Strong law of large numbers for branching diffusions

www.imstat.org/aihp 2010, Vol. 46, No. 1, 279–298

DOI:10.1214/09-AIHP203

© Association des Publications de l’Institut Henri Poincaré, 2010

Strong Law of Large Numbers for branching diffusions

János Engländer

, Simon C. Harris

and Andreas E. Kyprianou

aDepartment of Statistics and Applied Probability University of California, Santa Barbara, CA 93106-3110, USA.

E-mail:englander@pstat.ucsb.edu

bDepartment of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK. E-mail:S.C.Harris@bath.ac.uk cDepartment of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK. E-mail:a.kyprianou@bath.ac.uk

Received 11 February 2008; revised 29 October 2008; accepted 20 March 2009

Abstract. LetXbe the branching particle diffusion corresponding to the operatorLu+β(u²−u)onD⊆R^d(whereβ≥0 and β≡0). Letλ_cdenote the generalized principal eigenvalue for the operatorL+βonDand assume that it is finite. Whenλ_c>0 andL+β−λ_csatisfies certain spectral theoretical conditions, we prove that the random measure exp{−λ_ct}X_tconverges almost surely in the vague topology asttends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of [Ann. Probab.

30(2002) 683–722,Ann. Inst. H. Poincaré Probab. Statist.42(2006) 171–185]. We extend significantly the results in [Z. Wahrsch.

Verw. Gebiete36(1976) 195–212,Math. Scand.39(1977) 327–342,J. Funct. Anal.250(2007) 374–399] and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and “spine” decompositions or “immortal particle pictures.”

Résumé. SoitXle processus de diffusion avec branchement correspondant à l’operateurLu+β(u²−u)surD⊆R^d(oùβ≥0 etβ≡0). La valeur propre principale généralisée de l’operateurL+βsurDest dénotée parλc et on la suppose finie. Quand λ_c>0 etL+β−λ_csatisfait certaines conditions spectrales théoriques, on montre que la mesure aléatoire exp{−λ_ct}X_tconverge presque sûrement pour la topologie vague quandttend vers l’infini. Ce résultat est motivé par un ensemble d’articles par Asmussen et Hering datant du milieu des années soixante-dix, ainsi que par des travaux plus récents [Ann. Probab.30(2002) 683–722, Ann. Inst. H. Poincaré Probab. Statist. 42(2006) 171–185] concernant des résultats analogues pour les superdiffusions. Nous généralisons de manière significative les résultats de [Z. Wahrsch. Verw. Gebiete36(1976) 195–212,Math. Scand.39(1977) 327–

342,J. Funct. Anal.250(2007) 374–399] et nous donnons quelques exemples clés de la théorie des processus de branchement. En ce qui concerne les démonstrations, nous faisons appel aux techniques modernes de martingales et aux “spine decompositions” ou

“immortal particle pictures.”

MSC:Primary 60J60; secondary 60J80

Keywords:Law of Large Numbers; Spine decomposition; Spatial branching processes; Branching diffusions; Measure-valued processes;

h-transform; Criticality; Product-criticality; Generalized principal eigenvalue

1. Introduction and statement of results 1.1. Model

LetD⊆R^dbe a nonempty domain and writeC^i,η(D)to denote the space ofitimes (i=1,2) continuously differen- tiable functions with all theirith order derivatives belonging toC^η(D). [HereC^η(D)denotes the usual Hölder space.]

ConsiderY = {Y_t;t≥0}, the diffusion process with probabilities{Px, x∈D}corresponding to the operator L=1

2∇ ·a∇ +b· ∇ onD, (1)

where the coefficientsa_i,j andb_i belong toC^1,η(D), i, j=1, . . . , d,for someηin(0,1],and the symmetric matrix a(x)= {ai,j(x)}is positive definite for allx∈D. At this point, we do not assume thatY is conservative, that is, the exit time fromDmay be finite with positive probability. Intuitively, this means thatY may get killed at the Euclidean boundary ofDor “run out to infinity” in finite time.

Furthermore let us first assume that 0≤β∈C^η(D)is bounded from above onDandβ≡0. The (strictly dyadic) (L, β;D)-branching diffusionis the Markov process with motion componentY and with spatially dependent rateβ, replacing particles by precisely two offspring when branching and starting from a finite configuration of individuals.

At each timet >0, the state of the process is denoted byX_t∈M(D)where M(D):=

_n

i=1

δx_i: n∈Nandxi∈Dfori=1, . . . , n

.

We will also use the following notation:X= {Xt: t≥0}has probabilities{Pμ: μ∈M(D)}, andEμis expectation with respect toP_μ. As usual, f, μ :=

Df (x)μ(dx)and f, g :=

Df (x)g(x)dx, where dxis Lebesgue measure, and so f, gdx = f g,dx = f, g.

Whenβ is not bounded from above, one may wonder if the(L, β;D)-branching diffusion is still well defined, in particular, whether the global (or even local) mass may blow up in finite time. LetMloc(D)denote the family of locally finite measures onDand consider the family of locally finite discrete measures onD:

M^∗loc(D):=

μ∈Mloc(D):

i

δx_i: xi∈D

.

