Law of large numbers for superdiffusions : the non-ergodic case

www.imstat.org/aihp 2009, Vol. 45, No. 1, 1–6

DOI: 10.1214/07-AIHP156

© Association des Publications de l’Institut Henri Poincaré, 2009

Law of large numbers for superdiffusions: The non-ergodic case

János Engländer

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

E-mail: [email protected]

Received 2 January 2007; revised 22 July 2007; accepted 23 August 2007

Abstract. In previous work of D. Turaev, A. Winter and the author, the Law of Large Numbers for the local mass of certain superdiffusions was proved under an ergodicity assumption. In this paper we go beyond ergodicity, that is we consider cases when the scaling for the expectation of the local mass is not purely exponential.Inter alia, we prove the analog of the Watanabe–Biggins LLN for super-Brownian motion.

Résumé. Dans un travail précédent, l’auteur, D. Turaev et A. Winter, ont prouvé la Loi des Grand Nombres pour la masse locale de certaines diffusions sous une hypothèse d’ergodicité. Dans cet article nous allons au delà de l’ergodicité, plus précisement nous considérons des cas où le scaling de l’espérance de la masse locale n’est pas purement exponentiel. Entre autres, nous prouvons l’analogue de la LGN de Watanabe–Biggins pour le super mouvement brownien.

MSC:60J60; 60J80

Keywords:Super-Brownian motion; Superdiffusion; Superprocess; Law of Large Numbers;H-transform; Weighted superprocess; Scaling limit;

Local extinction

1. Introduction and statement of results

Since this paper extends some results of [5], it will be helpful if the reader has [5] at hand, especially regarding notation and the notion ofH-transformed (or weighted) superprocesses.

LetD⊆R^d be a domain. We use the standard notationMf(D),μ, μ∈Mf(D),C_b⁺(D),C⁺_c(D),C^k,η(D) andC^η(D)exactly as in [5]. For example,C^k,η(D)is the usual space of functions whosekth order derivatives are η-Hölder. Furthermore,A⊂⊂Dmeans that the closure of the bounded domainAis inD. LetLbe an elliptic operator onDof the form

L:=1

2∇ ·a∇ +b· ∇, (1)

wherea_i,j, b_i ∈C^1,η(D),i, j =1, . . . , d, for some η∈(0,1], and the matrixa(x):=(a_i,j(x)) is symmetric, and positive definite for allx∈D. In addition, letα, β∈C^η(D), and assume thatαis positive andβ is bounded from above. In this paper X denotes the (L, β, α;D)-superdiffusion (for the definition, even with time-inhomogeneous coefficients, see [5]). Hereαis the ‘variance’ or ‘intensity’ parameter andβ is the ‘mass creation’ or ‘growth bias’

term. The corresponding probabilities will be denoted by{P^μ;μ∈Mf(D)}.

In order to understand what follows, we need a brief review on some concepts from thecriticality theory of second order elliptic operators. Let

λc=λc(L+β, D):=inf{λ∈R: ∃u >0 satisfying(L+β−λ)u=0 inD}

denote the generalized principal eigenvaluefor L+β on D. By standard theory, λ_c<∞ whenever β is upper bounded. The operatorL+β−λcis 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−λc

is also one dimensional. Suppose thatφandφ are representatives of these two spaces. We say thatL+β−λ_c is product-critical, if, in addition to criticality,

Dφφdx <∞holds (in which case one usually picksφandφwith the normalization

Dφφdx=1).

Our principal interest is in establishing a Weak Law of Large Numbers (WLLN) for the local mass of certain superdiffusions. Since Pinsky proved thatXexhibits local extinction if and only ifλ_c≤0, we can only hope to have the WLLN if we assume thatλc>0. In fact, the following has been shown in [5]:

Proposition 1 ([5], Theorem 1). In addition to the assumptionλ_c>0,also assume thatL+β−λ_c is product- critical,thatαφ_cis bounded and thatXstarts in a stateμwithμ, φ_c<∞.Letf∈C_c⁺(D).Iff ≡0andμ =0, then there exists a nonnegative non-degenerate random variable denoted byW_∞satisfying that

t→∞lim

Xt, f

E^μXt, f= W_∞

μ, φc, inP^μ-probability. (2)

(The precise definition ofW_∞is given in the paragraph preceding Theorem 1 of this paper.)

In fact, product-criticality is equivalent to the property that the semigroup corresponding toL+β scales precisely exponentially (see again [5], or see [4]). For example a simple case of a superdiffusion is whenD=R^d, d≥1, L=

1

2Δ, withα, β positive constants (supercritical super-Brownian motion). Thenλ_c=β, φ=φ≡1 and sothis case is not included in the setupof [5]. On the other hand, the corresponding (Strong) LLN is well known fordiscrete particle systems. Based on ideas in [7], Biggins [1] proved the Strong LLN for the case when branching-Brownian motion is replaced by branching random walk (in discrete time).

The purpose of this paper is to prove the WLLN for a class of superprocesses thatincludes supercritical super- Brownian motion. Instead of trying to adapt the Watanabe–Biggins approach to our setting, our method will use some ingredients from [5] and some results from [3,6] too. Note that the Watanabe–Biggins result gives an a.s. limit; thus, it would be interesting to see if our main result (see Theorem 1 later) can be strengthened to an a.s. result.

Throughout the paper the following, fairly mild assumption will be in force.

Assumption 1. LetS= {S_t}t≥0denote the semigroup corresponding toL+β−λ_conD.

