Longtime behavior of a branching processcontrolled by branching catalystsDonald A. Dawson�;y Klaus Fleischmannz;x(WIAS preprint No. 261 of August 22, 1996)AMS Subject Classi�cation Primary 60J80; Secondary 60J55, 60G57Keywords catalytic reaction di�usion equation, super-Brownian motion, super-process, branching functional, critical branching, measure-valued branching, per-sistence, super-Brownian medium, random medium, catalyst process, catalyticmedium, Brownian collision local time, self-similarity, random ergodic limitRunning head: Branching controlled by branching catalystsper2.tex typeset by LATEX�Supported by an NSERC Grant.yFields Institute, Toronto, Canada M5T3J1. E-mail: ddawson@�elds.utoronto.cazSupported by the DFG.xWeierstrass Institute for Applied Analysis und Stochastics (WIAS), Mohrenstr. 39,D{10117 Berlin, Germany. E-mail: eischmann@wias-berlin.de

Branching controlled by branching catalysts 1AbstractThe model under consideration is a catalytic branching model con-structed in [DF96], where the catalysts themselves su�er a spatialbranching mechanism. Main attention is paid to dimension d = 3: Thekey result is a convergence theorem towards a limit with full intensity(persistence), which in a sense is comparable with the situation forthe \classical" continuous super-Brownian motion. As by-products,strong laws of large numbers are derived for the Brownian collisionlocal time controlling the branching of reactants, and for the catalyticoccupation time process. Also, the catalytic occupation measures areshown to be absolutely continuous (in space).Contents1 Introduction and main results 22 Brownian collision local time 42.1 Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42.2 Catalyst process % : : : : : : : : : : : : : : : : : : : : : : : : : : : 52.3 Brownian collision local time L[W;%] : : : : : : : : : : : : : : : : : : 62.4 A strong law of large numbers for L[W;%] in d = 3 : : : : : : : : : : 73 Occupation times 93.1 Catalytic SBM : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93.2 Absolutely continuous occupation time states : : : : : : : : : : : : 103.3 A strong law of large numbers for Y % in d = 3 : : : : : : : : : : : 124 Self-similarities in dimension d = 2 134.1 A scaling property of L[W;%] : : : : : : : : : : : : : : : : : : : : : : 134.2 Self-similarities of [X%; Y %] : : : : : : : : : : : : : : : : : : : : : : 144.3 A random ergodic limit : : : : : : : : : : : : : : : : : : : : : : : : 165 Persistence in dimension d = 3 175.1 Main result : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 185.2 Proof of the main theorem : : : : : : : : : : : : : : : : : : : : : : : 18

2 D.A. Dawson & K. Fleischmann1 Introduction and main resultsConsider two types of \particles" situated in Rd, one of them we call the cat-alysts, the others the reactants. The catalysts perform a continuous super-Brownian motion (SBM) % with constant branching rate > 0: The reac-tants are also super-Brownian, however given %; their branching rate at timet in the volume element db of Rd is just given by %t(db): In other words, �rst% is realized, and then a continuous SBM X = X% = (X%; P %s;�) evolves withvarying branching rates %t(db) (quenched approach). More precisely, the rateof branching of a reactant with (Brownian) pathW is controlled (in the senseof Dynkin's [Dyn91] additive functional approach to superprocesses) by theBrownian collision local time (BCLT) L[W;%] of %, formally described byL[W;%](dr) := drZ %r(db) �b(Wr); (1)which exists non-trivially for dimensions d � 3 (cf. Barlow et al. [BEP91]). Inhigher dimensions instead, W and % do not collide (see Barlow and Perkins[BP94, Proposition 1.3]), and therefore branching should not occur, whichmeans that X% degenerates to the heat ow. That catalytic SBM X% in Rd;d � 3; was constructed as a continuous process in detail in [DF96].It might be useful at this point to recall the longtime behavior of SBMwith constant branching rate, starting with a Lebesgue measure ` ([Daw77]).In dimension one, it su�ers local extinction almost surely, in dimension twoin probability, whereas in d � 3 it converges in law to a non-trivial steadystate with expectation ` (persistence).The study of the longtime behavior of the catalytic SBM X% was initi-ated also in [DF96], but restricted only to dimension d = 1. In this case,X% behaves quite di�erent from the usual spatial branching models in lowdimensions. In fact, if both the catalyst process % and the catalytic SBM X%start o� with the Lebesgue measure `, then, for almost all catalyst processrealizations, X%t converges in probability to the starting Lebesgue measure `(persistence), caused by the clumping features of the one-dimensional cata-lyst ([DF88]).Here we continue the study of this model X% in the time-space catalyticmedium %. In dimension d = 2; we get only some partial results, namely someself-similarity properties (Proposition 13) and a random ergodic limit (The-orem 15). The question whether or not persistence occurs in this \delicate"