Let

λ_c=λ_c(L+β, D):=inf

λ∈R: ∃u >0 satisfying(L+β−λ)u=0 inD

denote thegeneralized principal eigenvalueforL+βonD. It is known (see Chapter 4 in [27]) thatλ_c<∞whenever βis upper bounded and that, for generalβ, there exists anh >0 satisfying that

(L+β−λ_c)h=0, (2)

wheneverλ_c<∞. From (2) it follows by standard approximation arguments (either by truncating the domain or by truncatingβ) that the(L, β;D)-branching diffusionX is well defined and isM^∗_loc(D)-valued, and, when weighted byh, it is even finite measure-valued. (W_t^h:=e^−λct

ih(X_tⁱ)is a supermartingale forXt=

iδ_Xi

t.) Therefore, from now on,we relax the assumption thatsup_Dβ <∞and replace it with the much milder assumptionλ_c<∞.

1.2. Motivation

This paper concerns the local growth of mass for branching particle diffusions. In doing so we address a gap in the literature dating back to the mid-seventies when the study of growth of typed branching processes on compact domains of the type space was popularized by Asmussen and Hering. Also we complement a recent revival in this field which has appeared amongst the superprocess community.

Before discussing main results, we shall introduce the topic in detail.

Definition 1 (Local extinction). Fixμ∈M(D).We say thatXexhibits local extinction underP_μif for every Borel setB⊂⊂D,there exists a random timeτ_Bsuch that

P_μ

τ_B<∞,andX_t(B)=0for allt≥τ_B

=1.

[HereB⊂⊂Dmeans thatBis bounded and its closure is a subset ofD.]

Local extinction has been studied by [15], [28] (for superprocesses) and [14] (for branching diffusions). To explain their results, recall that we assume that the generalized principal eigenvalue forL+β onDis finite. In fact,λ_c≤0 if and only if there exists a functionh >0 satisfying(L+β)h=0 onD– see Section 4.4 in [27]. Following the papers [15,28] where similar issues were addressed for superprocesses, in [14] the following was shown.

Theorem 2 (Local extinction versus local exponential growth). Let0=μ∈M(D).

(i) X underP_μexhibits local extinction if and only if there exists a functionh >0satisfying(L+β)h=0onD, that is,if and only ifλ_c≤0.

(ii) Whenλc>0,for anyλ < λcand∅=B⊂⊂Dopen, P_μ

lim sup

t↑∞ e^−λtX_t(B)= ∞

>0, P_μ

lim sup

t↑∞ e^−λctX_t(B) <∞

=1.

In particular,local extinction/local exponential growth does not depend on the initial measure0=μ∈M(D).

(In [14] it is assumed thatβ is upper bounded, whereas in [15] only the finiteness ofλ_cis assumed. The proofs of [14] go through for this latter case too.) On closer inspection this last theorem says that whenλc≤0 mass “escapes out ofB” even though the entire process may survive with positive probability. (IfYis conservative inDfor example then it survives with probability one). Further, whenλ_c>0 mass accumulates on all nonempty bounded open domains and in such a way that with positive probability this accumulation grows faster than any exponential rateλ < λ_c. On the other hand, mass will not grow faster than at the exponential rateλc. It is natural then to ask whether in factλc

gives an exact growth rate or not. That is to say, for each∅=B⊂⊂Ddo the random measures{exp{−λ_ct}X_t: t≥0} converge in the vague topology almost surely? (The latter we henceforth refer to as the SLLN, the use of the word

“strong” here pertains to a.s. convergence.) Further, can one identify the limit? This is precisely the object of interest of a variety of previous studies for both branching diffusions and superprocesses which we shall now review.

We note already here that the process in expectation is given by the linear kernelcorresponding to the operator L+β onD. Therefore, trusting in the Law of Large Numbers for branching processes, one should expect that the process itself grows like the linear kernel too. On the other hand, it is easy to see that the linear kernel does not in general scale precisely with exp{−λct}but rather withf (t )exp{−λct}, wheref grows to infinity ast→ ∞and at the same time is subexponential. (Take, for example,L=/2 andβ >0 constant onR^d, thenf (t )=t^d/2.) In fact the growth is pure exponential if and only ifL+β is product-critical (see the definition later in this subsection). Proving SLLN seems to be significantly harder in the general case involving the subexponential termf.

In the late seventies Asmussen and Hering wrote a series of papers concerning weak and strong laws of large numbers for a reasonably general class of branching processes which included branching diffusions. See [1] and [2]. In the context of the branching diffusions we consider here one can summarize briefly their achievements by saying that, whenDisbounded, for a special class of operatorsL+β, the rescaled process{exp{−λ_ct}X_t: t≥0}converges in the vague topology, almost surely for branching diffusions. Further, for the same class ofL+βwhenDis unbounded they proved that there exists the limitin probabilityof exp{−λct}Xt ast↑ ∞(in the vague topology). The class of L+β alluded to they called “positively regular.” The latter is a subclass of the classP_p^∗(D)(the class that we shall work with) given below.

