviside iniial condiions, we noe ha he ineface is conained in he se L(u, v ), R(u, v ), an ineval whose widh we call he diamee of he ineface (his noion is well defined due o he compacness esul of EF4). I is poved in Ti95 fo ϱ = 1 and fo iniial condiions u = 1 v which saisfy < L(u, v ) R(u, v ) < ha unde Bownian escaling, he moion of he posiion of he igh endpoin of he ineface R(u n, 1 u n )/n,, conveges o a Bownian moion as n. The fis cenal idea in he poof in Ti95 is o conside he following measue valued pocesses µ n (dx) = u n (nx)dx and ν n (dx) = v n (nx)dx, (6) and show ha he sequences of hese pocesses ae igh. In fac, since in he case of ϱ = 1, v = 1 u, i suffices o conside only one of hese pocesses. The second sep is o idenify he limi and i is shown ha fo ϱ = 1, (µ n ) conveges in law o he measue-valued pocess (1l x B ) fo (B ) a sandad Bownian moion. In his noe, we ake he fis sep in his pogamme and show ighness of he measuevalued pocess defined in (6). Hee, he measue-valued pocesses ae eaed as elemens of C((, ), M em) he space of coninuous pocesses aking values in he space of (pais of) empeed measues, see also he Appendix A.1. Theoem 1.5. Assume ϱ < ϱ(4) = 1. Le (u, v ) be a soluion o he symbioic banching iniial value poblem wih complemenay Heaviside iniial condiions. Then, he pocesses (µ n, ν n ) ae igh in C((, ), M em). I would be ineesing o see if he poin ϱ(4) is eally significan o meeley due o echnicaliies. Theefoe, one should check whehe ϱ affecs ohe, fine, popeies of he ineface. The essenial sep poof of he ighness esul 1.5, is a fouh momen esimae. In he case ϱ = 1, MT97 exploi he coesponding esul o ge a esimae on he momens of he widh of he ineface R(u, v ) L ( u, v ). Howeve, his momen esimae heavily elies on he fac ha hee ae no holes in he sysem whee boh u and v ae zeo. In ou case, we can imiae he easoning o ge an esimae fo he appoximae ineface defined in he following way. Fo any ε >, define an appoximae lef end poin of he ineface as L (ε) = inf { x : and similaly, fo he igh end poin { R (ε) = sup x : x x } u (y)v (y)dy ε R(u, v ). } u (y)v (y)dy ε L(u, v ). 4

5 Since R(u, v ), L(u, v ) ae almos suely finie, R (ε), L (ε) ae well-defined. Ou nex esul saes ha his appoximae widh of he ineface emains small unifomly in in he following way. Theoem 1.6. Supose (u, v ) = (1l R, 1l R +) and ε >. Then fo any ϱ < ϱ(4) = 1, fo any p (, 1), hee exiss a consan C = C(p, ε, γ, ϱ) such ha fo all >, E 1l R,1l R+ ((R (ε) L (ε)) + ) p C. Remak 1.7. Open poblems. Ideally, one would like o show ha he measue-valued pocess ae no only igh, bu also convege in a suiable opology. One can show ha fo n fixed he densiies (u (n), v (n) ) saisfy a maingale poblem which is he coninuous analogue of he discee space infinie ae symbioic poblem, see DM11b, Pop. 3.3 (whee one has o apply he opeao o he es funcion). Theefoe, one way of showing convegence, is o show ha he coninuum poblem has a unique soluion (unde some condiion which says ha he wo measues have essenially disjoin suppo), which is howeve sill open. A second poblem is o impove he esul of Theoem 1.6 and eplace he appoximae lef and igh end poins by he exac bounds on he ineface L(u, v ) and R(u, v ) and cay ou he emaining pogamme of MT A coloued paicle momen dual Thee ae seveal dual pocesses fo he symbioic banching model. descibe he asympoic behaviou of mixed momens of he fom Hee, we aim o E u,v u (x 1 ) u (x n )v (x n+1 ) v (x n+m ). The dual woks as follows fo ϱ ( 1, 1). Conside n + m paicles in R which can ake on wo colous, say colou 1 and. Each paicle moves like a Bownian moion independenly of all ohe paicles. A ime, we place n paicles of colou 1 a posiions x 1,..., x n, especively, and m paicles of colou a posiions x n+1,..., x n+m. As soon as wo paicles mee, hey sa collecing collision local ime. If boh paicles ae of he same colou, one of hem changes colou when hei collision local ime exceeds an (independen) exponenial ime wih paamee γ. Denoe by L = he oal collision local ime colleced by all pais of he same colou up o ime, and le L be he colleced local ime of all pais of diffeen colou up o ime. Finally, le l := (l 1, l ),, be he coesponding paicle pocess, ha is, l 1 (x) denoes he numbe of paicles of colou 1 a x a ime and l (x) is defined accodingly fo paicles of colou. Ou mixed momen dualiy funcion will hen be given, up o an exponenial coecion involving boh L = and L, by a momen dualiy funcion (u, v) l l := 1(x) v(x) l (x). u(x) x R: l 1 (x) o l (x) Noe ha since hee ae only n + m paicles he poenially uncounably infinie poduc is acually a finie poduc and hence well-defined. The following lemma is aken fom Secion 3 of EF4. 5