Branching controlled by branching catalysts 3dimension is an open problem (see also Remark 14).But our main result concerns dimension d = 3: Here we alternativelyallow % to start o� with the ergodic steady state (of the catalyst process)leading to a time-stationary (in law) medium. Then with respect to theannealed distribution, Xt converges in law to some random measure of fullintensity and �nite variance (convergence and persistence Theorem 18 (a) atp.18). From this point of view, the time averaged process should obey a stronglaw of large numbers (Theorem 10). Both can be considered as a randommedium analog of properties of the classical SBM in higher dimensions. Butnote that it can be expected that the limit is di�erent from the classical one.To complete the picture, we establish a strong law of large numbers forthe BCLT L[W;%] (Theorem 5). We also show that the catalytic (weighted)occupation time process Yt := R t0dr Xr has absolutely continuous states.The log-Laplace functional vt = v%t � 0 of the catalytic SBM X%t at timet satis�es (formally) the following reaction di�usion equation� @@s vt(s; a) = 12 �vt(s; a)� %s(da) v2t (s; a); s � t; a 2 Rd; (2)with a terminal condition vt(s; � )js=t� = f � 0: (The backward setting re- ects the fact that, for % �xed, the deterministic process v% is \dual" tothe stochastic process X%; realized by the log-Laplace functional.) Via thisconnection, our results can also be understood as a probabilistic contribu-tion to the study of that equation with a (random) singular reaction coef-�cient %s(da); describing a spatially heterogeneous catalytic reaction. Actu-ally, our results give information on the longtime behavior of the L1{normR `(da) v(s; a) of the solution to (2) as s ! �1 if it \starts" at time twith a �nite mass R `(da) f(a). In fact, we proved in [DF96] that in theone-dimensional case one has convergence to the starting mass R `(da) f(a)(persistence). Dimension two is open. But the main result of the presentpaper establishes in dimension three a.s. convergence to a non-zero limit(possibly depending on the medium %):Note that the one-dimensional case resembles a bit a (two-dimensional)reaction di�usion process of electrically charged species studied by Glitzkyet al. [GGH96]. They got convergence to an equilibrium with exponentialvelocity. But our three-dimensional model behaves di�erent in that we donot get an equilibrium at the equation level.

4 D.A. Dawson & K. Fleischmann2 Brownian collision local timeIn this section we rigorously introduce the Brownian collision local time L =L[W;%] , and state in dimension d = 3 a strong law of large numbers (Theorem5 at p.7).2.1 PreliminariesFix a constant p > d with d � 1 the dimension of space, and introduce thereference function �p(b) := (1 + jbj2)�p=2; b 2 Rd: (3)Let B p denote the space of all measurable functions f de�ned on Rd suchthat jf j � cf �p for some constant cf : Write C p;` for the subset of allcontinuous functions f in B p such that f(b)=�p(b) has a �nite limit as jbj !1: Equipped with the norm kfk := kf=�pk1 ; the Banach space C p;` isseparable.Set I := [0; T ]; T � 0: Write C p;I for the set of all continuous functions de�ned on I � Rd such that j sj � c �p ; s 2 I; for some constant c :Let Mp refer to the cone of all (non-negative) measures � de�ned on Rdsuch that k�kp := h�; �pi := Z �(db)�p(b) < +1: (4)Mp is endowed with the coarsest topology such that the maps � ! h�; fiare continuous where f = �p or f 2 Ccomp+ : Here Ccomp denotes the space ofcontinuous functions on Rd with compact support (and the index + indicatesthe subset of all non-negative members). Recall that each Lebesgue measure` belongs to Mp : Write i` (> 0) for the unit volume in Rd measured withrespect to `:Let W = (W;�s;a) denote the Brownian motion in Rd on canonical pathspace of continuous functions, with \generator" 12 �: Let pt(a; b) = pt(b� a)refer to its continuous transition density function, and S = fSt : t � 0gto its semigroup. Set �s;� := R�(da)�s;a : We also introduce the (time-inhomogeneous) Brownian potential kernelqs;t(a; b) = qs;t(b� a) := Z ts dr pr(a; b); 0 � s � t; a; b 2 Rd: (5)

