AN EXTENDED CLASS OF TIME-CONTINUOUS BRANCHING PROCESSES. Rong-Rong Chen. ( University of Illinois at Urbana-Champaign)

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1 AN EXTENDED CLASS OF TIME-CONTINUOUS BRANCHING PROCESSES Rong-Rong Chen ( Unversty of Illnos at Urbana-Champagn Abstract. Ths paper s devoted to studyng an extended class of tme-contnuous branchng processes, motvated by the study of stochastc control theory and nteractng partcle systems. The unqueness, extncton, recurrence and postve recurrence crtera for the processes are presented. The man new pont n our proofs s the use of several dfferent comparson methods. The resultng pcture shows that the methods are effectve and hence should also be meanngful n other stuatons.. Introducton Branchng processes (or the Galton-Watson processes form one of the classcal felds of probablty theory and have a very wde range of applcatons. There are several specalzed books devoted to the subect (See [2], [3] and [7], for nstance. As s well-known, a branchng process can be descrbed as follows. Let α>andlet(p : be a probablty dstrbuton. Then the process has death rate αp : ( and growth rate αp + : ( >. Note that the process absorbs at state. Yamazato (975 [] consdered a modfed model, called a return branchng process, n whch the process startng from can stll ump to wthrateq, q := q <. The last model was then analyzed and mproved by Pakes and Tavaré (98 [9]. For the reader s convenence and also for later use, we summarze the related prevous results [7],[9],[] as follows. Theorem.. The branchng process s unque ff one of the followng condtons holds. ( M := k= kp k <. (2 M = and ε p(s = = p s. p(s s ds = for some (equvalently, for all <ε<, where Moreover, when p > and p > for some 2, the extncton probablty of the process s equal to ff M. Next, for the return branchng process, the condtons for the unqueness are the same as above. Fnally, assume that the process s unque and rreducble and set h = q /q. (4 Then the process s recurrent ff M. (5 It s postve recurrent ff ether M < and = h log < or M =but stll where h(s = = h s. h(s ds <, (. p(s s 99 Mathematcs Subect Classfcaton. 6J8, 6J85. Key words and phrases. Branchng processes, unqueness, recurrence, postve recurrence.

2 then the process s unque. Especally, under the assumpton M (,, thecondtons are fulflled (by settng w = r and w respectvely f r α + β ( for some α, β >, or = /r = and {r } s non-decreasng. 2 RONG-RONG CHEN We remark that the condtons ( and (2 can be unfed nto a sngle condton. That s, the ntegraton gven n (2 dverges for ε suffcently close to. But the above presentaton s more sutable for our purposes. The present study s motvated by two sources. The frst s the stochastc control theory. Recently, Gao (992 [6] has studed a tme-dscrete lnear controlled branchng process. In our context, the model smply replaces the above coeffcents α wth α + β for some β and to take q = p or p +,. Here, the appearance of the constant β s due to the control. The second source s the nteractng partcle systems. The dual of a measure-valued process often leads to a modfed model of the branchng processes. For nstance, the followng model comes from a typcal measure-valued process (the Flemng-Vot process. It was ntroduced to us by D. A. Dawson: q = p ( and the coeffcent α s replaced by 2 (. The above background leads us to consder an extended class of branchng processes wth Q-matrx: r p +, and r ( p, = q = q, > = (.2 q, = =, else,, Z +. where r for all. Of course, the typcal case we are nterested n s where q = p or p + ( and r s a polynomal wth degree θ. Before movng further, let us compare our model wth the classcal (or the return branchng processes. The Q-processes we are consderng dffer qute a lot from the classcal process whch absorbs at, and ts partcles splt ndependently. Hence, the process has a straghtforward expresson f (t = f r (tf r2 (t f r (t, (.3 r +r 2 + +r = where (f (t s the mnmal Q-process determned by ts Q-matrx. As for the return process, by usng the method of taboo probablty, we stll have an explct expresson. However, the formula (.3 does not hold for our model even n the lnear case (r = α + β, β > because when r α, the partcles no longer splt ndependently. Thus, the prevous method s useless now. Furthermore, t s mpractcal to apply the known crtera drectly due to the dffcultes n solvng the correspondng systems of equatons or nequaltes. Consequently, new methods must be adopted. Throughout ths paper, we use several comparson methods, ether comparng wth the classcal branchng processes or comparng wth some carefully desgned brth-death processes. It s nterestng that through these methods we stll obtan relatvely complete results. It s to be hoped that these methods wll have more applcatons n other stuatons. We now state the man results of the paper. Theorem.2 (Unqueness. Let Q =(q be the Q-matrx gven by (.2. ( If M, then the process s unque. (2 Let M (,. Gven a non-negatve sequence {w k } such that k= w k/r k =. Set J = { : p + > }. If sup sup J k= /( r w +k + r +k k= w k <, r k