A more detailed comparison with [1,2] as well as the discussion on related results on superprocesses is deferred to Section2.

Before we give the definition of the basic classes of operators that we shall use, Pp(D)andP_p^∗(D), we need to recall certain concepts of the so-calledcriticality theory of second-order operators. The operatorL+β−λ_c is calledcriticalif the associated space of positive harmonic functions is nonempty but the operator does not possess a (minimal positive) Green’s function. In this case the space of positive harmonic functions is in fact one-dimensional.

Moreover, the space of positive harmonic functions of the adjoint ofL+β−λ_cis also one dimensional.

Assumption 3. Suppose we choose representatives of these two spaces to beφ andφ respectively.Throughout the paper and without further reference,we will always assume thatL+β−λ_cis product-critical,i.e. φ,φ<∞,and in this case we pickφandφwith the normalization φ,φ =1.

We now define the classesPp(D)andPp^∗(D). Since we want to talk about spatial spread on a generic domainD, we fix, for the rest of the paper, an arbitrary family of domains{D_t, t≥0}withD_t⊂⊂D, D_t↑D. (ForD=R^d,D_t can be thet-ball, but we can take any other family withDt⊂⊂D, Dt↑Dtoo.)

Definition 4 (Pp(D)andPp^∗(D)). Forp≥1,we writeL+β∈Pp(D)if (i) λ_c=λ_c(L+β;D) >0.

(ii) φ^p,φ<∞,in which case we say thatL+β−λcis productp-critical.

Letq(x, y, t)be transition density ofL+β and Q(x, y, t):=q(x, y, t)−e^λctφ(y)φ(x).

We writeL+β∈P_p^∗(D)when the following additional conditions hold for each givenx∈Dand∅=B⊂⊂D.

(iii) There exists a functiona:[0,∞)→ [0,∞)such that for allδ >0, P_δx

∃n0,∀n > n0: supp(Xnδ)⊂D_anδ

=1.

(iv) There exists a functionζ:[0,∞)→ [0,∞)such that,ast↑ ∞, (1) ζ (t)↑ ∞,

(2) ζ (a_t)=O(t), (3) α_t:= sup

z∈D_t,y∈B

|Q(z, y, ζ (t))| φ(y)φ(z) =o

e^λct .

Letp(x, y, t)denote the transition density of the diffusion corresponding to the operator(L+β−λ_c)^φ.Then p(x, y, t)=e^−λctφ(y)φ⁻¹(x)q(x, y, t),and thus, (iv)is equivalent to

(iv*) With the sameζ as in(iv),

tlim→∞ sup

z∈Dt,y∈B

p(z, y, ζ (t )) φφ(y) −1

=0.

Note thatadepends onxandζ, αdepend onxandBthrough (2) and (3). For notational efficiency and on account of the fact that in our proofs no uniformity inxandBis required, we have chosen not to emphasize this dependency.

Moreover, it is often the case thatζ andαin factdo not dependonxorB, as we shall see in the examples of Section3 where explicit cases of these quantities are discussed.

Remark 5 (Ergodicity). Note that criticality is invariant underh-transforms.Moreover,an easy computation shows thatφandφtransforms into1 andφφrespectively when turning from(L+β−λ_c)to theh-transformed(h=φ) operator(L+β−λ_c)^φ=L+aφ⁻¹∇φ· ∇.Therefore,product criticality is invariant underh-transforms too(this is not the case with productp-criticality whenp >1).Further,for operators with no zeroth-order term,it is equivalent to positive recurrence(ergodicity)of the corresponding diffusion process.In particular,(L+β−λ_c)^φ corresponds to an ergodic diffusion process provided(L+β−λ_c)is product critical(see[27],Section4.9).

1.3. Main results

With the following theorem we wish to address the issue of almost sure convergence in the vague topology of {exp{−λ_ct}X_t: t≥0}for branching diffusions withL+β belonging toPp^∗(D), p >1 thus generalizing the results of Asmussen and Hering.

Note that sinceL+β−λ_cis critical,φis the unique (up to constant multiples)invariant positive functionfor the linear semigroup corresponding toL+β−λ_c(Theorem 4.8.6. in [27]). Let{S_t}t≥0denote the semigroup correspond- ing toL+β. It is a standard fact (sometimes called “the one particle picture”) that

S_t(g)(x)=E_δx g, X_t (3)

for all nonnegative bounded measurableg’s. Even thoughφis not necessarily bounded from above,St(φ)makes sense and (3) remains valid whengis replaced byφ, becauseφcan be approximated with a monotone increasing sequence ofg’s and the finiteness of the limit is guaranteed precisely by the invariance property ofφ. By the invariance ofφ,

E_δxe^−λct φ, X_t =e^−λctS_t(φ)(x)=φ(x), which, together with the branching property, is sufficient to deduce that W^φis a martingale where

W_t^φ:=e^−λct φ, X_t, t≥0.

