SUPERCRITICAL BRANCHING DIFFUSIONS IN RANDOM ENVIRONMENT

Elc. Comm. in Proa. 6 (), 78 79 ELECTRONIC COMMUNICATIONS in PROBABILITY SUPERCRITICAL BRANCHING DIFFUSIONS IN RANDOM ENVIRONMENT MARTIN HUTZENTHALER ETH Zürich, Dparmn of Mahmaics, Rämisrass, 89 Zürich. mail: hunhalr.marin@gmx.d Sumid July 5,, accpd in final form Ocor 3, AMS Sujc classificaion: 6J8; 6K37; 6J6. Kywords: Branching procss; random nvironmn; BDRE; suprcriicaliy; survival proailiy. Asrac Suprcriical ranching procsss in consan nvironmn condiiond on vnual xincion ar known o sucriical ranching procsss. Th cas of random nvironmn is mor sul. A suprcriical ranching diffusion in random nvironmn (BDRE) condiiond on vnual xincion of h populaion is no a BDRE. Howvr h law of h populaion si of a suprcriical BDRE (avragd ovr h nvironmn) condiiond on vnual xincion is qual o h law of h populaion si of a sucriical BDRE (avragd ovr h nvironmn). As a consqunc, suprcriical BDREs hav a phas ransiion which is similar o a wll-known phas ransiion of sucriical ranching procsss in random nvironmn. Inroducion and main rsuls Branching procsss in random nvironmn (BPREs) hav aracd considral inrs in rcn yars, s.g. [3,, 7] and h rfrncs hrin. On h on hand his is du o h mor ralisic modl compard wih classical ranching procsss. On h ohr hand his is du o inrsing propris such as a phas ransiion in h sucriical rgim. L us rcall his phas ransiion. In h srongly sucriical rgim, h survival proailiy of a BPRE (Z () ) scals lik is xpcaion, ha is, P Z () > cons E Z () as whr cons is som consan in (, ). In h wakly sucriical rgim, h survival proailiy dcrass a a diffrn xponnial ra. Th inrmdia sucriical rgim is in wn h ohr wo cass. Undrsanding h diffrncs of hs hr rgims is on moivaion of h liraur cid aov. Th main osrvaion of his aricl is a similar phas ransiion in h suprcriical rgim. L us inroduc h modl. W considr a diffusion approximaion of BPREs as his is mahmaically mor convnin. Th diffusion approximaion of BPREs is du o Kur (978) and had n conjcurd (slighly inaccuraly) y Kiding (975). W follow Böinghoff and Hunhalr () and dno his diffusion approximaion as ranching diffusion in random nvironmn (BDRE). For vry n N := {,,...}, l (Z (n) ) k k N a ranching procss in h UNIVERSITY OF MUNICH (LMU), GERMANY. 78

78 Elcronic Communicaions in Proailiy random nvironmn Q (n),q(n),... which is a squnc of indpndn, idnically disriud offspring disriuions. If m Q (n) (n) k dnos h man offspring numr for k N, hn S := k k n i= log m Q (n) i, k N := {,,,...} dnos h associad random walk whr n N. S := max{m N : m } for. L h nvironmn such ha S (n) n / n convrgs o a Brownian moion (S ) wih infinisimal drif α R and infinisimal sandard dviaion [, ) as n. Furhrmor assum ha h man offspring varianc convrgs o [, ), ha is, lim E n k= k m Q (n) (n) Q (k) =. () If Z (n) /n [, ) as n and if a hird momn condiion holds, hn Z (n) n n (n) S, n n w n Z, S () in h Skorohod opology (s.g. [8]) whr h limiing diffusion is h uniqu soluion of h sochasic diffrnial quaions (SDEs) d Z = Z d + Z ds + ds = αd + () dw Z dw () for whr Z = and S =. Th procsss (W () ) and (W () ) ar indpndn sandard Brownian moions. Throughou h papr h noaions P and E rfr o Z = and S = for [, ). Th diffusion approximaion () is du o Kur (978) (s also [5]). No ha h random nvironmn affcs h limiing diffusion only hrough h man ranching varianc and hrough h associad random walk. W dno h procss (S ) as associad Brownian moion. This procss plays a cnral rol. For xampl i drmins h condiional xpcaion of Z E Z (S s ) s = xp S (4) for vry [, ) and. Morovr h infinisimal drif α of h associad Brownian moion drmins h yp of criicaliy. Th BDRE (3) is suprcriical (i.. posiiv survival proailiy) if α >, criical if α = and sucriical if α <, s Thorm 5 of Böinghoff and Hunhalr (). W will rfr o α as criicaliy paramr. Afanasyv (979) was h firs o discovr diffrn rgims for h survival proailiy of a BPRE in h sucriical rgim (s [3, 4, 3, ] for rcn aricls). Th following characrisaion for h BDRE (3) is du o Böinghoff and Hunhalr (). Th survival proailiy of (Z ) dcays lik h xpcaion, ha is, P(Z > ) cons E(Z ) = cons xp (α+ ) as, if and only if α < (srongly sucriical rgim). In h inrmdia sucriical rgim α =, w hav ha P(Z > ) cons xp as. Finally h survival proailiy dcays lik P(Z > ) cons 3 xp α as in h wakly sucriical rgim α (, ). This aricl concnras on h suprcriical rgim α >. Our main osrvaion is ha hr is a phas ransiion which is similar o h sucriical rgim. Such a phas ransiion has no (3)