3 AN EXTENDED CLASS OF TIME-CONTINUOUS BRANCHING PROCESSES 3 In partcular, f r const. θ and r > for all, then the process s unque ff (4 θ> and M or (5 θ =and condtons ( or (2 of Theorem. hold. We menton that the unqueness for the exceptonal case of M = and = /r = s actually ndefnte. See for nstance part (2 of Theorem.. The next result can be deduced from Theorem., t s not new but ncluded here for the sake of thoroughness. We wll also present a new and short proof of the theorem, based on the same comparson dea. Theorem.3. Let the process be unque. ( (Recurrence. Assume addtonally that the process s rreducble. Then t s recurrent ff M. (2 (Extncton. Let q =, p >, r > for all and p k > for some k 2. Then the extncton probablty of the process s equal to ff M. Theorem.4 (Postve Recurrence. Let the process be rreducble and M. ( Then the process s postve recurrent f = r < and h ( + r = = = <. The condtons are fulflled provded r const. θ,etherθ>2 or θ (, 2] but stll = h <. (2 If r const. θ (, θ, then the process s postve recurrent ff r h(s p(s s ( sθ ds <. (.4 In partcular, f θ =, then the same concluson of part (5 of Theorem. holds. (3 If h <, M < and lm r / >, then the process s exponentally ergodc. As a straghtforward consequence of the above theorems, we obtan the followng complete result whch refers to our typcal models. Corollary.5. Let r const. θ for some θ and r > for all and take q = p or p +,. Then the process s unque ff ether θ> and M, orθ =but condtons ( or (2 of Theorem. stll hold. Next, assume that the process s unque and rreducble. Then t s recurrent ff M. Furthermore, t s postve recurrent ff one of the followng condtons holds. ( θ>. (2 θ =and M <. (3 θ =, M =and s ds <. p(s s The respectve proofs of Theorems.2.4 are gven n the subsequent three sectons. 2. Unqueness Proof of Theorem.2. a We show that for the unqueness, t s enough to consder the case where q =. Todoso,let( q denote for a moment the Q-matrx whch concdes wth our Q-matrx (q except q =. Clearly, the unqueness of ( q -process follows from that of the (q -process snce the former s domnated by the latter. In detal, the maxmal soluton of the equaton u = q u, u,

4 4 RONG-RONG CHEN domnates that of the equaton u = q λ + q u, u,. Hence the asserton follows from the unqueness crteron. Conversely, let the ( q -process be unque. Then, the correspondng process (X t has at most a fnte number of umps n every fnte tme-nterval. Now, consder the mnmal process (X t determned by (q. After (X t reaches, t stays at satsfyng the exponental law (havng parameter q andthenumpstosome accordng to the dstrbuton (h. Ths s the only way n whch (X t yelds more umps than (X t. Due to the condtonal ndependence of the umps, the process (X t may have at most fntely many more umps than (X t n every fnte tme-nterval and so the (q -process should also be unque. The above observaton s due to [9]. Certanly, the same concluson holds f the sngle absorbng state s replaced by a fnte set of absorbng states. From now on, we gnore ( q and smply assume that q =. b To prove part (5 of Theorem.2, by a, we may choose α and ᾱ so that α r ᾱ,. Then, the concluson follows by comparng our process wth two classcal branchng processes wth coeffcents α and ᾱ respectvely, n the same way as we explaned n the frst paragraph of a. c We now prove parts ( and (2 of Theorem.2. The frst part s easy snce the process s ndeed recurrent as we wll see n the next secton. An alternatve proof goes as follows. Let Γ = k= kp k+. It s easy to check that Γ = M + p. Hence M Γ p and M = Γ=p.Takeϕ = +. Then q (ϕ ϕ = q, + =+ q ( =r (Γ p,. Hence q (ϕ ϕ ϕ for all. Now, as an applcaton of [4; Theorem.] or [5; Theorem 2.25] wth E n = {, 2,..., n} and the above choce of ϕ, we obtan the unqueness of the Q-process. Smlarly, one can prove part (2 of the theorem wth ϕ =+ k= w k/r k. d It remans to prove part (3. Let M (, ] and = /r <. The man dea s to compare our process wth a brth-death process. For ths, let Γ (p, Γ, whch wll be determned later, and consder the brth-death process ( p (t wth brth rate b =,b = r Γ and death rate a = r p,. Snce ( k= b k + k = a + a k b b k ( = Γ r k k= t follows that ( p (t s not unque. Hence the equaton k= ( p Γ k <, a (ϕ ϕ +b (ϕ + ϕ =cϕ ϕ =,c>, has a non-negatve bounded soluton. Moreover, t s easy to check that ϕ + ϕ for all. Obvously, we also have ϕ + ϕ p Γ (ϕ ϕ,. (2. Next, snce p k ( l p kp =Γ (, ], as Γ p,