6 Lemma 1.8. Le (u, v ) be a soluion of dsbm(ϱ, γ) u,v wih ϱ ( 1, 1). Then, fo any x R,, E u,v u (x 1 ) u (x n )v (x n+1 ) v (x n+m ) = E (u, v ) l e γ(l= +ϱl ), (7) whee he dual pocess {l } behaves as explained above, saing in l = (l 1, l ) wih paicles of colou 1 locaed in (x 1,..., x n ) and paicles of colou especively in (x n+1,..., x n+m ). Noe ha if u = v 1 he fis faco in he expecaion of he igh-hand side equals 1. Poofs The poof of Theoem 1.5 splis ino hee main pas. Fis, we need o pove an analogue of Ti95, Lemma.1 o obain a bound on inegaed mixed fouh momens. Alhough he esul is simila o Tibe s, he poof is vey diffeen since we have o wok wih he coloued paicle momen dual wih exponenial coecion insead of he sysem of coalescing Bownian moion available in he case of he hea equaion wih Wigh-Fishe noise. This will be done in Secion.1 Nex, we pove ighness of he diffusively escaled coodinae pocesses wih he help of he fouh momen bound obained above in Secion.. Then, we check ighness of he measue-valued pocesses on pah-space. Hee, we employ a vaian of Jakubowski s cieion, which equies o check a compac conainmen condiion. This is ivial in Tibe s case, bu equies exa wok in he case ϱ ( 1, 1 ). See Secion.3 Finally, in Secion.4, we show he momen esimae on he widh of he ineface of Theoem 1.6. Noaion: We have collec some of he basic facs and noaions abou measue-valued pocess in Appendix A.1. Moeove, Appendix A. is a collecion of esimaes fo Bownian moion and is local ime. Thoughou his pape, we will denoe by c, C geneic consans, whose value may change fom line o line. If he dependence on paamees is essenial we will indicae his coespondingly..1 A bound on inegaed fouh mixed momens Lemma.1 (Mixed momens). Le (u, v ) be a soluion o he symbioic banching iniial value poblem (1) wih inial values u = 1l R, v = 1l R +. Then, fo ϱ < ϱ(4) = 1, unifomly fo all. E u,v u (x)u (y)v (x)v (y) dx dy C(u, v ; γ) 6

7 Noe ha by Fubini s heoem and a simple subsiuion, i is sufficien o pove ha fo z >, E u,v u (x)u (x z)v (x)v (x z) dx, is inegable in z. Ou Ansaz is o use he momen dualiy fom Lemma 1.8 and combine i wih he momen bounds of Theoem 1.1. Howeve, Theoem 1.1 equies consan iniial condiions, which simplifies he momen dualiy consideably. In ou case, we have o be much moe caeful. Then, he dualiy in (7) eads E 1 R,1 R + u (x)u (x z)v (x)v (x z) = E l 1 =(x,x z),l =(x,x z) (u, v ) l e γ(l= +ϱl ) To descibe he dynamics of {l }, we inoduce a sysem of fou independen Bownian moions {B, i i = 1,..., 4} wih especive ypes (colous) c i () {1, } a ime. A possible ype change may occu when wo paicles of he same ype collec a subsanial amoun of collision local ime. Iniially, we have locaions B 1 =, B =, B3 = z, B4 = z and colous c 1 () = c 3 () = 1, while c () = c 4 () =. Defining f 1 := u = 1l R, f := v = 1l R +, we can wie he dualiy, by anslaion invaiance and symmey as E u,v u (x)v (x)u (x z)v (x z) = E l 1 =(,z),l =(,z) 4 i=1 f c i() (x B i )e γ(l= +ϱl ). We now inegae ove x and esimae he inegal. Noe ha he exponenial em does no depend on x. Hence, we may esic ou aenion o 4 f ci() (x B) i dx, (8) i=1 fo diffeen ype configuaions. Fis obseve ha f 1 (x B ) = 1l{x < B } and f (x B ) = 1l{x > B }, (9) so ha one should hink of he inegal in (8) as an inegal ove a poduc of Heaviside funcions cened a B i, whee he ype deemines he shape. Now, if we denoe by () he index of he lef-mos Bownian moion of ype 1, i.e. c () () = 1 and B () B i fo all i such ha c i () = 1, (whee we choose he smalle index o esolve ies). Similaly by l() he index of he igh-mos Bownian moion of ype, i.e. c l() () = and (wih he smalle index o esolve ies). B l() B i fo all i such ha c i () =, Obseve ha, due o he definiion of ou dual paicle sysem {l }, if we sa wih fou paicles and wo colous, hee will always be a leas one paicle of ype 1 and a leas 7

8 paicle of ype aound a any ime, no mae wha he acual ype changes wee (ype changes can only occu if wo paicles of he same colou mee). Moeove, wih he above noaion, he inegal in (8) is unless B () we obain 4 i=1 f c i() (x B i ) dx = (B () > B l() and since he poduc is eihe o 1, B l() ) +. Alogehe, we aive a E u,v u (x)u (x z)v (x)v (x z) dx = E (,z),(,z) (B () B l() ) + e γ(l= +ϱl ) (1) and need o show ha, fo z >, his expession is inegable in z. We pepae his wih a lemma which coves he impoan case whee a leas wo paicles in he middle ae a he same locaion. Lemma.. Le < x < y < z <, fo ϱ < ϱ(4), and δ >. Then, fo any iniial configuaion l = x ha conains fou paicles in posiions x, y, y, z and wo of each colou, i.e. { } x (x, y), (y, z); (y, z), (x, y); (x, z), (y, y); (y, y), (x, z), we have E x (B () ) + e γ(l= +ϱl ) { (z y + 1)(y x + 1) C(ϱ, γ, δ) min B l() 1 δ }, 1 δ. Poof. Pick ϱ so ha ϱ < ϱ < ϱ(4) and le δ (, 1). Using he (genealized) Hölde inequaliy wice fo p 1, p, p 3 1 wih p 3 = (1 δ) 1 and p 1 = p such ha 1 p p + 1 p 3 = 1, we obain E x (B () B l() ) + e γ(l= +ϱl ) 1 E x ((B () B l() ) + ) p 1 p 1 E x e p γ(l = +ϱ L ) 1 p E x e p 3γ(ϱ ϱ)l ) 1 p 3. The second expecaion in (11) coesponds o he fouh mixed momen of a sysem wih banching ae p and coelaion paamee ϱ. Since ϱ < ϱ(4), his expession is bounded by a consan (depending only on p, γ) unifomly in, see also Remak 1.. Fo he fis expecaion on he igh hand side in (11), we claim ha (11) E x ((B () B l() 1 ) + ) p 1 p 1 C(p 1 ) 1 (1) The claim follows if we can show ha he expecaion on he lef hand side does no depend on he disances of he saing poins z y, y x. We ecall ha he paicles ae labelled fom lef o igh accoding o he iniial posiions. In paicula, 3 ae he labels of he paicles saed in y. Also, we can always assume ha B l < B () since his is he 8