Suprcriical ranching diffusions in random nvironmn 783 n rpord for BPREs y. W condiion on h vn {Z = } = {lim Z = } of vnual xincion and propos h following noaion. If P(Z > Z = ) cons E(Z Z = ) as, hn w say ha h BDRE (Z, S ) is srongly suprcriical. If h proailiy of survival up o im condiiond on vnual xincion dcays a a diffrn xponnial ra as, hn w rfr o (Z, S ) as wakly suprcriical. Th inrmdia rgim is rfrrd o as inrmdia suprcriical rgim. Our firs horm provids h following characrisaion. Th BDRE is srongly suprcriical if α >, inrmdia suprcriical if α = and wakly suprcriical if α (, ). Thorm. Assum α,, (, ). L (Z, S ) h uniqu soluion of (3) wih S =. Thn α 3 lim P Z > Z = = 8 3 f (a)φ β (a) da > if α (, ) (5) lim P Z > Z = = lim α P Z > Z = for vry (, ) whr β := α φ β (a) = for vry a (, ). π Γ > if α = π (6) = α > if α > (7) and whr φ β : (, ) (, ) is dfind as β + a a β/ u (β )/ u sinh(ξ) cosh(ξ)ξ dξ du (8) (u + a(cosh(ξ)) )(β+)/ Th proof is dfrrd o Scion. L us rcall h havior of Fllr s ranching diffusion, ha is, (3) wih =, which is a ranching diffusion in a consan nvironmn. Th suprcriical Fllr diffusion condiiond on vnual xincion agrs in disriuion wih a sucriical Fllr diffusion. This is a gnral propry of ranching procsss in consan nvironmn, s Jagrs and Lagrås (8) for h cas of gnral ranching procsss (Crump-Mod-Jagrs procsss). Knowing his, Thorm migh no surprising. Howvr, h cas of random nvironmn is diffrn. I urns ou ha h suprcriical BDRE (Z, S ) condiiond on {Z = } is a wo-dimnsional diffusion which dos no saisfy h SDE (3) and is no a ranching diffusion in homognous random nvironmn if > and if >. Mor prcisly, h associad Brownian moion (S ) condiiond on {Z = } has drif which dpnds on h currn populaion si. Thorm. L (, ), l, [, ) and assum + >. If Z, S is h soluion of (3) wih criicaliy paramr α (, ), hn Z, S Z = = Ž, Š (9) whr Ž, Š is a wo-dimnsional diffusion saisfying Ž = Z, Š = and d Ž = α Ž Ž + d + Ž dš + dw () Ž dš = Ž α α Ž + d + () dw ()