5 x π k x k, x (3.2 AN EXTENDED CLASS OF TIME-CONTINUOUS BRANCHING PROCESSES 5 we can fnd some Γ (p, Γ so that k= By (2. and (2.2,wehavefor, k ( p p k+ l= q (ϕ ϕ =a (ϕ ϕ +r Γ l Γ. (2.2 p k+ k= k (ϕ +l ϕ +l l= a (ϕ ϕ +r (ϕ + ϕ a (ϕ ϕ +b (ϕ + ϕ = cϕ. k= k ( l p p k+ By the comparson lemma ([4; Lemma 3.] and [; 2 Lemma ], or [5; Lemma 3.4] and the unqueness crteron, we have thus proved that (p (t s not unque. 3. Recurrence and extncton From now on, unless otherwse stated, let us assume that the process (p (t s unque and rreducble. Proof of Theorem.3. a We prove that (p (t s recurrent provded M. Let (π be the embeddng chan of (p (t. It s well known that the Q-process (p (t s recurrent ff the equaton π y y, (3. has a fnte soluton (y satsfyng lm y = (cf. [5; Theorem 4.34 and Theorem 4.24]. Take y =,. Then, we have π y = π, y + π,+k y +k = (p Γ = y,. p k= We see that (3. holds and so (p (t s recurrent. b We now consder the case that M >. Clearly, there exsts some <s < satsfyng s >p(s. To prove the transent asserton of the theorem, the goal s to compare our process wth a brth-death process. For ths, let M = p /s and consder the brth-death process ( p (t wth brth rate b =(α + βm and death rate a =(α + βp. The process ( p (t s certanly unque. It s transent snce the equaton x = k π k x k, x l= Γ has a non-trval soluton: x = s and x = s (, where ( π s the embeddng chan of ( p (t ([; 5, Lemma ] or [5; Theorem 4.35]. On the other hand, by the comparson lemma mentoned above, we know that the equaton x = k π k x k, x has a non-trval soluton ff the nequalty

6 6 RONG-RONG CHEN also has a non-trval soluton. Thus, to prove the transence of (p (t, t s suffcent to show that ( x s a soluton to (3.2. Ths can be done by some elementary computatons (remember that s >p(s. c To prove part (2 of the theorem, note that underlyng our assumptons, the extncton probablty of the Q-process (p (t s equal to ff the mnmal soluton of the equaton x = k π k x k + π, (3.3 s equal to dentcally. Because the last property s ndependent of whether the state at s absorptve or not, the proof s exactly the same as those for the recurrence. 4. Postve Recurrence Proof of Theorem.4. a Frst we prove part ( of the theorem. Consder the brth-death process ( p (t wth rates b = a = r p. The process ( p (t s unque and postve recurrent. Hence, by [8; Theorem 9.4.] or [; Theorem 6.2.3], the equaton b (x + x +a (x x +=, (4. has a fnte non-negatve soluton. Wthout loss of generalty, assume that x =. Then, t s easy to check that a soluton to (4. s as follows: x = p ( = Clearly, x x as. Thus,for, Next, by assumpton, we have + r = r,. q (x x q, (x x + q,+k k(x + x = q x = q p k= a (x x +b (x + x. h ( = = + r = <. Combnng these facts wth the result ust quoted above, we see that (p (t s postve recurrent. b Next, we prove the man concluson of part (2. Because we are n the recurrent stuaton, up to a postve constant, the equaton μ q = μ q, : has a unque postve soluton. We show that when r = θ (, θ, (μ s bounded. Frst, we have μ 2 = μ q μ q q 2 μ q q 2 = μ p 2 θ p μ. Assume that μ μ μ.thenfor 2, we have ( q μ = μ q μ q μ q q. r