9 only scenaio when when we obseve a posiive conibuion o he expecaion. Case 1: {(), l()} = {, 3}. Then E x ((B () B l() 1 ) + ) p 1 p 1l 1 {(),l()}={,3}} E y,y B B 3 p 1 1/p 1 C(p 1 ) 1, (13) by he scale invaiance of Bownian moion. Nex, we conside Case : {(), l()} = {1, 4}. By definiion of he labels (), l(), hee ae no paicles in beween 1, 4 a ime and heefoe eihe one of he paicles, 3 ends up o he igh of () and he ohe o he lef of l() and we can bound (B () B l() ) + B B 3 and hen poceed as in (13). The ohe possibiliy is ha boh and 3 end up o he lef of 1, 4 o boh o he igh. Say boh, 3 end up o he lef of 1 = l (he ohe possibiliies wok analogously). Then, one of he paicles o 3 mus have collided befoe ime wih paicle 4, fo ohewise he hee paicles on he lef do no ineac wih 4 and heefoe, canno all have he same ype, which conadics he assumpions ha B l < B (). Hence, if τ i,j is he fis collision ime of paicles i, j, we can assume ha τ,4 and in paicula we can esimae he lef hand side of (1) using he song Makov popey by E x 1l {τ,4 } B 4 B 1 p 1 1 p 1 E x 1l {τ,4 }E sup Bτ+s Bτ+s 4 p 1 F(τ) 1 p 1 s τ 1 E, sup Bs Bs 4 p 1 p 1 C(p 1 ) 1 s (14) Finally, we conside Case 3, whee he labels {(), l()} coespond o one paicle saed a y and he ohe saed a x o z, wih loss of genealiy we assume ha {(), l()} = {1, }. If τ 1,, hen we can ague as in (14) o ge he igh bound. Ohewise, if τ 1, >, hen necessaily l() = 1, () = and if paicle 3 ends up o he lef of 1, hen we can esimae (B l() B () ) + B B 3 and he agumen in 13 gives he equied bound, while if 1 does no mee 3, necessaily 4 has o mee 1 (ohewise he paicles on he igh canno all have he same ype) and he agumen befoe (14) applies. These hee cases combined yield he esimae (1). Thus, we can conclude fom (11) ha E x (B () B l() ) + e γ(l= +ϱl ) C(p 1, p, γ, ϱ) 1 Ex e γp 3(ϱ ϱ)l ) 1 p 3. Thus, ecalling ha 1 p 3 = 1 δ, we see ha in ode o complee he poof i suffices o show ha fo any s > hee is a consan C(s), such ha fo all, { E x e sl (z x + log( e))(y x + log( e)) } C(s) min, (log( e)) 1, (15) whee we noe ha he log( e) em can be bounded by δ 1. Fis, ecall ha fo fo he collision local ime L up o ime of wo independen Bownian moions, saed in posiions x y, we have he classic bound ha fo all 1, P x,y {L < α log } (α log + y x) 1, α >, (16) see fo example Coollay A.5. Now, fix s > and le c = s. We disinguish he hee cases: 9

10 (i) L c log, (ii) L (iii) L < c log, bu L o := L = + L c log, < c log and L o < c log. Regading (i), we can esimae E x e sl 1l{L c log } sc. Fo (ii), we have in paicula ha L = c log. Now, fom ou classic fouh momen bounds (Theoem 1.1 and Remak 1. fo he sysem wih banching ae s/ϱ) we can deduce ha E x e sl 1l{L <c log cs ϱ E,Lo c log } x e s ϱ (L= +ϱl ) 1l {L <c log,lo c log } cs ϱ E x e s ϱ (L= +ϱl ) C(s, ϱ) cs ϱ C(s, ϱ) cs. Finally, conside case (iii). Hee, noe ha if he oal collision local ime is small, hen in paicula he collision local ime beween he wo Bownian moions saed a y is small. Tha is, using (16), E x e sl 1lL <c log,lo <c log P y,y {L < c log } c(log ) 1. A diffeen bound can be eached by consideing he collision local imes beween each pai of Bownian moions saed in y, z and y, x especively, leading o (again using (16)) E x e sl 1lL = P x,y {L c log }P y,z {L < c log } <c log,lo <c log (4c log + y x)(4c log + z y) 1. This complees he poof since we noice ha by ou choice of c = s, he dominaing conibuion is obained by aking he minimum in he las scenaio. Poof of Lemma.1. Fix < ε < 1. By (1), i suffices o show ha hee exiss a consan C such ha fo all z >, E (,z),(,z) (B () which is clealy inegable in z. B l() ) + e γ(l= +ϱl ) C(1 z (1 ε) ), (17) We condiion on he ime of he fis collision of ceain pais of he fou Bownian moions. Indeed, le τ i,j denoe he fis hiing ime of Bownian moions wih index i and j, and conside he sopping ime τ := τ 1,3 τ 1,4 τ,3 τ,4, which is he fis ime ha a moion saed in mees wih a moion saed in z. Noe ha we can always assume ha τ, fo ohewise he expecaion in (17) is. Then, if (F()) denoes he filaion of he dual pocess, we can apply he song 1