Branching controlled by branching catalysts 52.2 Catalyst process %For convenience, we expose the following de�nition of the catalyst process %:De�nition 1 (catalyst process %) Write % = (%; IPs;�) for the continuousSBM in Rd with constant branching rate > 0: Consequently, for �xed t � 0;the log-Laplace functional of % is given by� log IPs;� exp h%t ;�fi = D�;�vt(s; � )E; s� t; �2Mp ; f 2B p; (6)where vt is the unique non-negative solution to (2) with %s(da) replacedby the constant , and with terminal condition vt(s; � )js=t� = f: Here wealways work with a mild solution, that is with a solution to the equation inthe integrated and Dynkin's probabilistic formvt(s; a) = �s;a�f(Wt)� Z ts dr v2t (r;Wr)�; s � t; a 2 Rd: (7)% is called the catalyst process. 3Recall the expectation formulaIPs;� h%t ; fi = h� ; St�s fi; s � t; � 2 Mp ; f 2 B p: (8)Recall also, that in dimensions d � 3; with respect to IP0;` ; the catalystprocess %t has a non-trivial limit %1 in law as t " 1 which has full intensitymeasure ` (see, e.g. [DP91, Proposition 6.1]). Hence, here we can form thetime-stationary continuous Mp{valued process % = f%t : t 2 Rg which one-dimensional laws L(%t) coincide with L(%1): In this case we write IP andsometimes IP�1;` for the law of %:The following mixing property is taken from a general result in Fleisch-mann [Fle82b], which was formulated in a time-discrete setting. In thepresent particular situation, a simpli�ed proof will be given.Lemma 2 (time-space mixing of all orders) In dimensions d � 3; thecatalyst process % with time-space-shift invariant law IP is time-space mixingof all orders. In particular, for f; g 2 Ccomp+ ; the vectorhh%t1 ; fi ; h%t2 ; gii is asymptotically independent as jt1 � t2j ! 1(mixing in time).

6 D.A. Dawson & K. FleischmannProof First of all, recall the following covariance formula for % :CCovs;�hh%t1 ; fi ; h%t2 ; gii = 2�s;� Z t1^t2s dr St1�rf (Wr)St2�rg (Wr); (9)s � t1; t2 ; � 2 Mp ; and f; g 2 Ccomp+ ; see e.g. [DF96, Proposition 12 (b)].Hence, the covariance density function of % at [[t1 ; b1]; [t2 ; b2]] with respectto IPs;` is given by 2 Z t1^t2s dr pt1+t2�2r(b1 ; b2):Letting s # �1; we arrive at the covariance density function at [t1 ; b1] and[t2 ; b2] of the catalyst process % with respect to IP: Since IP is invariant withrespect to the time-space shift and in�nitely divisible, it su�ces to show thatthis covariance density function converges to 0 as j[t1 ; b1]� [t2 ; b2]j ! 1 onthe sets fjb1 � b2j � "g; " > 0; see the remark after Theorem 2.0.2 in[Fle82a]. Here we may set [t2 ; b2] = 0 without loss of generality. Thus it issu�cient to demonstrate thatZ 0�1dr pt�r(b) �! 0 as [t; jbj]! +1 on R+ � fjbj � "g; " > 0:But the latter integral equals R1t dr pr(b) and can be estimated from aboveby � const [jbj2�d ^ t�1=2] with a constant const depending on ": This�nishes the proof.2.3 Brownian collision local time L[W;%]Assumption 3 (catalyst process) From now on we restrict our attentionto dimensions d � 3; and assume, if not otherwise indicated, that the catalystprocess % is distributed according to IP0;` or to the stationary IP, the latterof course only if d = 3: 3For " > 0 and given %; consider the following continuous additive func-tional L"[W;%] of Brownian motion W :L"(dr) := L"[W;%](dr) := dr Z %r(db) p"(Wr ; b) ; (10)describing the collision local time of the measure-valued path % with the\"{vicinity" of the Brownian path W:

Branching controlled by branching catalysts 7Lemma 4 (Brownian collision local time L[W;%]) Suppose Assumption3, and �x a constant � 2 (0; 14): Then for almost all paths % of the catalystprocess, there exists a continuous additive functional L = L[W;%] of the Brow-nian motion W; called the Brownian collision local time (BCLT) of %; withthe following properties.(a) (convergence) If is a (strictly) positive function in C p;I ; I = [0; T ];T > 0; thensups2I; a2Rd �s;a sups�t�T ���� Z ts L"(dr) r(Wr)� Z ts L(dr) r(Wr)����2 �!"#0 0:(b) (�rst two moments) For measurable : R+ � Rd 7! R+ ; and s � t;a 2 Rd;�s;a Z ts L[W;%](dr) r(Wr) = Z ts dr Z %r(db) pr�s(a; b) r(b);�s;ah Z ts L[W;%](dr) r(Wr)i2= 2 Z ts dr Z tr dr0Z %r(db)Z %r0(db0) pr�s(a; b) pr0�r(b; b0) r(b) r0(b0):Proof This follows from Proposition 38 and Theorem 42 in [DF96].2.4 A strong law of large numbers for L[W;%] in d = 3In this subsection we assume that Brownian motionW is distributed accord-ing to �0;0 : First we recall that in dimension d = 1 the total BCLT L[W;%](R+)of % is �nite, for almost all [W;%] ([DF96, Proposition 4.8]). Next we mentionthat in d = 2 we have a self-similarity property for L = L[W;%] ; see Corollary12 below. But in dimension d = 3; a strong law of large numbers holds (recalli` denotes the unit volume):Theorem 5 (strong LLN for the BCLT) If d = 3; thenT�1L[W;%][0; T ] ��!T"1 i` ; �0;0� IP0;`�a:s: and �0;0 � IP�a:s:

8 D.A. Dawson & K. FleischmannProof Step 1� First of all, for s � 0; by the expectation formula inLemma 4 (b),�0;0 � IPs;` T�1L[0; T ] = IPs;` T�1 Z T0 dr h%r ; pri � i` (11)since by the expectation formula (8)IPs;` %r � ` (12)(independent of the dimension d): Hence, the claimed a.s. convergence followsif we show that the �0;0 � IPs;` {variance of T�1L[0; T ] is of order O(T�1=2)as T " 1; uniformly in s � 0 (covering the cases s = 0 and s = �1corresponding to IP0;` and IP; respectively, we are interested in).Step 2� In view of the second moment formula in Lemma 4 (b),�0;0(L[0; T ])2 = 2 Z T0 dr Z Tr dr0Z %r(db)Z %r0(db0) pr(b) pr0�r(b; b0):Therefore, by (12), byZ `(db)Z `(db0) pr(b) pr0�r(b; b0) = i2̀;and by step 1�;�0;0� IPs;`���T�1L[0; T ]� i`���2= 2T�2 Z T0 drZ Tr dr0 CCovs;` h%r(db); %r0(db0)ipr(b) pr0�r(b; b0): 9>>=>>; (13)But by (9), the latter covariance expression equals2�s;` Z rs dt Z db pr�t(Wt ; b) pr(b) p2r0�r�t(Wt ; b):Therefore we may continue the r.h.s. of (13) with= 4 i`T�2 Z T0 drZ Tr dr0 Z rs dt p2r0�2t(0):However, the internal integral is of order O((r0�r)�1=2); uniformly in s � 0:Hence, altogether we get an order O(T�1=2) uniformly in s � 0; �nishing theproof.

Branching controlled by branching catalysts 93 Occupation timesHere we rigorously introduce the catalytic SBM X%, verify that its catalyticoccupation time Y % has absolutely continuous states, and satis�es a stronglaw of large numbers, the latter in the case d = 3:3.1 Catalytic SBMSince the BCLT L = L[W;%] of Lemma 4 is a locally admissible additivefunctional with \small" increments one can conclude for the existence of thecatalytic SBM X% in the catalytic medium % :Lemma 6 (catalytic SBM X%) Under Assumption 3, for almost all real-izations % of the catalyst process, the following statements hold:(a) (existence) There exists the continuous SBM X = X% = (X%; P %s;�)with branching functional given by the BCLT L = L[W;%] :(b) (log-Laplace functional) The log-Laplace functional of X% is givenby � logP %s;� exp hXt ;�fi = h�;�vt(s; � )i; (14)s� t; �2Mp ; f 2B p; where vt is the unique non-negative solution toDynkin's log-Laplace equationvt(s; a) = �s;a�f(Wt)� Z ts L(dr) v2t (r;Wr)�; (15)s � t; a 2 Rd:(c) (moments) Expectation and covariance of X% are given byP %s;� hXt ; fi = h�; St�s fi; s � t; � 2 Mp ; f 2 B p;andCov%s;�hhXt1; fi ; hXt2 ; gii= 2 Z �(da) Z t1^t2s dr Z %r(db) p(r � s; a; b)St1�rf(b)St2�rg(b); 9>>=>>;s � t1; t2 ; � 2 Mp ; and f; g 2 Ccomp+ :Proof See [DF96], De�nition 44 which is based on Theorem 18 (b), andformula (95).Note that (15) is the precise meaning of the catalytic reaction di�usionequation (2) with reaction rate %s(da):

10 D.A. Dawson & K. Fleischmann3.2 Absolutely continuous occupation time statesSince X% is pathwise continuous, we may introduce the catalytic (weighted)occupation time process Y = Y % = fY %t : t � 0g related to X = X%; de�nedby Yt := R t0 dr Xr :Recall that for the \classical" continuous SBM, say X`; in dimensionsd � 3; the related occupation measures Yt̀ are absolutely continuous, i.e.density functions yt̀ exist (see e.g. [Fle88]). This property is shared also bythe SBM X% in the catalytic medium %: For a convenient formulation, weintroduce the annealed law P` := IP0;`P %0;` : That is, the laws P %0;` of X% given% are averaged by means of the distribution IP0;` of %:Theorem 7 (occupation densities) Under Assumption 3, for T > 0 andz 2 Rd �xed, the following statements hold.(a) (densities of Yt) With respect to the annealed law P` we obtain:(a1) (L2{densities) The L2(P`){limit of hYT ;p"(z; � )i as " # 0 existsand is denoted by yT (z):(a2) (absolutely continuous states) With respect to P` ; the ran-dom measure YT is absolutely continuous with density functionyT : P`�YT (db) = yT (b) db� = 1:(a3) (�rst two moments) The following formulas hold:P` yT (z) � i` ;Var` yT (z) = 2 i2̀Z T0 drZ T�r0 dtZ T�r0 dt0 pt+t0(0) > 0:(b) (densities of Y %t ) For IP0;` {a.a. catalyst process realizations % we ob-tain:(b1) (L2{densities) The L2(P %0;`){limit of hY %T ;p"(z; � )i as " # 0 ex-ists and is denoted by y%T (z):(b2) (absolutely continuous states) With respect to P %0;` ; the ran-dom measure Y %T is absolutely continuous with density functiony%T : P %0;`�Y %T (db) = y%T (b) db� = 1:

Branching controlled by branching catalysts 11(b3) (�rst two moments) The following formulas hold:P %0;` y%T (z) � i` ;Var%0;` y%T (z) = 2 i`Z T0 drZ %r(db) q20;T�r(b; z)(recall de�nition (5) of the Brownian potential kernel q):Proof We start with the proof of (b). According to [DF96, Proposition24] it su�ces to show that for almost all %�0;` Z T0 L[W;%](dr) q2r0;"+r0(Wr ; z) �!"#0 0 for r0 = 0 and r0 = T � r:(16)By the expectation formula in Lemma 4 (b)�s;` Z Ts L[W;%](dr) r(Wr) = i` Z Ts drZ %r(db) r(b): (17)Hence, the l.h.s. in (16) equalsi` Z T0 dr Z %r(db) q2r0;"+r0(b; z):Since this is monotone in "; the limit as " # 0 exists (for each %): Thus it issu�cient to show that even the expectation over % converges to 0 as " # 0:But by (12), the latter IPs;` {expectation, s � 0; is independent of s and equalsi` Z T0 drZ `(db) q2r0;"+r0(b; z) = i2̀ T Z "+r0r0 dt Z "+r0r0 dt0 pt+t0(0):Because the integrand is monotone decreasing in t and t0; we may replace r0by 0; and since T is �xed we continue the latter formula line with� const Z "0 dt Z "0 dt0 (t+ t0)�d=2 � const "1=2 �!"#0 0 since d � 3:This �nishes the proof of (b). But then (a) immediately follows by averagingover the medium.In dimension d = 3; let P denote the annealed law IPP %0;` of the catalyticSBM concerning the time-space stationary medium % with law IP:

12 D.A. Dawson & K. FleischmannRemark 8 (occupation densities in the stationary case) Theorem 7remains valid, if in dimension d=3 we replace IP0;` by IP, thus P` by P. 3Remark 9 (occupation density �eld) It can be expected that for thecatalytic occupation time process Y % in all dimensions (d � 3) a jointly con-tinuous catalytic occupation density �eld y% exists, as it is for the \classical"continuous SBM, established by Sugitani [Sug89] (and reproved in [DF96,Lemma 25]). 33.3 A strong law of large numbers for Y % in d = 3First we recall that in dimension d = 1; the catalytic SBM X%t converges inP %0;` {probability to ` as t " 1; for IP0;`{almost all %; see [DF96, Theorem51]. This of course implies a law of large numbers for the related catalyticoccupation time process Y %: In d = 2; self-similarity properties hold instead,see Proposition 13 below. Here now we restrict our attention to dimensiond = 3: Recall that P` and P denote the annealed laws of the catalyticSBM arising by averaging P %0;` over the medium % by means of IP0;` and thestationary IP; respectively.Theorem 10 (LLN for Y ) If d = 3; thenT�1Y %T ��!T"1 `; P`�a:s: and P�a:s:Consequently, in dimension d = 3; the time-averagedX%{process behavesfor almost all % just as for the \classical" SBM. That is, here the averagingprinciple holds: Finally only the expectation ` of the medium %t is \e�ective"leading to the expectation of X%.Proof Since P %0;`T�1Y %T � ` by the expectation formula in Lemma 6 (c),it su�ces to show that for �xed f � 0 in the separable Banach space C p;`;Ps;`���T�1 hYT ; fi � h`; fi ���2 � const T�1=2; (18)uniformly in s � 0: Here we wrote Ps;` for IPs;`P %0;` ; s 2 [�1; 0]: ButVar%0;` hYT ; fi = 2�0;` Z T0 L(dr)h Z Tr dt �r;Wr f(Wt)i2; (19)