784 Elcronic Communicaions in Proailiy for. Th proof is dfrrd o Scion. I is rahr inuiiv ha h condiiond procss is no a sucriical BDRE if >. Th suprcriical BDRE has a posiiv proailiy of xincion. Thus xincion dos no rquir h associad Brownian moion o hav ngaiv drif. As long as h BDRE says small, xincion is possil dspi h posiiv drif of h associad Brownian moion. No ha if Ž is small for som, hn h drif rm of Š is clos o α. Bing doomd o xincion, h condiiond procss (Ž ) is no allowd o grow o infiniy. If Ž is larg for som, hn h drif rm of Š is clos o α which lads a dcras of (Ž ). Th siuaion is rahr diffrn in h cas =. Thn h xincion proailiy of h BDRE is ro. So h drif of (Š ) nds o ngaiv in ordr o guaran Ž as. I urns ou ha if =, hn h drif of (Š ) is α and (Ž, Š ) is a sucriical BDRE wih criicaliy paramr α. W hav sn ha condiioning a suprcriical BDRE on xincion dos in gnral no rsul in a sucriical BDRE. Howvr, if w condiion (Z, S ) on {S = }, hn h condiiond procss urns ou o a sucriical BDRE wih criicaliy paramr α. Thorm 3. L (, ), l, [, ) and assum + >. L Z (α), S (α) h soluion of (3) wih criicaliy paramr α R. If α >, hn, S (α) S = =, S ( α) () whr Z ( α) = Z (α). Z (α) Z ( α) Th proof is dfrrd o Scion. Now w com o a somwha surprising osrvaion. W will show ha h law of (Z ) condiiond on vnual xincion agrs in law wih h law of h populaion si of a sucriical BDRE. Mor formally, insring h scond quaion of () ino h quaion for dž w s ha d Ž = α Ž d + Ž dw () + dw () Ž () for. This is h SDE for h populaion si of a sucriical BDRE wih criicaliy paramr α. As h soluion of () is uniqu, his provs h following corollary of Thorm. Corollary 4. L (, ), l, [, ) and assum + >. L Z (α), S (α) h soluion of (3) wih criicaliy paramr α for vry α R. If α >, hn h law of h BDRE wih criicaliy paramr α condiiond on xincion agrs wih h law of h BDRE wih criicaliy paramr α, ha is, (Z (α) ) Z = = (Z ( α) ) (3) whr Z ( α) = Z (α). So far w considrd h vn of xincion. Nx w condiion h BDRE on h vn {Z > } := { lim Z = } of non-xincion. Dfin U : [, ) [, ) y U() := + for [, ). W agr on h convnion ha α (4) c := if c (, ] if c = c := for c [, ) and ha =. (5)

Suprcriical ranching diffusions in random nvironmn 785 Thorm 5. L (, ), l, [, ) and assum + >. L Z, S soluion of (3) wih criicaliy paramr α >. Thn Z, S Z > = Ẑ, Ŝ whr Ẑ, Ŝ is a wo-dimnsional diffusion saisfying Ẑ = Z, Ŝ = and d Ẑ = dŝ = + α Ẑ + Ẑ α + α Ẑ + U Ẑ U() U Ẑ d + Ẑ dŝ + dw () Ẑ Ẑ U Ẑ U() U Ẑ d + () dw for. Th law of (Z ) condiiond on non-xincion saisfis ha (Z ) Z > = (Ẑ ) h (6) (7) (8) whr Ẑ is h soluion of h on-dimnsional SDE saisfying Ẑ = Z and for. dẑ = + α + α U Ẑ U() U Ẑ d + Ẑ dw () + dw () Ẑ (9) Ẑ Th proof is dfrrd o Scion. On h vn of non-xincion, h populaion si Z of a suprcriical BDRE grows lik is xpcaion E(Z S ) as. Thorm 6. L (, ), l, [, ) and assum + >. L (Z, S ) h soluion of (3) wih criicaliy paramr α R. Thn Z / S is a nonngaiv maringal. Consqunly for vry iniial valu Z = [, ) hr xiss a random varial Y : Ω [, ) such ha Z S Y as almos surly. () Th limiing varial is ro if and only if h BDRE gos o xincion, ha is, P (Y = ) = P (Z = ). In h suprcriical cas α >, h disriuion of h limiing varial Y saisfis ha E xp ( λy ) = E xp for all, λ [, ) whr G ν is gamma-disriud wih shap paramr ν (, ) and scal paramr, ha is, P(G ν d x) = Γ(ν) x ν x d x () for x (, ). Th proof is dfrrd o Scion. In paricular, Thorm 6 implis ha Z := lim Z xiss almos surly and ha Z {, } almos surly. G α + λ ()