11 Makov popey and use ha up o ime τ hee ae no paicles of he same ype ha accumulae local ime. In paicula, none of he paicles have swiched ype up o ime τ, so he posiions of Bτ i a ime τ and he ype configuaion a ime τ saisfy he assumpions of Lemma., and we hus obain ha fo δ = ε 4, hee exiss a consan C(ϱ, γ, ε) such ha E (,z),(,z) (B () B l() ) + e γ(l= +ϱl ) = E (,z),(,z) E (B () B l() ) + e γ(l= L= τ +ϱ(l L τ )) F(τ) e γ(l= τ +ϱl τ ) (18) 4C(ϱ, γ, ε)e (,z),(,z) 1l {τ=τ,3 } min { (Bτ 4 B3 τ +1)(B τ B1 τ +1), ( τ) δ 1 } e ϱγ(l 1, τ +L 3,4 τ ). ( τ) 1 δ Hee, we also used ha he fou possible cases τ = τ 1,3, τ 1,4, τ,3, τ,4 ae all equally likely and in all cases we obain he same bound fom Lemma.. Moeove, in his scenaio L τ = L 1, τ + L 3,4 τ. In he analysis of he igh hand side of (18), we disinguish fou cases (whee we always assume τ ): (i) τ z ε, (ii) τ > z ε and (z ε > 1 4 o ) (iii) τ > z ε, bu z ε 1 4 and τ 1/ δ fo δ = ε 4,. (iv) τ > z ε, z ε 1 4, bu τ > 1/ δ fo δ = ε 4,. Case (i). On he even ha τ z ε, we obain E (,z),(,z) 1l {τ=τ,3 z ε } min { (Bτ 4 B3 τ +1)(B τ B1 τ +1), ( τ) δ 1 } e ϱγl ( τ) 1 δ E (,z),(,z) 1l {τ=τ,3 z ε }(Bτ 4 Bτ 3 + 1)(Bτ Bτ 1 + 1) E (,) max s z ε(b s Bs 1 + 1) P (,z) {τ 1, z ε } 1 C(1 z ε 1 4 ε )e 1 8 zε, whee we used in he penulimae sep Cauchy-Schwaz and fo he esimae of he fis collision ime ha if τ() denoes he fis hiing ime of fo a single Bownian moon saed a z, we have ha P (,z) {τ 1, z ε } = P z {τ() z ε } = P { max s z ε B z} = P {B z ε z} 1 π z 1 ε e zε 4, whee we used a sandad Gaussian esimae, see e.g. MP1, Lemma 1.9, in he las sep. This shows ha in case (i) we obain an uppe bound on (17) ha is inegable in z. τ 11

12 Case (ii). In his scenaio, we can find an uppe bound on he expecaion on he igh hand side in (18) E (,z),(,z) 1l {z ε <τ=τ,3 } min { (Bτ 4 Bτ 3 +1)(Bτ Bτ 1 +1), ( τ) δ 1 } e ϱγ(l 1, τ +L 3,4 τ ) ( τ) 1 δ E (,z),(,z) 1l {z ε <τ,3 =τ }(1 δ )e γϱ(l1, τ +L 3,4 τ ) (1 δ )E, e γϱl 1, (z ε ) C(1 δ )(1 z +ε ), whee we used he independence of he wo pais of Bownian moions and hen Lévy s equivalence, see Lemma A.3, o calculae he asympoics. Howeve, if we assume ha eihe o z ε > 1 4, hen his lae expession can be bounded by C(1 z +ε+4δ ), which by ou choice of δ = ε 4 is of he equied fom. Case (iii). In his case, we can assume ha so ha in paicula we can esimae hee ha since τ 1 δ, ( τ) ( 1 δ) ( 1 δ ) ( 1 δ) C 1 +δ. Hence, we can use he fis esimae in (18) and deduce ha E (,z),(,z) 1l z ε τ 1/ δ min { (Bτ 4 Bτ 3 +1)(Bτ Bτ 1 +1), ( τ) δ 1 } e ϱγ(l 1, τ +L 3,4 τ ) ( τ) 1 δ CE (,z),(,z) 1l {z ε τ=τ,3 1 δ } C 1 +δ E (,z),(,z) max (B 4 τ B 3 τ +1)(B τ B 1 τ +1) ( τ) 1 δ e γϱ(l 1, τ +Lτ 3,4 ) s 1 δ ( B 4 s B 3 s + 1)( B s B 1 s + 1)e γϱ(l 1, z ε+l3,4 ε) z Now, applying Hölde s inequaliy, wih p = 1 1 ε and q is conjugae, and hen using he independence, we obain an uppe bound C 1 +δ E (,z),(,z) max ( B s 1 s 4 Bs 3 + 1)( Bs Bs 1 + 1)e γϱ(l 1, z ε+l3,4 ε) z δ C 1 +δ E (,) max ( B s 1 s Bs 1 + 1) q δ q E, e γϱpl 1, z ε p C 1 +δ E (,) max ( 1 ( 1 δ) Bs Bs 1 + 1) q q E, e γϱpl 1, z ε p s 1 C(p, q)(1 z ( ε) 1 p ) whee we used Bownian scaling (and ) o evaluae he fis em and Lévy s equivalence, see Lemma A.3, fo he second em. In paicula, we obain ha he lae expession is bounded by C(1 z ( ε) 1 p ) (1 z (1 ε) ), by ou choice of p. Case (iv). Fo he emaining case (whee by (ii) we can assume ), we can use he second esimae in (18) and he independence of he Bownian moions o ge an uppe bound on (18) E (,z),(,z) 1l 1 min { (B { δ τ 4 B3 τ +1)(B τ B1 τ +1), ( τ) δ 1 } e ϱγ(l 1, τ +L 3,4 τ ) τ=τ,3 } ( τ) 1 δ C(1 δ )E (,z),(,z) exp { γϱ(l 1, 1/ δ + L 3,4 1/ δ ) } C(1 1 +δ ) C(1 z 4( ε)( 1 +δ) ), 1