Indeed note that Eδ_x

e^−λc(t+s) φ, Xt+s|Ft

=e^−λctEX_t

e^−λcs φ, Xs

=e^−λct φ, Xt.

Being a positive martingale, P_δx-almost sure convergence is guaranteed, and the a.s. martingale limit W_∞^φ :=

limt→∞W_t^φappears in the following main conclusion.

LetC_c⁺(D)denote the space of nonnegative, continuous and compactly supported functions onD.

Theorem 6 (SLLN). Assume thatL+β∈P_p^∗(D)for somep∈(1,2]and βφ^p,φ<∞.Then,

tlim↑∞e^−λct g, Xt = g,φW_∞^φ, g∈C_c⁺(D) (4)

holdsPδ_x-a.s.for allx∈D,andEδ_x(W_∞^φ)=φ(x).Moreover,ifsup_Dβ <∞then the restrictionp∈(1,2]can be replaced byp >1.

We close this subsection with the Weak Law of Large Numbers. Here we change the classP_p^∗(D)to the larger class Pp(D)and getL¹(P_δx)-convergence instead of a.s. convergence (hence the use of the word “weak”). It is important to point out, however, that the classPp^∗(D)is already quite large – see Section3, where we verify that key examples from the literature are in fact inPp^∗(D)and thus obey the SLLN.

Theorem 7 (WLLN). Suppose thatL+β∈Pp(D)for somep∈(1,2]and βφ^p,φ<∞.Then for allx∈D, (4) holds in the L¹(P_δx)sense andE_δx(W_∞^φ)=φ(x).Moreover,ifsup_Dβ <∞ then the restrictionp∈(1,2]can be replaced byp >1.

In closing, we would like to mention an important concept that will play a central role in the proof of Theorems6 and7: the so called “spine” will intuitively represent a “typical” particle within the branching diffusion whose motion is governed by the operator (L+β−λ_c)^φ=L+aφ⁻¹∇φ· ∇. In Section4, we give the spine construction and probabilistic interpretation of the branching diffusion under a change of measure using martingaleW^φ. The reader may find it helpful to familiarize themselves with this notion even before reading the proofs. Also see, for example, Section 4.1 of [12] and [22].

1.4. Outline

The rest of this paper is organized as follows. In Section2we embed our results into the literature, while in Section3 we discuss some key examples for the SLLN. The proofs are given in Section4.

2. Detailed comparison with some older results

The methods of Asmussen and Hering were based for the most part on classical techniques of truncation and ap- plications of the Borel–Cantelli lemma. Using this method, they proved the convergence of e^−λct X_t, g for all 0≤g∈L¹(φ(x)dx). It is also worth noting that the generic strength of their method extended to many other types of branching processes; discrete time, discrete space and so on.

Interestingly, preceding all work of Asmussen and Hering is the single article [29] (later improved upon by [4]).

Watanabe demonstrates that when a suitable Fourier analysis is available with respect to the operatorL+β, then by spectrally expanding anyg∈C_c⁺(D),one can show that{ g, Xt: t≥0}is almost surely asymptotically equivalent to its mean. From this the classic Strong Law of Large Numbers for dyadic branching Brownian motion inR^d is

recovered. Namely that whenL= /2 andβ >0 is a constant,

tlim↑∞t^d/2e^−βtX_t(B)=(2π)^d/2|B| ×N_μ,

whereBis any Borel set,|B|is its Lebesgue measure andN_μis a strictly positive random variable depending on the initial configurationμ∈M(R^d). The operator 1/2+βdoes not fall into the classP1(D). (This is important since [16,17] and also [8] assumes that the operator is inP1(D).) For an analogous result on supercritical super-Brownian motion see [13].

Let us discuss now how our assumptions relate to the assumptions imposed in the article [1].

In [1] the domain is bounded (and even one dimensional) when the Strong Law of Large Numbers is stated for branching diffusions; on general domains, only convergence in probability was obtained. Furthermore, in [1] the notion ofpositively regularoperators was introduced. In our context it first means that:

(A) λ_c>0 (in [1] this property is called “supercriticality”), (B) φis bounded from above,

(C) φ,1<∞.

Obviously, (B–C) is stronger than the assumption φ,φ<∞(product-criticality).

Secondly, {S_t}t≥0, the semigroup corresponding toL+β (the so called “expectation semigroup”) satisfies the following condition. Ifηis a nonnegative, bounded measurable function onR^d, then

(D) S_t(η)(x)= η,φφ(x)

e^λct+o e^λct

ast↑ ∞,uniformly inη.

LetT_t be the semigroup defined byT_t(f ):=S_t^φ(f )=_φ¹S_t(φf ),for all 0≤f measurable withφf being bounded.

ThenT_t correspond to theh-transformed (h=φ) operatorL^φ₀. Recall that L^φ₀ corresponds to a positive recurrent diffusion process. Then, assuming thatφis bounded, it is easy to check that the following condition would suffice for (D) to hold:

tlim↑∞sup

x∈D

sup

g≤1

g, φφ⁻¹T_t(g)− g, φφ=0, (5)

where · denotes sup-norm. However this is not true in most cases on unbounded domains (or even on bounded domains with general unbounded coefficients) because of the requirement on the uniformity inx. (See our examples in Section3– neither of the examples onR^dsatisfy (5).)