Branching controlled by branching catalysts 13see [DF96, formula (46)]. Hence, the l.h.s. in (18) equals2T�2 IPs;`�0;` Z T0 L(dr)h Z Tr dt �r;Wr f(Wt)i2:Using the expectation formulas (17) and (12), we may continue with= 2 i`T�2Z T0 drZ `(db)h Z Tr dtZ dz pt�r(b; z) f(z)i2:Interchanging the order of integrations, and calculating the `(db){integral,this can be estimated from above by� 2 h`; fi2 T�2 Z T0 drZ Tr dtZ Tr dt0 pt+t0�2r(0):As in the end of proof of Theorem 5, the internal integral can be estimatedby � const (t� r)�1=2; and the claim follows.4 Self-similarities in dimension d = 2Recall that in dimension d = 2 the \classical" SBM % with law IP0;` is self-similar: For K > 0;nK�1%Kt(K1=2� ) : t � 0o L= f%t : t � 0g: (20)This has some consequences for the BCLT L[W;%] and the catalytic SBM X%:4.1 A scaling property of L[W;%]We start with a scaling property of the Brownian collision local time:Lemma 11 (scaling of the BCLT) Fix d = 2; s � 0; a 2 R2; K > 0:Then for �s;a � IP0;` {almost all [W;%]; and measurable g : R+ 7! R+ ;Z L[W;%](dr)K�1g(K�1r) = Z L[K�1=2WK� ;K�1%K� (K1=2 � )](dr) g(r): (21)

14 D.A. Dawson & K. FleischmannProof Recalling the de�nition (10) of L", by de�nition of the BCLT L =L[W;%] it su�ces to verify the claim with L replaced by LK" and L"; respec-tively. Then by (10),Z LK"[W;%](dr)K�1g(K�1r) = Z drZ %r(db) pK"(Wr ; b)K�1g(K�1r):By a change of variables, and using the self-similarity of the Brownian tran-sition density p; the r.h.s. can be written asZ drZK�1%Kr(K1=2db) p"(K�1=2WKr ; b) g(r):Again by (10), we arrive at the r.h.s. of (21) with L replaced by L"; �nishingthe proof.Combining Lemma 11 with the self-similarities of Brownian motion Wand % (recall (20)) we get the following result.Corollary 12 (self-similarity of the BCLT) For d = 2 and K > 0; withrespect to �0;0� IP0;` ; K�1L[W;%](K� ) L= L[W;%] :4.2 Self-similarities of [X%; Y %]Instead of the well-known self-similarity of the \classical" SBM in d = 2 (asin (20)), for the catalytic SBM we have the following versions (recall thecatalytic occupation time process Y introduced in the beginning of x 3.2):Proposition 13 (scaling and self-similarity of [X;Y ]) Suppose d = 2,K > 0; and T � 0:(a) (scaling) For IP0;` {almost all % the following holds. If [X;Y ] is formedwith respect to P %0;` ; then the pairhK�1XKT (K1=2� ); K�2YKT (K1=2� )ihas the same law as [XT ; YT ] formed with respect to PK�1%K� (K1=2 � )0;` :

Branching controlled by branching catalysts 15(b) (self-similarities) Formed either with respect to the random law P %0;` ;or the annealed law P` = IP0;`P %0;` ;hK�1XKT (K1=2� ); K�2YKT (K1=2� )i L= [XT ; YT ]:Proof By [DF96, Hypothesis 13 and notation (47)], for f; g in B p+; andIP0;` {a.a. %;� log P %0;` exp h hXT ; �fi + hYT ; �gi i = D`; vT (0; � )Ewith vT (s; a) = �s;a�f(WT ) + Z Ts dr g(Wr)� Z Ts L[W;%](dr) v2T (r;Wr)�;0 � s � T; a 2 R2: Hence� log P %0;` exp h DXKT ; �K�1f(K�1=2� )E + DYKT ; �K�2g(K�1=2� )E i= D` ; vKT (0; � )E = D` ; KvKT (0;K1=2� )E (22)withKvKT (Ks;K1=2a) = �Ks ;K1=2a�f(K�1=2WKT ) + Z KTKs dr K�1g(K�1=2Wr)� K�1 Z KTKs L[W;%](dr)K2v2KT (r;Wr)�:Setting uT (s; a) := KvKT (Ks;K1=2a) (for the �xed K); by a change ofvariables and using the scaling of BCLT Lemma 11 (with [s; a] replaced by(Ks;K1=2a)); the latter equation can be written asuT (s; a) = �Ks ;K1=2a�f(K�1=2WKT ) + Z Ts dr g(K�1=2WKr)� Z Ts L[K�1=2WK� ;K�1%K� (K1=2 � )](dr)u2T (r;K�1=2WKr)�:ButW distributed according to �Ks ;K1=2a implies by scaling that the processt ! K�1=2WKt has the law �s;a : Therefore the latter formula lines can becontinued with= �s ; a�f(WT ) + Z Ts dr g(Wr)� Z Ts L[W ;K�1%K� (K1=2� )](dr)u2T (r;Wr)�:

16 D.A. Dawson & K. FleischmannHence, by uniqueness of the solution to the log-Laplace equation, we concludethat the r.h.s. of (22) equals� logPK�1%K� (K1=2� )0;` exp hhXT ; �fi+ hYT ; �gii:This proves the claim (a). The statement (b) concerning the random lawP %0;` then immediately follows from the self-similarity (20) of %, and the claimconcerning the annealed law P̀ results by integration with IP0;` : This �n-ishes the proof.Remark 14 (open problems) By the self-similarity of X% with respectto the annealed law P` ; the distribution of X%T coincides with the law ofTX%1 (T�1=2 � ): Passing formally to T " 1; we arrive at X%1 and x%1(0): Thisrelates the questions of existence of a non-trivial limit X%1 and of absolutecontinuity of X%1 in the critical dimension d = 2: But whether or not a non-trivial limit X%1 exists remains open. 34.3 A random ergodic limitRecall that for the continuous SBM X` in R2 with constant branching rateand with law P0̀;` we have the following \random" ergodic limit:T�1YT̀ ��!T"1 y1̀(0) ` in law;where y1̀(0) is the random density of the occupation measure Y1̀ at time 1at the origin 0; see e.g. [Fle88]. The two-dimensional catalytic SBM X% hasa similar property:Theorem 15 (random ergodic limit) Let d = 2 and consider the cat-alytic SBM X% with annealed law P` = IP0;`P %0;` ; or with IP0;` {random lawP %0;` :(a) (annealed approach) T�1Y %T converges in P`{law as T " 1 towardsthe multiple y1(0)` of Lebesgue measure `; where y1(0); is the L2(P`){density at 0 of the catalytic occupation measure Y1 at time 1; accordingto Theorem 7 (a1).

Branching controlled by branching catalysts 17(b) (quenched approach) The IP0;` {random law of T�1Y %T converges inIP0;` {law as T " 1 towards the IP0;` {random law of the multiple y%1(0)`of Lebesgue measure `; where y%1(0); given %; is the L2(P %0;`){density at 0of the catalytic occupation measure Y %1 at time 1; according to Theorem7 (b1).Proof We start by proving part (b). Using the random self-similarity inProposition 13 (b), the IP0;` {random law of T�1Y %T coincides with that ofT Y1(T�1=2 � ): But by Theorem 7 (b1), for f 2 C p;`+ and IP0;` {almost all %DT Y %1 (T�1=2 � ); fE ��!T"1 y%1(0) h`; fi in L2(P %0;`);implying the claim.Part (a) follows analogously from the annealed self-similarity in Propo-sition 13 (b), and Theorem 7 (a1).Note that opposed to dimensions 1 and 3; here the limit remains randomafter the averaging procedure, since the P` {variance of y1(0) is (strictly)positive, and the P %0;` {variance of y%1(0) is positive with IP0;` {probabilityone.5 Persistence in dimension d = 3In this �nal section we pay attention to the following situation (which oc-curred already in Remark 8).Assumption 16 (time-space-shift invariance) Let d = 3 and assumethat the catalyst process % is distributed according to the time-space-shiftinvariant distribution IP (introduced in x 2.2), and consider the annealed lawP = IPP %0;` : 3Remark 17 (approximation) Working with the non-stationary catalystprocess % distributed according to IP0;` would require some additional ap-proximation. 3

18 D.A. Dawson & K. Fleischmann5.1 Main resultNow we are in a position to formulate our main result :Theorem 18 (convergence and persistence of second order) ImposeAssumption 16.(a) (annealed convergence) With respect to the annealed distribution P;the catalytic SBM XT converges in law as T " 1 to some limit X1with full intensity ` and �nite variance (persistence of second order).(b) (random convergence) The IP{random distribution ofP %0;`(XT 2 � ) =: Q%Tconverges in IP{law as T " 1 to some IP{random distribution Q%1 withfull intensity and �nite variance (persistence of second order): WithIP{probability one,ZQ%1(d�) � = `; ZQ%1(d�) ���h�; fi � h`; fi���2 <1; f 2 B p+ :Consequently, at the �rst sight, our catalytic SBM X% behaves similar tothe classical continuous SBM X`. However, the main di�erence should bethat a new limit occurs. For instance, the limiting random measure X1 of(a) should be di�erent from the classical steady state X1̀ . Such statementswill be investigated in a forthcoming paper (the tools in the present paperare insu�cient for this).5.2 Proof of the main theoremThe key of proof will be some backward technique: By the time-stationarityof the random medium % we may start X% at time �T with `, and observethe state at time 0: Then we may continue for �xed realization %; sending�T to �1; by exploiting some backward monotonicities.1� (convergence) First of all, for IP{almost all %; the law Q%T coincides withthe law P %T+��T;` (X0 2 � ): Here %T+� is the catalyst process shifted by time T:Hence, by the time-shift invariance of the catalyst process %; the distributionof the random law Q%T coincides with that of the random law P %�T;`(X0 2 � ):