13 whee we used Lévy s equivalence again and finally ha z ε 1 4. Hence, he esuling expession is of he fom (17), since δ = 1 ε < 1 4. These cases exhaus all possibiliies so ha he Lemma is poved via (18).. Tighness of he coodinae funcions of he escaled ineface Recall ha fo n N and >, x R he escaled soluions ae u (n) (x) = u n (nx) and similaly v (n) (x) = v n (nx). We fis esablish ighness of u (n) and v (n) inegaed agains suiable es funcions. Fo a discussion of he spaces involved, see Appendix A.1. Lemma.3. Suppose ϱ < 1 and (u, v ) = (1l R, 1l R +). If φ C ap, hen he coodinae pocesses { φ, u (n) : } n Z+ and { φ, v (n) : } n Z+ ae igh in he space D (, ) (R). Having esablished he fouh momen bound in Lemma.1, he poof of he ighness follows closely he poof of Ti95, Lemma 4.1. Poof. We denoe by (S ) he hea semigoup. By he scaling popey of he model, see EF4, Lemma 8, (u (n), v (n) ) is a soluion of he oiginal model when he banching ae γ is eplaced by nγ. In paicula, he Geen s funcion epesenaion fo he symbioic banching model, see Poposiion A., yields fo φ C ap, φ, u (n) = S φ, u (n) + n1/, R φ, v (n) = S φ, v (n) + n1/, R u (n) s u (n) s (x)v (n) (x)s s φ(x)dw 1 s(x) s (x)v (n) (x)s s φ(x)dw s(x) fo a pai of Gaussian whie noises (W 1 s(x), W s(x)) wih coelaion given by (). Noe ha fo ou iniial condiion, he fis em on he igh hand in (19) side is equal o S φ, 1 (,. We check Kolmogoov ighness cieion fo he sochasic inegal in (19). Fo < s <, and i = 1,, n 1/, R u (n) = n 1/ (x)v (n) (x)s φ(x)dw i (x) n 1/ s, R + n 1/ u (n),s R,s R s u (n) (19) (x)v (n) (x)s s φ(x)dw i (x) (x)v (n) (x)s φ(x)dw i (x) () u (n) (x)v (n) (x)(s φ(x) S s φ(x))dw i (x) Fo he fouh momen of he fis em on he igh hand side in () we obain using fis he Bukholde-Davis-Gundy inequaliy, hen Jensen s inequaliy and finally he fouh 13

14 momen bound, Lemma.1 fo ϱ < 1, E (n 1/ s, R u (n) ) (x)v (n) (x)s φ(x)dw i 4 (x) ( 1 ) C φ 4 ( s) E nu (n) (x)v (n) (x)ddx s s, R ( ) C(φ)( s)e u n (x)v n (x)dx d C(φ, γ, ϱ)( s) s Similaly, using he bound S φ S s φ φ ( s s 1 1) and he Bukholde-Davis- Gundy inequaliy, we have ha he expecaion of he fouh powe of he second em on he igh hand side in ( is bounded above by E (n 1/,s R u (n) ( φ s E R ) (x)v (n) (x)(s φ(x) S s φ(x))dw i 4 (x) Now, noe if s, ), ha by an explici calculaion R ) nu n (nx)v n (nx)dx ( s (s ) 1)d (1) s s (1 s (s ) )d = ( s) ( s) s ( s). () In paicula, if we define f() = 1 s (s ), hen we can ewie he lef hand side (1), and hen apply Jensen and finally he fouh momen bound, φ ( s f()d ) ( 1 s ) E s f()d nu n (nx)v n (nx)dx f()d R s s ( ) φ f()d E u n (x)v n (x)dx f()d R ( s C(φ, γ, ϱ) f()d) 4C(φ, γ, ϱ) s, by he esimae (). Moeove, if s, 1, so ha in paicula s s, we find ha s f()d = s and he same agumen shows ha he lae expession is bounded by C(φ, γ, ϱ)s C(φ, γ, ϱ)( s). Combining he fouh momen esimaes of he wo ems in (), one can deduce ha ) E (n 1/ u (n) (x)v (n) (x)s φ(x)dw i 4 (x) C(φ, ϱ, γ)( s), s, R confiming ha he sochasic inegal saisfies Kolmogoov s ighness cieion, and hus compleing he poof. 14

15 .3 Tighness of he measue-valued pocesses on pah space In his secion we will pove ighness of he measue-valued pocesses (µ n ) and (ν n ) in he Skoohod pah space on he space of empeed measues, see Appendix A.1 fo a discussion of hese spaces. A nice exposiion of he geneal saegy in he same seing of empeed measues can be found in DEF +, Secion 4.1. We sa wih a unifom bound on he fis momens of u (n) inegaed agains a suiable es funcion. Lemma.4. Fo any ϱ and fo each T > and ϕ C ap, sup E n and analgously fo u (n) eplaced by v (n). sup T u (n), ϕ <, Poof. We can assume ha ϕ C λ (R) fo some λ >, hen i suffices o veify he saemen fo ϕ λ (x) = e λ x, x R since φ λ φ λ φ λ, see also he discussion in Appendix A.1. In fac, i even suffices o check he claim fo ψ λ defined via (9) as he mollified vesion of φ λ (by inequaliy (3)). Recall, ha he escaled soluion u (n) is a soluion of he symbiic banching model, whee he banching ae γ is eplaced by nγ, see EF4, Lemma 8. In paicula, u (n) saisfies a suiable maingale poblem, see EF4, Definiion 3), moe pecisely M 1,n (ϕ) := u (n), ϕ u (n), ϕ u (n) s, 1 ϕ ds, is a coninuous squae-inegable maingale wih a covaiance sucue given by M 1,n = γ Hence, we can esimae he fis momen by E sup T u (n), ψ λ E sup T nϕ (x)u (n) M 1,n (ψ λ ) + u (n), ψ λ (x)v (n) (x) dx d + T Eu (n) s, 1 ψ λ ds. (3) We deal wih each of he summands on he igh hand side sepaaely. The second summand is bounded since he iniial densiy is bounded by 1. The las summand is conolled, since fis of all, by (3) hee exiss c λ such ha ψ λ (x) c λ e λ x fo all x R. Secondly, Eu (n) s (x) = S n su (nx) 1. Finally, we conside he fis summand in (3). Using fis Bukholde-Davis-Gundy and hen Jensen, we obain E sup T M 1,n (ψ λ ) E ( M 1,n (ψ λ ) ) 1/ ( = E R ( ) 1/ E M 1,n (ψ λ ) 1/. nψλ (x)u(n) (x)v (x)dxd) (n) 15