786 Elcronic Communicaions in Proailiy Proofs If = and Z >, hn h procss (Z ) dos no hi in fini im almos surly. So h inrval (, ) is a sa spac for (Z ) if =. Th following analysis works wih h sa spac [, ) for h cas > and wih h sa spac (, ) for h cas =. To avoid cas-y-cas analysis w assum > for h rs of his scion. On can chck ha our proofs also work in h cas = if h sa spac [, ) is rplacd y (, ). Insring h associad Brownian moion (S ) ino h diffusion quaion of (Z ), w s ha (Z ) solvs h SDE d Z = α + Z d + Z () dw + Z dw () (3) for [, ). On-dimnsional diffusions ar wll-undrsood. In paricular h scal funcions ar known. For h rason of complnss w driv a scal funcion for (3) in h following lmma. Th gnraor of (Z, S ) is h closur of h prgnraor : C ([, ) R) C([, ) R) givn y f (, s) := α + f (, s) + α s f (, s) + + + s f (, s) + f (, s) s for all [, ), s R and vry f C [, ) R. f (, s) Lmma 7. Assum,, α (, ). Dfin h funcions U : [, ) (, ) and V : R (, ) hrough U() := + α and V (s) := xp α s (5) for [, ) and s R. Thn U is a scal funcion for (Z ) and V is a scal funcion for (S ), ha is, U and V so U(Z ) and V (S ) ar maringals. (4) Proof. No ha U is wic coninuously diffrnial. Thus w g ha + α = α + α + α + α α + + = + α 4 + α α α + 4α + α (6) = for all [, ). Morovr V is wic coninuously diffrnial and w oain ha xp α s = α xp α α s + xp α α s = (7)

Suprcriical ranching diffusions in random nvironmn 787 for all s R. This shows U V. Now Iô s formula implis ha du(z ) = U(Z ) d + U (Z ) Z dv (S ) = V (S ) d + V (S ) dw () () dw + Z dw () (8) for all. This provs ha U(Z ) and V (S ) ar maringals. Lmma 8. Assum,, α (, ). Thn h smigroup of h BDRE (Z, S ) condiiond on xincion saisfis ha E (,s) f (Z, S ) Z = = E(,s) U(Z )f (Z, S ) U() h smigroup of h BDRE (Z, S ) condiiond on {S = } saisfis ha, (9) E (,s) f (Z, S ) S = = E(,s) V (S )f (Z, S ) V (s) (3) and h smigroup of h BDRE (Z, S ) condiiond on {Z > } saisfis ha E (,s) f (Z, S ) Z > = E(,s) U() U(Z ) f (Z, S ) U() U() (3) for vry [, ), s R, and vry oundd masural funcion f : [, ) R R. Proof. Dfin h firs hiing im T x (η) := inf{ : η = x} of x R for vry coninuous pah η C [, ), R). As V is a scal funcion for (S ), h opional sampling horm implis ha P s T N (S) < = lim P s T N (S) < T K (S) V (K) V (s) = lim K K V (K) V ( N) = V (s) V ( N) for all s R and N N, s Scion 6 in [] for mor dails. Thus w g ha E (,s) f (Z, S ) S = = lim E (,s) f (Z, S ) T N (S) < N E (,s) f (Z, S )P S T N (S) < = lim N P (,s) T N (S) < = E(,s) f (Z, S )V (S ) V (s) (3) (33) for all [, ), s R and. Th proof of h assrions (9) and (3) is analogous. No for h proof of (3) ha for vry [, ). P Z > = P lim Z = = lim P T N (Z) < T (Z) U() U() = N U() (34)

788 Elcronic Communicaions in Proailiy Proof of Thorm. I suffics o idnify h gnraor ˇ of h condiiond procss. This gnraor is h im drivaiv of h smigroup of h condiiond procss a =. L f C [, ) R, R fixd. Dfin f (, s) := f (, s), f s(, s) := f (, s), f s (, s) := f (, s), f ss (, s) := f (, s) and f s s (, s) := f (, s) for [, ) and s R. Lmma 8 implis ha s ˇ f (, s) E (,s) U(Zh )f (Z h, S h ) U()f (, s) /U() (U f )(, s) = lim = h h U() = α + U f + U f (, s) + α U fs (, s) + U fss (, s) U() + + U f + U f + U f (, s) + U f s + U f s (, s) = f (, s) U() + ( U()) f (, s) U() U() + + U U U f (, s) + U f s (, s) (35) for all [, ) and s R. Now w xploi ha U and ha U () = U α/( + ) for [, ) o oain ha ˇ f (, s) = f (, s) α f (, s) α + f s (, s) = α + f (, s) + + f (, s) + α α + f s (, s) + f ss(, s) + f s(, s) (36) for all [, ), s R and all f C [, ) R, R. This is h gnraor of h procss (). Thrfor h BDRE condiiond on xincion has h sam disriuion as h soluion of (). Proof of Thorm 3. As in h proof of Thorm w idnify h gnraor condiiond on {S = }. Similar argumns as in (35) and V rsul in f (, s) = f (, s) + V V f s (, s) + V = f (, s) αf s (, s) α f (, s) = α + V f f (, s) αf s (, s) + + (, s) of h BDRE f (, s) + f ss(, s) + f s(, s) for all [, ), s R and all f C [, ) R, R. This is h gnraor of h BDRE wih criicaliy paramr α.