Branching controlled by branching catalysts 19(Note that here we made a transition in IP{law, but after this we will continuefor a �xed %:)Given %; we turn to the log-Laplace functional according to Lemma 6 (b):For f 2 B p+ ; writing h`; fi =: kfk1 ;� logP %�T;` exp hX0 ;�fi = kv0(�T; � )k1where by the log-Laplace equation (15),kv0(�T; � )k1 = ��T;`hf(W0)� Z 0�T L[W;%](dr) v20(r;Wr)i:Using the expectation formula in Lemma 4 (b), this formula line can be con-tinued with = kfk1 � i` Z 0�TdrZ %r(db) v20(r; b):But this non-negative expression is non-increasing1) in T: Hence, the limitof kv0(�T; � )k1 exists (for the �xed %) and determines (note that the familyfQ%T : T � 0g is relatively compact, see (23) below) a log-Laplace functionalof a random measure, its law denoted by Q%1 : This gives the convergenceclaim in (b). By averaging over %; also the convergence claim of (a) follows.2� (expectation bounds) For almost all %; from the expectation formula inLemma 6 (c), ZQ%T (d�)� = P %0;`XT � `; (23)which implies for the limits thatZQ%1 (d�)� � `; hence Z IPQ%1 (d�)� � `: (24)Consequently, the limiting intensity measures in (b) and (a) are boundedby `:3�(variance bounds) Let again f 2 B p+ : Given %; by the variance formulain Lemma 6 (c),Var%�T;` hX0 ; fi = 2 i`Z 0�T drZ %r(db) [S�rf(b)]2;1) Note that this monotonicity would be violated if we started % at time �T with%�T = `: That is, the present method only works for the time-stationary process % on thewhole time axis R:

20 D.A. Dawson & K. Fleischmannwhich monotonously converges to2 i`Z 0�1 drZ %r(db) [S�rf(b)]2 as T " 1: (25)Integrating additionally % with IP, by the expectation formula (12) we getthe monotone convergenceIPVar%0;` hXT ; fi %T"1 2Z `(dx) f(x) Z `(dy) f(y) Z 10 dr p2r(x; y): (26)Note that by (12), the l.h.s. in (26) is the variance of hXT ; fi with respect tothe annealed law P: On the other hand, the r.h.s. is the variance expressionrelated to the classical steady state X1̀ (see, e.g., [Daw77]), hence is �nite.Therefore also the limit (25) is �nite IP{a.s. But this implies that in (24)even equalities must hold (persistence). Moreover, the variance of hX1 ; fiis �nite with respect to IPQ%1 ; hence also with respect to the laws Q%1 ; given%: In other words, in both cases (a) and (b), we get persistence of secondorder. This �nishes the proof.

Branching controlled by branching catalysts 21References[BEP91] M.T. Barlow, S.N. Evans, and E.A. Perkins. Collision local times andmeasure-valued processes. Can. J. Math., 43(5):897{938, 1991.[BP94] M.T. Barlow and E.A. Perkins. On the �ltration of historical Brownianmotion. Ann. Probab., 22:1273{1294, 1994.[Daw77] D.A. Dawson. The critical measure di�usion process. Z. Wahrsch. verw.Gebiete, 40:125{145, 1977.[DF88] D.A. Dawson and K. Fleischmann. Strong clumping of critical space-time branching models in subcritical dimensions. Stoch. Proc. Appl.,30:193{208, 1988.[DF96] D.A. Dawson and K. Fleischmann. A continuous super-Brownian motionin a super-Brownian medium. WIAS Berlin, Preprint Nr. 165, 1995,Journ. Theoret. Probab., to appear 1996.[DP91] D.A. Dawson and E.A. Perkins. Historical processes. Mem. Amer. Math.Soc., 454, 1991.[Dyn91] E.B. Dynkin. Branching particle systems and superprocesses. Ann.Probab., 19:1157{1194, 1991.[Fle82a] K. Fleischmann. Mixing properties of in�nitely divisible random mea-sures and an application in branching theory. Carleton MathematicalLecture Notes, 43, 1982.[Fle82b] K. Fleischmann. Space-time mixing in a branching model. In Advancesin Filtering and Optimal Control, volume 42 of Lecture Notes in Controland Information Sciences, pages 125{130, Cocoyoc, Mexico, 1982.[Fle88] K. Fleischmann. Critical behavior of some measure-valued processes.Math. Nachr., 135:131{147, 1988.[GGH96] A. Glitzky, K. Gr�oger, and R. H�unlich. Free energy and dissipation ratefor reaction di�usion processes of electrically charged species. ApplicableAnalysis, 60:201{217, 1996.[Sug89] S. Sugitani. Some properties for the measure-valued branching di�usionprocess. J. Math. Soc. Japan, 41(3):437{462, 1989. printed October 17, 1996