16 Now, using he paicle dualiy we can wie he lae expecaion as E nψλ (x)u(n) (x)v (n) (x)dxd = nψλ (x)e (,)1 R (B (1) n s + nx)1 R + (B () + nx) n exp{γϱl1, s n s }dxds R = ψλ (x)e (,)1 R (B s (1) + x)1 R+ (B s () + x)n exp{γϱnl 1, s }dxds, R whee we used he Bownian scale invaiance in he las sep. We can coninue o esimae using a simple applicaion of Tanaka s fomula, see Lemma A.6, o ge an uppe bound sup x {ψ λ (x)} sup x E (,) 1 R (B s (1) + x)1 R+ (B s () + x)n exp{γϱnl 1, s } dxds R {ψλ (x)} E (,) (B () s B s (1) ) + n exp{γϱnls 1, } ds sup{ψλ (x)} x = sup{ψλ (x)}e x E (Bs ) + n exp{γϱnl s} ds 1 γ ϱ (1 eϱγnl ), whee (L s) s is he local ime of a single Bownian in zeo. This expession is clealy bounded since ϱ, which complees he poof. Now, we can combine he pevious lemma wih he ighness of he coodinae funcions o show he ighness of he measue-valued pocesses. Lemma.5. The measue-valued pocesses {µ n, } n N and {ν n, } n N ae igh on he Skoohod space D, ) (M em ) on he space of empeed measues. Poof. By a sandad agumen, known as Jakubowski s cieion, see fo example Daw93, Thm , ighness follows in he Skoohod space if we can show a a compac conainmen condiion ogehe wih ighness of he coodinae funcions. To show he compac conainmen condiion, we define he elaive compac subse K = K((c m ) m 1 ) := {ν M em : ν, φ 1/m c m, m 1}, whee (c m ) m 1 is a sequence of posiive numbes. Then, given ε > and any m N, we can find by Lemma.4, a numbe c m > such ha fo all n N, { } P u (n), φ 1/m c m ε m, sup T In paicula, i follows ha fo all n N The same saemen also holds fo v (n). P { u (n) K((c m ) m 1 ) fo all, T } 1 ε. (4) 16

17 Secondly, we need ighness of { φ, u (n) : } n Z+ and { φ, v (n) : } n Z+ (5) fo any es funcion φ C ap, which we aleady showed in Lemma.3. Hence, he compac conainmen condiion (4) combined wih he ighness of he coodinae funcions (5) yields ighness of he measue-valued pocesses (µ n ) and (ν n ) on he space D((, ), M em ). Since all ou pocesses ae coninuous, ighness also follows in he C- space..4 Bounds on he widh of he ineface In his secion, we will pove he ph momen esimae on he appoximae widh of he ineface (R (ε) L (ε)) of Theoem 1.6 using he fouh momen esimaes esablished in Lemma.1 gives a bound on he widh of he ineface. We ecall ha { x } L (ε) = inf x : u (y)v (y)dy ε R(). s and similaly, fo he igh end poin { R (ε) = sup x : x } u (y)v (y)dy ε L(). Poof of Theoem 1.6. Fis, we ecall fom (17) in he poof of Lemma.1 ha since ϱ < 1, we have ha fo any ε (, 1 ), hee exiss a consan C = C(γ, ϱ) > such ha fo all z > and all, E 1l R,1l R + u (x)v (x)u (x + z)v (x + z) dx R = E 1l R,1l R + u (x)v (x)u (x z)v (x z) dx (6) If we define fo q (, 1), I q () = R R C(1 z (1 ε) ). R x y q u (x)v (x)u (y)v (y) dx dy, and choosing ε = 1 4 (1 q), he esimae in (6) shows ha E 1l R,1l R+ I q () = z q E 1l R,1l R + C R u (x)v (x)u (x + z)v (x + z) dx dz ( z q (1 z (1 ε) ) dz C(γ, ϱ) ) z + ε+q dz < fo all, since by ou choice of ε, we have ha ε + 1 = q < 1. Fix z >, hen on he even ha R (ε) L (ε) > z, we can esimae using he definiion of L (ε), R (ε) ha L(ε) I q () z q u (x)v (x)dx u (y)v (y)dy ε z q. R (ε) 17