Turning to superprocesses, there would seem to be considerably fewer results of this kind in the literature (see the references [10,11,18] for superprocesses in general). The most recent and general work in this area we are aware of are [9,13,16,17].

In [16] it was proved that (in the vague topology){exp{−λ_ct}X_t: t≥0}converges in law whereXis the so called (L, β, α,R^d)-superprocess (with αbeing the “activity parameter”) satisfying that L+β ∈P1(D) and thatαφ is bounded from above. (An additional requirement was that φ, μ<∞ whereμ=X0 is the deterministic starting measure.) The long and technical proof relied heavily on the theory of dynamical systems applied to the Laplace transforms of{e^−λctφX_t, t≥0}.

In [17] the convergence in law was replaced by convergence in probability. Furthermore, instead ofR^da general Euclidean domainD⊆R^dwas considered. The heavy analytic method of [16] was replaced by a different, simpler and more probabilistic one. The main tool was the introduction of a “weighted superprocess” obtained by a “space–time H-transform.”

Very recently, in [8], almost sure limits were proven for a class of Markov branching processes,¹ using mostly functional analytic methods. The main difference between the setup in [8] and our setup are that

(i) [8] used anL²-approach and therefore the setting had to be restricted to symmetric operators, whereas our results are applicable for non-symmetric operators as well, as long as the other conditions are satisfied (a concrete class of such operators is given in Example13of the next section).

1A similar result for superprocesses has been obtained in [9].

(ii) Even within the symmetric case, our milder spectral assumptions include e.g. Examples10and11of the next section, which do not satisfy the assumptions in [8]. To be more specific, Example 10 does not satisfy the assumption that the ground stateφis upper bounded. In Example11, sinceβ is constant, the assumption thatβ belongs to the classK_∞(Y )is not satisfied. (The classK_∞(Y )depends on the motion processY, and is defined in [8] with the help of standard Kato classes; it contains rapidly decaying functions.)

(iii) While [8] uses a functional analytic (Hilbert space) approach, our method is more probabilistic and is based on a

“spine-decomposition.”

3. Examples

In this section we give examples which satisfy all the assumptions we have, and thus, according to Theorem6, obey the SLLN. (Those examples do not fall into the setting in [1,2] and two of them are not covered by [8] either.)

Before we turn to the specific examples, we give some heuristics. Although these are not actually needed for understanding the examples, we feel that the reader “gets a more complete picture” by first reading them.

Remark 8 (Expectation calculations and local vs. global growth rates). From(3),we have E_x 1_{·∈dy}, X_t =e^λctφ(x)

φ(y)p(t, x,dy) and then,by ergodicity,

e^−λctE_x 1, Xt =φ(x)

R^d

p(t, x, y)

φ(y) dy→φ(x)

R^d

φ(y)dy, ast→ ∞.

Hence,if φ,1<∞,then the global population growth is the same as the local population growth(in expectation), whereas,if φ,1 = ∞the global growth rate exceeds the local growth rate.

Remark 9 (Heuristics foraandζ). One may wonder how one can find the functionsaandζ as in Definition4(iii)–

(iv).In fact,it will often be straightforward to find them.

Fixx∈D.If,for example,we can pick a deterministic increasing functionasuch that,for allδ >0, ∞

n=1

P_δx

supp(Xnδ)⊂D_anδ

<∞,

then Borel–Cantelli says that the functionais an appropriate choice,if alsoζ (a_t)=O(t)holds.Since the probability one particle is present in a set is trivially dominated by the expected particle number in that set,it will be much easier to check that

∞ n=1

E_δx 1D_anδ^c , X_nδ<∞.

IfD=R^dandD_t=B_t and we can choosea_t such that,for someε >0,

|y|>at

p(x, y, t)

φ(y) dy <e^−(λc+ε)t, then we will have satisfied

∞ n=1

Ex 1D_anδ^c , Xnδ<∞.

Heuristically,at least for a nicely decaying φ,if the transition density² p(t, x, y)converges to its equilibrium φ(y)φ(y)sufficiently quickly even for very largey,we might hope to take

a_t≈F⁻¹ e^λct

, whereF (α)=

|y|>α

φ(y)dy.

If the spine starts at a very large position,since it is ergodic it will tend to move back toward the origin,albeit taking a potentially large time,and Ventcel-Freidlin large deviation theory suggests that it will “closely” follow the path of a deterministic particle with the same drift function.We can use this to guess for a suitable form forζ (t).At least heuristically,to find out how far away the spine particle may start in order that it both returns to the vicinity of the origin and then ergodizes towards its invariant measure before large timet,we can consider the deterministic differential equation

f (t )˙ =μ f (t )

−σ²

f (t )∇φ(f (t)) φ(f (t))

whenL=¹₂σ²(x)−μ(x)· ∇,and,for example,often takeζ (t)a little larger than|f⁻¹(t)|in one dimensional settings.