7 Notce that AN EXTENDED CLASS OF TIME-CONTINUOUS BRANCHING PROCESSES 7 q +, q + q = p [( + θ θ ] = θ θ p θ θ k=2 = θ θ (p Γ, [ θ ( + k θ ]p k θ k=2 k=+ k=+ (k p k θ θ (k p k here the nequalty comes from the mean value theorem. We obtan μ + μ. By nducton, we have proved that (μ s bounded. Assume that = μ r s < for every s<. Snce we have μ q = μ q + μ + q +,,, = μ r ( p =μ q + μ r p + + μ + r + p,. (4.2 = If we multply both sdes of (4.2 by s, then sum both sdes over, we fnd that = μ r s = μ q h(s p(s s. (4.3 We now prove the man asserton of part (2 n the partcular case that r = θ (, θ. From and, t follows that = μ θ s = μ q h(s p(s s. (4.4 If we multply both sdes of (4.4 by ( s θ and ntegrate from to, we obtan μ θ s ( s θ ds = μ q = h(s p(s s ( sθ ds. Notcng that s ( s θ ds = B(, θ = Γ(Γ(θ Γ( + θ and usng Strlng formula, we get θ B(, θ θ /2 e Γ(θ const.,. ( + θ +θ /2 e θ We have thus proved that = μ < ff μ θ B(, θ =μ q = h(s p(s s ( sθ ds <. p k

8 By Strlng formula, one can check that the ntegral s fnte. Acknowledgement. The author would lke to acknowledge Prof. Mu-Fa Chen and Prof. Guang-Lu Gong for ther gudance and encouragement durng the wrtng of the paper, and 8 RONG-RONG CHEN v For general r const. θ (, θ, choose c,c > such that c θ r c θ. Recall that an rreducble recurrent process (p (t wth Q-matrx (q s postve recurrent ff the mnmal soluton (x of the equaton x = q x +, (4.5 q q, s fnte ([8; Theorem 9.4.]. Next, let (x c denote the mnmal soluton of (4.5 replacng r wth c θ. Then, by the comparson theorem ([8; Theorem 3.3.] or [4; Theorem 2.5], we have x c x x c,. Next, by the lnear combnaton theorem ([8; Theorem 3.3.2] or [4; Theorem 2.4], Hence, x c,x,xc c x c = c x c = x,. and x are all fnte or nfnte smultaneously. We have thus reduced the general case to the partcular one treated above. c Fnally, let r const.. When M =, the condtons (.4 and (. are the same. When M <, we have Notce that h(s ds < p(s s h(s s ds = h = l= h(s ds <. s s l ds = = h l +. We have h(s s ds < h log <. Thus, when M <, the last asserton of Theorem.4 now follows from (.4. d The last asserton of the theorem follows from [5; Corollary 4.49]. We have ust proved that when r const. θ (, θ, the process s postve recurrent ff (.4 holds. On the other hand, part ( of Theorem.4 says that the process s postve recurrent whenever θ>2. To prove the consstence of these two conclusons, one has to show that (.4 holds for all θ>2. The case where M < ssmple: h(s p(s s ( sθ ds = ( s θ ds const. p(s s l= ( s θ 2 ds <. When M =,let p(s =p +( 2p s + p s 2. Then, t s easy to check that p(s p(s, s<. Thus we have h(s p(s s ( sθ ds h(s p(s s ( sθ ds = h ( s ( s θ 3 ds p = = p = ( h θ 2 B(, θ. θ 2

9 AN EXTENDED CLASS OF TIME-CONTINUOUS BRANCHING PROCESSES 9 References [] Anderson, W. J. (99, Contnuous-Tme Markov Chans, Sprnger-Verlag. [2] Asmussen, S. and Herng, H. (983, Branchng Processes, Brkhäuser. [3] Athreya, K. B. and Ney, P. E. (972, Branchng Processes, Sprnger-Verlag. [4] Chen, M. F. (99, On three classcal problem for Markov chans wth contnuous tme parameters, J. Appl. Prob. 28, [5] Chen, M. F. (992, From Markov Chans to Non-Equlbrum Partcle Systems, World Scentfc. [6] Gao, S. Z. (992, Lnear controlled branchng processes (In Chnese, Wuhan J. Math. 92:4, [7] Harrs, T. E. (963, Branchng Processes, Sprnger-Verlag. [8] Hou, Z. T. and Guo, Q. F.(978, Tme-Homogeneous Markov Processes wth Countable State Space (In Chnese, Beng Sc. Press, Englsh translaton (988, Beng Sc. Press and Sprnger-Verlag. [9] Pakes,A.G.andTavaré, S. (98, Comments on the age dstrbuton of Markov processes, 3, 68 73, Adv. Appl. Prob. [] Yamazato, M. (975, Some results on contnuous tme branchng process wth state-dependent mmgraton, 7, , J. Math. Soc. Japan. [] Yan, S. J. and Chen, M. F. (986, Multdmensonal Q-processes, 7(B :, 9, Chn. Ann. Math.