18 Hence, we can conclude ha P 1l R,1l R+ {R (ε) L (ε) > z} ε z q E 1l R,1l R+ I q ()1l {R(ε) L(ε)>z} ε z q E 1l R,1l R+ I q () C(q, γ, ϱ)ε z q. Thus, we have by Fubini ha fo any < p < q < 1, E 1l R,1l R + ((R (ε) L (ε)) + ) p = p z p 1 P{R (ε) L (ε) > z} dz C(q)pε z p q 1 dz, which shows ha he p-h momen is finie. A Appendix A.1 Maingale poblems and Geen funcion epesenaions The following wo chaaceizaions of soluion o he symbioic banching model can be found in EF4 and will be impoan ools in ou invesigaion. To sae hem popely, howeve, we fis need o collec a consideable amoun of noaion. Fo λ R, le φ λ (x) := e λ x, x R, and fo f : R d R le f λ = f/φ λ, whee is he supemum nom. Denoe by B λ he space of all measuable funcions f : R d R wih f λ <, and such ha f(x)/φ λ (x) has a finie limi as x. Inoduce he spaces B ap = B ap (R d ) = λ> B λ and B em = B em (R d ) = λ> B λ, (7) of exponenially deceasing and empeed measuable funcions on R d especively. We wie C λ, C ap, C em fo he especive subspaces of coninuous funcions. Fo each λ R, he linea space C λ equipped wih he nom λ is a sepaable Banach space, and he space C ap is opologized by he meic d C ap(f, g) = n ( f g n 1), f, g C ap (8) n=1 which uns i ino a Polish space. Finally, C em is Polish if we opologize wih he analogous meic wih f, g C em. We also need o use he smoohed vesion of φ λ, see e.g. Secion.1 in DEF + fo a discussion of he elevan facs. fo his eason conside he mollifie ϱ(x) = c ϱ 1l { x 1} exp{ 1(1 x )}, x R, whee c ϱ is such ha ϱ is a pobabiliy densiy. Then, he ψ λ, he mollfied vesion of φ λ is defined as ψ λ (x) := φ λ (y)ϱ(y x) dy. (9) R 18

19 We will also need he following esimae fo he deivaives of ψ λ : fo any λ >, n N, hee exis consans c λ,n, c λ,n > such ha c λ,n φ λ (x) n x n ψ λ(x) c λ,nφ λ (x) fo all x R. (3) Le M = M(R d ) denoe he se of non-negaive Radon measues µ on R d and le d be a complee meic on M inducing he vague opology. We idenify µ wih is densiy if i exiss, and use he noaion µ, f fo he inegal of he funcion f wih espec o he measue µ. Denoe by M F (R d ) he space of finie non-negaive Radon measues µ on R d. We need he space M em = M em (R d ) of all measues µ in M such ha µ, φ λ < fo all λ >, and opologize his se of empeed measues by he meic d M em = d (µ, ν) + n ( µ ν 1/n 1), µ, ν M em (31) n=1 whee µ ν λ = µ, φ λ ν, φ λ. Noe ha (M em, d M em) is also Polish. Wie C = C((, ), (C + em ) ) fo he se of all coninuous pahs f in (C + em ) whee ((C + em ), (d C em) ) is defined as he Caesian poduc of (C + em, dc em). When endowed wih he meic d C (f, f ) = ( ( n sup (d C em) (f, f ) 1) ), f, f C, (3) 1/n n n=1 C is a Polish space. Le M 1 (C) denoe he se of all pobabiliy measues on C. Equipped wih he Pohoov meic d M1 (C), M 1 (C) is also a Polish space. Define C((, ), (C + ap) ) analogously. Similaly, given any Polish space S, one can un he space D, ) (S) of càdlàg pahs on S ino a Polish space using he usual Skoohod meic, see e.g. EK86. We define andom objecs ove a sufficienly lage sochasic basis (Ω, F, F, P) saisfying he usual hypoheses. If Y = {Y : } is a sochasic pocess, he law of Y is denoed P Y, and we use F Y o denoe he compleion of he σ-field ɛ> σ {Y s : s + ɛ},. Le p = p κ denoe he hea kenel in R elaed o 1, p (a) = } 1 exp { a, >, a R d, (33) (π) 1/ wie S = {S : } fo he semigoup of he associaed Bownian moion. Definiion A.1. The Symbioic Banching model in R is chaaceized via he following maingale poblem. Fix ϱ 1, 1 and (u, v ) (B em + ) (esp. (B ap) + ). A sochasic pocess (u, v ), wih law P (u,v ) on he pah space C((, ), (C em + ) ) (esp. C((, ), (C ap) + )) is a soluion o he maingale poblem fo Symbioic Banching if fo each es funcion φ C ap () (esp. C () em ), 19

20 M u (φ) = φ, u φ, u κ φ, u s ds,, (analogously fo v) is a pai (M u (φ), M v (φ)) of coninuous squae-inegable maingales null a zeo wih covaiance sucue whee M k (φ), M l (φ) = ϱ kl γ φ(x) L u,v (ds, dx) { 1 k = l (i.e. k = l = u o k = l = v), ϱ kl = ϱ k l. We poceed wih he Geen funcion epesenaion, see EF4, Coollay 19. Poposiion A.. Fo φ C ap (esp. C em ), k = 1,, and, φ, u = S φ, u + M u (d(s, a))s s φ(a) (34), R (simila fo M v ) whee M u (d(s, a)), M v (d(s, a)) is a pai of zeo-mean maingale measues wih covaiance sucue M k (d(s, a)))fs k (a), M l (d(s, a))fs(a) l = γϱ kl ds u s v s, fs k fs, l (35), R, R fo T and k, l {u, v} and f u, f v belong o he se of pedicable funcions f defined on Ω R + R such ha E x ds u s v s, (f s ) <, T. (36) A. Sandad esimaes fo Bownian moion and is local ime In his secion, we ecall some of he sandad facs (and is vaiaions) on Bownian moion in a fomulaion adaped o ou needs. Lemma A.3. If (B ) is a Bownian moion saed in x R wih local ime (L ) in, hen (L ) d = ((M ) + ), whee (M ) is he maximum pocess of a Bownian moion saed in x. Poof. We adap he poof of Theoem 7.38 in MP1. By Tanaka s fomula MP1, Thm. 7.33, we find ha B() x = sign(b(s))db(s) + L ().