Indeed,these heuristics appear to the correct form for botha_t andζ (t)in the examples considered below. Example 10 (OU process with quadratic breeding rate). Letσ, μ, a, b >0and consider

L:=1

2σ²−μx· ∇ onR^d

corresponding to an(inward)Ornstein–Uhlenbeck process and letβ(x):=bx²+a.SinceLcorresponds to a recur- rent diffusion andβ is a smooth function withβ≥0andβ≡0,it follows thatλ_c>0 (see Chapter4in[27]).The equilibrium distribution forLis given by a normal density,

π(x)= μ

πσ² d/2

exp

−μ σ²x²

.

Suppose thatμ > σ√

2b.Definingγ^±:= _2σ¹2(μ±

μ²−2bσ²),for the principal eigenvalue problem with(L+ β)φ=λ_cφwe can take

λc:=σ²γ⁻+a, φ(x):=c⁻exp

γ⁻x² and φ(x)=c⁺exp

−γ⁺x² ,

wherec⁻:=(1−(2bσ²/μ²))^d/8,c⁺:=c⁻(μ/(π σ²))^d/2.Note thatL+βis a self-adjoint operator with respect to π,and for theh-transform of a second-order operatorA,

φ_Ah=hφ_A.

Calculations using the “one-particle picture”(Eq. (3))reveal that,in expectation,the support of the process grows like

λct /γ⁺and one can pickat=

λt/γ⁺for anyλ > λcand condition(iii)in Definition4will hold.

The spine is also an(inward)Ornstein–Uhlenbeck process with parameterα:=μ−2γ⁻σ²=

μ²−2bσ²with (L+β−λc)^φ=L+σ²∇φ

φ · ∇ =1

2σ²−αx· ∇ onR^d, and transition density

p(x, y, t)=

α

π σ²(1−e^{−2(α/σ2)t}) d/2

exp

−α_d

i=1(y_i−x_ie^−(α/σ2)t)² σ²(1−e^{−2(α/σ2)t})

.

2Later we will see that this transition density corresponds to the “spine.”

We see that the drift of the inward OU reduces the influence of any starting position exponentially in time.Indeed,one can takeζ (t)=(1+ε)(σ²/2α)logtfor anyε >0for condition(iv^∗)in Definition4to hold.Finally,we trivially note thatζ (a_t)=O(t)(in fact,onlylogt growth),hence,forpsufficiently close to1,all necessary conditions are satisfied for our strong law theorem to hold.

Note that a strong law for a generalization of this model can be found in[23]where the convergence is proved using a martingale expansion for continuous functionsg∈L²(π )(rather than compactly supportedg).Almost sure asymptotic growth rates(and a.s.support)for the same model are studied in[20].

This is certainly a non-trivial model and it highlights the strength of our general result.In particular,a quadratic breeding rate is critical in the sense that a BBM with breeding rateβ(x)=const·x^pexplodes in a finite time a.s.if and only ifp >2,with explosion in the expected population size whenp=2.When a branching inward OU process with quadratic breeding is considered here,a strong enough drift withμ > σ√

2b can balance the high breeding, whereas any lower drift would lead to a dramatically different behavior.

Example 11 (Outward OU process with constant breeding rate). Letσ, μ >0;b > dμand consider L:=1

2σ²+μx· ∇ onR^d

corresponding to an “outward” Ornstein–Uhlenbeck process and let β(·)≡b.As the spatial movements have no affect on the branching,the global population grows likee^βt and this is achieved “naturally” with particles moving freely.This corresponds to(L+β)φ=bφwithφ≡1.On the other hand,the principal eigenvalue isλc=b−dμ < b withφ(x)=const·exp{−(μ/σ²)x²},it being associated with thelocal,as opposed to global,growth rate.

After some similar expectation calculations to the inward OU in quadratic potential, an upper bound on the process’ spread is roughly the same as for an individual outward OU particle,that is,we can takea_t =exp{(1+ δ)(μ/σ²)t}for anyδ >0.

Despite the transient nature of the original motion,the spine is an inward OU process

(L+β−λ_c)^φ=L+σ²∇φ

φ · ∇ =1

2σ²−μx· ∇onR^d,

with equilibrium φφ(x)∝exp{−(μ/σ²)x²}.Intuitively,this is the motion that maximizes the local growth rate at λ_c(here it is the original motion “conditioned to keep returning to the origin”).We can therefore takeζ (t)=(1+ ε)(σ²/μ)logt for anyε >0and hence still find thatζ (a_t)=(1+ε)(1+δ)t=O(t).All the conditions required for the strong law to hold are again satisfied.

Example 12 (BBM withβ∈C_c⁺(R^d)andβ≡0 ford=1,2). Consider the(¹₂+β)-branching diffusion where β∈C_c⁺(R^d)andβ≡0ford=1,2.Since Brownian motion is recurrent in dimensiond=1,2,it follows thatλ_c>0 and in fact,the operator ¹₂+β−λ_cis product-critical and even product-p-critical for allp >1 (see Example22 in[16]).