Suprcriical ranching diffusions in random nvironmn 789 Proof of Thorm. Th assrion follows from Corollary 4 and from Thorm 5 of Böinghoff and Hunhalr (). Proof of Thorm 6. Iô s formula implis ha d Z S = S d Z S Z ds + S Z d S Z d = S Z d + S Z dw () + S Z d S Z d = S Z dw () (37) for all. Thrfor Z / xp(s ) is a nonngaiv maringal. Th maringal convrgnc horm implis h xisnc of a random varial Y : Ω [, ) such ha Z S Y as almos surly. (38) If α, hn Z = almos surly, which implis Y = almos surly. I rmains o drmin h disriuion of Y in h suprcriical rgim α >. Fix [, ) and λ [, ). Dufrsn (99) (s also [4]) showd ha xp αs W () d s ds = G α. (39) Morovr w xploi an xplici formula for h Laplac ransform of h BDRE (3) condiiond on h nvironmn, s Corollary 3 of Böinghoff and Hunhalr (). Thus w g ha E xp ( λy ) = lim E xp λ Z S = lim E E xp λ Z Ss S s [,] = lim E xp xp xp(s S s ds + ) xp( S λ ) (4) = E xp xp αs W () s ds + λ = E xp. G α/ + λ This shows (). Ling λ w conclud ha P (Y = ) = E xp Th las qualiy follows from Thorm 5 of [5]. G α/ = P Z =. (4) Proof of Thorm 5. Analogous o h proof of Thorm, w idnify h gnraor ˆ of h BDRE condiiond on {Z > }. No ha U () U() U() = α U() + U() U() (4)

79 Elcronic Communicaions in Proailiy for all [, ). Similar argumns as in (35) and U rsul in ˆ f (, s) = f (, s) + + U () U() U() f (, s) + U () U() U() f s(, s) U() = α + α U() U() + f (, s) + α + α U() + f s (, s) U() U() + + f (, s) + f ss(, s) + f s(, s) for all [, ), s R and all f C [, ) R, R. Comparing wih (7), w s ha ˆ is h gnraor of (7) which implis (6). Insring dŝ ino h quaion of d Ẑ for [, ) shows ha (Ẑ ) solvs h SDE (9). Acknowldgmn W hank wo anonymous rfrs for vry hlpful commns and suggsions. Rfrncs [] AFANASYEV, V. I. On h survival proailiy of a sucriical ranching procss in a random nvironmn. Dp. VINITI (979), No. M794 79 (in Russian). [] AFANASYEV, V. I., BÖINGHOFF, C., KERSTING, G., AND VATUTIN, V. A. Limi horms for a wakly sucriical ranching procss in a random nvironmn. o appar in J. Thor. Proa., DOI:.7/s959--33-6 (). [3] AFANASYEV, V. I., GEIGER, J., KERSTING, G., AND VATUTIN, V. A. Criicaliy for ranching procsss in random nvironmn. Ann. Proa. 33, (5), 645 673. MR36 [4] AFANASYEV, V. I., GEIGER, J., KERSTING, G., AND VATUTIN, V. A. Funcional limi horms for srongly sucriical ranching procsss in random nvironmn. Sochasic Procss. Appl. 5, (5), 658 676. MR65338 [5] BÖINGHOFF, C., AND HUTZENTHALER, M. Branching diffusions in random nvironmn. hp://arxiv.org/as/7.773v (). [6] DUFRESNE, D. Th disriuion of a prpuiy, wih applicaions o risk hory and pnsion funding. Scand. Acurial. J. 99, (99), 39 79. MR994 [7] DYAKONOVA, E. E., GEIGER, J., AND VATUTIN, V. A. On h survival proailiy and a funcional limi horm for ranching procsss in random nvironmn. Markov Procss. Rlad Filds, (4), 89 36. MR8575 [8] ETHIER, S. N., AND KURTZ, T. G. Markov procsss: Characriaion and convrgnc. Wily Sris in Proailiy and Mahmaical Saisics: Proailiy and Mahmaical Saisics. John Wily & Sons Inc., Nw York, 986. MR83885

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