21 I is clea ha he sochasic inegal is in disibuion equal o a Bownian moion saed in, so if we se W () = ( x + sign(b(s))db(s) ), hen W is a linea Bownian moion saed a x and we have ha B = W + L, (37) Le (M ) denoe he maximum pocess of (W ). We wan o show ha fo all, we have ha M = L. I follows immediaely fom (37) ha fo any s, W s L s L, so ha by aking he maximum we obain ha M L. Now, suppose hee exiss a ime such ha M() + < L(). Le s = inf{ < : L() = L()}. Since L only inceases on he se {s : B(s) = }, by coninuiy and since L () >, we have ha u > and so B(u) =. In paicula, i follows W (u) = L (u) and u <. Thus, we can deduce ha M(u) W (u) = L (u) = L () > M(s), which yields a conadicion since u < s and M is obviously inceasing. Hence, M + = L as claimed. Lemma A.4. Le B be a Bownian moion saed in z R and denoe by L ime in. Then, fo all > P z {L α log + z α log }. π 1 is local Poof. Using Lemma A.3, we find ha if M denoes he maximum pocess, hen we can esimae P z {L α log } = P z {M + α log } = P {M α log + z } α log + z = P { B α log + z }, π whee we used he eflecion pinciple, see e.g. MP1, Thm..1, in he second o las sep. Coollay A.5. Suppose ha (B 1 ), (B ) ae indepeden Bownian moions saed in x < y especively and denoe he collision local ime as (L 1, ). Then, P x,y {L 1, α log } 1 α log + y x. π Poof. This follows immediaely fom Lemma A.4. Noe ha W := B B 1, is by definiion a Bownian moion (wih quadaic vaiaion and saed in y x) and hus B = W / (y x), is a sandad Bownian moion. Moeove, L 1, = L (B B 1 ) = L (W ). Now, L 1 (W ) = lim ε ε d 1 = lim ε ε 1 1l { Ws ε}ds = lim ε ε 1 1l ds = 1 { Bs+y x ε} L 1 1 1l { Bs +y x ε}ds x y (B).

22 Hence, x y P x,y {L 1, α log } = P {L α log } = P y x {L α log } α log + 1 (y x), π which poves he coollay. Lemma A.6. Fo (B ) a Bownian moion and L is local ime in, we have ha fo x >,, E x B + e βl = 1 β E x(1 e βl ) Poof. Fis of all, inegaion by pas yields B + e βl = x + 1 e βl db + s Now, by Tanaka s fomula we have ha B + +B i wih he inegaion by pas and he fac B + β B + s e βl s dl s. = B = sign(b s) db s +L. Combining = d B, we obain E x B + e βl = x + 1 E x e βl ssign(b s ) db s + E x e βl s( 1 βb+ s ) dl s, = x + 1 β E x(1 e βl ) whee we used in he las sep ha all he ohe expessions ae eihe zeo o have zeo expecaion (he sochasic inegal). Refeences BDE11 Daw93 J. Blah, L. Doeing, and A. Eheidge. On he momens and he wavespeed of he symbioic banching model. Ann. Pobab., 39(1):5 9, 11. D. A. Dawson. Measue-valued Makov pocesses. In École d Éé de Pobabiliés de Sain-Flou XXI 1991, volume 1541 of Lecue Noes in Mah., pages 1 6. Spinge, Belin, DEF + D. A. Dawson, A. M. Eheidge, K. Fleischmann, L. Mynik, E. A. Pekins, and J. Xiong. Muually caalyic banching in he plane: infinie measue saes. Elecon. J. Pobab., 7:No. 15, 61 pp. (eleconic),. DM11a DM11b L. Döing and L. Mynik. Longime behavio fo muually caalyic banching wih negaive coelaions. Pepin, 11. axiv: L. Döing and L. Mynik. Muually caalyic banching pocesses on he laice and voe pocesses wih sengh of opinion. Pepin, 11. axiv: DP98 D. A. Dawson and E. A. Pekins. Long-ime behavio and coexisence in a muually caalyic banching model. Ann. Pobab., 6(3): , 1998.

23 EF4 EK86 A. M. Eheidge and K. Fleischmann. Compac ineface popey fo symbioic banching. Sochasic Pocess. Appl., 114(1):17 16, 4. S. N. Ehie and T. G. Kuz. Makov pocesses. Wiley Seies in Pobabiliy and Mahemaical Saisics: Pobabiliy and Mahemaical Saisics. John Wiley & Sons Inc., New Yok, Chaaceizaion and convegence. KM1 A. Klenke and L. Mynik. Infinie ae muually caalyic banching. Ann. Pobab., 38(4): , 1. KM11a KM11b A. Klenke and L. Mynik. Infinie ae muually caalyic banching in infiniely many colonies: consucion, chaaceizaion and convegence. Pobabiliy Theoy and Relaed Fields, 11. To appea. 1.17/s A. Klenke and L. Mynik. Infinie ae muually caalyic banching in infiniely many colonies: The longime behaviou. Ann. Pobab., 11. To appea. AXiv: KO1 A. Klenke and M. Oele. A Toe-ype appoach o infinie ae muually caalyic banching. Ann. Pobab., 38(): , 1. MP1 MT97 Ti95 P. Möes and Y. Pees. Bownian moion. Cambidge Seies in Saisical and Pobabilisic Mahemaics. Cambidge Univesiy Pess, Cambidge, 1. Wih an appendix by Oded Schamm and Wendelin Wene. C. Muelle and R. Tibe. Finie widh fo a andom saionay ineface. Elecon. J. Pobab., :no. 7, 7 pp. (eleconic), R. Tibe. Lage ime behavio of ineface soluions o he hea equaion wih Fishe-Wigh whie noise. Pobab. Theoy Relaed Fields, 1(3):89 311,