We now show how to find aζ that satisfies(iv^∗)in Definition4.We do it ford=1,thed=2case is similar.

Letb >0be so large that supp(β)⊂ [−b, b]and letM:=max_Rβ.Recall thatp(t, x, y)denotes the(ergodic) kernel corresponding to(¹₂+β−λ_c)^φ.In this exampleP will denote the corresponding probability.By comparison with the constant branching rate case,it is evident that a_t :=√

2M·t is an appropriate choice,because it is well known that a BBM with constant rateMhas velocity√

2M.Therefore,we have to find aζ which satisfies that for any fixed ballB,

tlim→∞sup

|z|≤t

p(z, B, ζ (t ))

Bφφ(y)dy −1 =0 together with the condition thatζ (a_t)=ζ (√

2M·t )=O(t)asn→ ∞.

An easy computation(see again Example22in[16])shows that onR\ [−b, b], 1

2+β−λ_c φ

=1

2−sgn(x)· 2λc

d dx,

wheresgn(x):=x/|x|, x=0.Fix anεand letτ_±bandτ₀denote the first hitting time(by a single Brownian particle) of[−b, b]and of0,respectively.We first show that ast→ ∞,

sup

b<|x|≤t

Px

τ_±b>t (1+ε)

√2λc

→0. (6)

Obviously,it is enough to show that,for example, Pt

τ₀>t (1+ε)

√2λc

→0, wherePcorresponds to ¹₂−√

2λc d

dx on[0,∞).Indeed,if Wdenotes standard Brownian motion starting at the origin with probabilityQ,then

Pt

τ0>t (1+ε)

√2λc

≤Pt[Y_{t (1+ε)/}^√_2λ

c>0] =Q

t− 2λc

t (1+ε)

√2λc

+Wt (1+ε)/√ 2λ_c>0

=Q[Wt (1+ε)/√

2λ_c> εt] →0 (the last term tends to zero by the SLLN forW).

We now claim thatζ (t):=^{t (1√+}_2λ^2ε)

c satisfies

tlim→∞sup

|z|≤t

p(z, B, ζ (t ))

Bφφ(y)dy −1 =0.

(The conditionζ (a_t)=O(t) is obviously satisfied.) By the ergodicity ofp(t, x, y),it is sufficient to show that ζ satisfies

tlim→∞ sup

b<|z|≤t

p(z, B, ζ (t ))

Bφφ(y)dy −1 =0.

Let,for example,b < x≤t.By the strong Markov property atτ_b(the hitting time ofb)and by(6), p(x, B, ζ (t ))

Bφφ(y)dy =p(b, B, ζ (t )−t (1+ε)/√ 2λc)

Bφφ(y)dy P_x

τ_b≤t (1+ε)

√2λc

+o(1), uniformly inb < x≤t.

Finally,

tlim→∞

p(b, B, ζ (t )−t (1+ε)/√ 2λc)

Bφφ(y)dy =1

becausep(t, x, y)is an ergodic kernel and

tlim→∞

ζ (t)−t (1+ε)

√2λc

= lim

t→∞

√t ε 2λc

= ∞,

completing the proof of our claim aboutζ.

Example 13 (Non-symmetric operator). For the sake of concreteness, we give here a simple example for a non- symmetric operator that satisfies our assumptions,by slightly modifying the setting of Example12. (For more on symmetric operators,see Section4.10in[27].)

In Example12set d=2.Now add a driftb(x, y)as follows.Letb=(b1, b2)^T,whereb1(x, y):=m(x)n(y)and b2(x, y):=p(x)q(y).Ifm, n, p, qare smooth compactly supported functions,then so isb,and the same argument

as in the previous example shows that the conditions are satisfied,but ifm(x)n(y)is not equal top(x)q(y)for all (x, y),that is,ifm/p(x)is not equal toq/n(y),then the operator is not symmetric,because thenbis not a gradient vector.

Hence,wheneverq/nis not a constant orm/pis not a constant,this setting constitutes a non-symmetric example for the SLLN.

Example 14 (Bounded domain). First note that whenDis bounded,an important subset ofPp(D), p >1is formed by the operators L+β which are uniformly elliptic on D with bounded coefficients which are smooth up to the boundary ofDand withλ_c>0.That is,in this caseL+β−λ_cis critical(see[27],Section4.7),and sinceφand φare Dirichlet eigenfunctions(are zero at the boundary ofD),it is even product-p-critical for allp >1.Theorem7 thus applies.

Although in this caseY is not conservative inD,in fact even Theorem6will be applicable whenever(iv^∗)can be strengthened to the following uniform convergence onD:

tlim→∞ sup

z∈D,y∈B

p(z, y, ζ (t )) φφ(y) −1

=0. (7) (Note that[1]has a similar global uniformity assumption – see the paragraph after(5).)Indeed,then the proof of Theorem6(which can be found later,in Section4)can be simplified,because the functionais not actually needed:

D_ancan be replaced byDfor alln≥1.

As far as(7)is concerned,it is often relatively easy to check.For example,assume thatd=1 (the method can be extended for radially symmetric settings too)and so letD=(r, s).Then the drift term of the spine isb+a(logφ). Now,if this is negative and bounded away from zero ats−ε < x < s and positive and bounded away from zero at r < x < r+εwith someε∈(0, s−r),then(7)can be verified by a method similar to the one in the previous example.

The above condition on the drift is not hard to check in a concrete example(note that sinceφsatisfies the Dirichlet boundary condition, logφtends to−∞at the boundary).

If we relax the regularity assumptions onL+β then,for example,φis not necessarily upper bounded,and so we are leaving the family of operators handled in[2] (see the four paragraphs preceding(5));nevertheless our method still works as long asL+β∈Pp^∗(D), p >1 (for the SLLN)orL+β∈Pp(D), p >1 (for the WLLN).

4. Proofs

4.1. A spine approach

To establish theL^p(P_δx)convergence ofW^φ forp >1 we appeal to a, by now, standard technique that have been introduced to the literature by [26] and by the references given there (in particular [25]) and involves a change of measure inducing a “spine” decomposition. Similar applications can be found in [3,5,14,21] to name but a few. See, for example, [18,19] as well as the discussion in [14] for yet further references.

It is important to point out that we will need the spine decomposition not only to establish the L^p-convergence mentioned above but also in the key lemma (Lemma18) in the proof of Theorem6. In both cases, we found the spine method to be indispensable and we were not able to replace it by otherL^pmethods.

Before we can state our spine decomposition, we need to recall some facts concerning changes of measures for diffusions and Poisson processes.

Girsanov change of measure

Suppose thatY is adapted to some filtration{Gt: t≥0}. Under the change of measure dP^φx

dPx

G^t =φ(Y_t) φ(x)e⁻

t

0(λ_c−β(Y_s))ds

(8) the process(Y,P^φx)corresponds to theh-transformed(h=φ)generator(L+β−λ_c)^φ=L+aφ⁻¹∇φ· ∇. Note now in particular that sinceL+β∈P1(D), it follows that(Y,P^φx)is an ergodic diffusion with transition densityp(x, y, t) and an invariant densityφφ.

Change of measure for Poisson processes

Suppose that given a non-negative continuous functiong(t), t≥0,the Poisson process(n,L^g)wheren= {{σ_i: i= 1, . . . , nt}: t≥0}has instantaneous rateg(t). Further, assume thatnis adapted to{Gt: t≥0}. Then under the change of measure

dL^2g dL^g

Gt

=2^ntexp

− _t

0

g(s)ds

the process(n,L^2g)is also a Poisson process with rate 2g. See Chapter 3 in [24].

Theorem 15 (The spine construction). Let{Ft: t≥0}be the natural filtration generated byX.Define the change of measure

dPδ_x

dPδ_x

Ft

=e^−λct φ, Xt φ(x) = W_t^φ

φ(x).

Then,underPδ_x,Xcan be constructed as follows:

• a single particle,Y = {Y_t}t≥0,referred to as the spine,initially starts atxand moves as a diffusion corresponding to theh-transformed operatorL+aφ⁻¹∇φ· ∇;

• the spine undergoes fission into two particles at an accelerated rate 2β(Yt)at timet,one of which is selected uniformly at random to continue the spine motionY;

• the remaining child gives rise to an independent copy of aP-branching diffusion started at its space–time point of creation.

A similar construction for BBM was established in Chauvin and Rouault [7]. See Theorem 5 in [14] on how to prove it.³

Remark 16 (The spine decomposition). Theorem15says that(X,P_δx)has the same law as a process constructed in the following way.A(Y,P^φx)-diffusion is initiated along which(L, β;D)-branching processes immigrate at space–

time points{(Yσ_i, σi): i≥1}where,givenY, n= {{σi: i=1, . . . , nt}: t≥0}is a Poisson process with lawL^{2β(Y )}.It will often be very useful to think of(X,P_δx)as being constructed in this richer way and it will be convenient to define the natural filtration of the spine and the birth process along the spine asGt:=σ (Ys, ns: s≤t ).Note that using the

“spine construction” of(X,P ), we can write

W_t^φ=e^−λctφ(Yt)+

nt

i=1

e^−λcσiWi,

where,conditional on the spine filtrationGt,W_iis an independent copy of the martingaleW_t^φstarted from positionY_σi and run for timet−σ_iwhereσ_iis theith fission time along the spine fori=1, . . . , nt.Remembering that particles off the spine behave the same as if under the original measureP and that the martingale property givesEδ_x(W_t^φ)=φ(x), we then have the so called “spine decomposition”:

E W_t^φ|Gt

=e^−λctφ(Y_t)+

nt

i=1

e^−λcσiφ(Y_σi). (9)

3Although the construction in [14] was for a branching process with killing on the boundary of a compact domain, the analysis applies almost verbatim with obvious changes to take account of the fact that there are no boundary conditions; this straightforward exercise is left to the reader.