(A.1) (Local survival)λ_c>0.

(A.2) (Scaling of linear semigroup)There exist two functionss:(0,∞)→(0,∞), andh:D→(0,∞), h∈C^2,η(D), and a locally finite measurer(dx)such that

(a) logs_t=O(logt )(i.e. sup_{t >0}^logs_logt^t <∞) and limt→∞(logs_t)=0, (b) sup_Dαh <∞,

(c) limt→∞μ(dx), st·S_t(f )(x) = r, f · μ, hfor allf ∈C⁺_c(D), andμ∈ ¹_hMf(D).

(A.3) (Spatial spread)There exist a function,ζ:(0,∞)→(0,∞)and a family of subdomains{Dt;t≥0}, Dt ⊂D such that

(a) limt→∞ζt= ∞,

(b) log(t+ζ_t)=o(t ), t→ ∞, (c) limt→∞st+ζ_t/sζ_t =1,

(d) limt→∞P^μ[X_t(D_t) >0] =0, (e) iff ∈C_c⁺(D), then

tlim→∞sup

x∈Dt

s_ζt

h(x)·Sζ_t(f )(x)− r, f =0.

Let H (x, t ):=exp(−λ_ct )h(x), where λ_c and h are as in Assumption 1 and consider the weighted superprocess (X^H,P^hμ)defined by

X^H_t (dx):=H (t, x)X_t(dx), t≥0.

LetXt

H := X_t^H.Following [5], we first show that

tlim→∞Var^hμ Xt

H

= _∞

0

dse^−2λcs μ,Ss

αh²

<∞, (3)

and that if the diffusion processY corresponding toL^h₀:=L+a^∇_h^h· ∇onDis conservative (that is, it never leaves Dwith probability one), thenX^H is a uniformly integrable (UI)P^hμ-martingale, whereas in general,X^H is a (non- negative)P^hμ-supermartingale.

LetS= {S_s}s≥0denote the semigroup corresponding toY(i.e. toL^h₀), that is,S:=S^h. Clearly,S_s1≤1.(IfY is conservative, then actuallySs1=1.) IfYis conservative, then consider the class

C(D):=

f ∈C²(D):∃⊂Dbounded s.t.⊂D; f =const onD\ . By Theorem A2 in [2], we have that for allf ∈C(D),

dX^H_t , f

= X^H_t , L^h₀f

dt+dMt(f ), (4)

where{M_t(f )}t≥0is a square-integrableP^hμ-martingale, and its quadratic variationM(f )is given by M(f )

t= _t

0

dse^−λcs X^H_s , αhf²

, t≥0. (5)

(The point is that one can take the function classC(D)instead of justC_c²(D)whenYis conservative.) Applying (4) to the functionf≡1, it follows thatX^H is aP^hμ-martingale. Furthermore, by (5),

E^hμ X_t^H,12

= μ, h²+ _t

0

dse^−λcs hμ,Ss[αh]

. (6)

That is Var^hμ

X^H_t

= _t

0

dse^−λcs hμ,S_s[αh]

= _t

0

dse^−2λcs μ, S_s αh²

. (7)

Lettingt→ ∞we obtain (3). Replacingtby∞in the first of the integrals in (7), we have from the fact thatSs1≤1 and from our assumptions that

Var^hμ X^H_t

≤ _∞

0

dse^−λcs hμ,Ss[αh]

≤λc−1αh∞μ, h<∞. Hence, by (6), sup_t≥0E^μh(X^H_t )²<∞,and consequentlyX^H is UI.

AbbreviateW:=X^Hand letWdenote thetotal mass process:W:=X^H= X^H.LetD:=D∪ {Δ}be theone- point compactificationofD(when the underlying diffusion processYis non-conservative onD,Δis thecemetery state forY). Relaxing the assumption on the conservativeness ofY, the argument in [2], pp. 726–727 shows that, although one cannot work directly with the function class C(D)(only with its subclass C_c²(D)), one can extend (W, P)appropriately and get(X, P)onDmakingXaP^hμ-martingale. Since the mass on the cemetery stateΔ is nondecreasing in time,W is aP^hμ-supermartingale. (In the non-conservative case, intuitively, mass is ‘lost’ at the Euclidean boundary ofDor at infinity.)

Now, ifW_∞:=limt→∞W_t, then one does not haveE^μW_∞= μ, hin general. Nevertheless, as we have seen, whenY is conservative onD,W is a UI martingale and then, of course,E^μW_∞= μ, h>0 forμ =0.

LetMc(D)⊂Mf(D)denote the subspace of finite measures with compact support inD. Our main result is as follows.

Theorem 1 (WLLN). With the notations of Assumption1,if0≡f∈C_c⁺(D)and0≡μ∈Mc(D),then inP^μ-pro- bability,

tlim→∞

Xt, f

E^μXt, f= W_∞ μ, h.

The limit is mean-one(and thus,it is not identically zero)whenL^h₀corresponds to a conservative diffusion.

(The proof is given in Section 2.)

In order to give a simple condition for the limit to be mean-one, let us recall that the superprocessXpossesses the compact support propertyifP^μ(

0≤s≤tsupp(Xs) ⊂⊂D)=1,for allμ∈Mc(D), t≥0.Since there are various conditions given in [2,3] for the compact support property to hold, the following result is useful. (The proof is given in Section 2.)

Theorem 2 (No loss of mass in the limit). If the compact support property holds,then the diffusion process corre- sponding toL^h₀onDis conservative,and consequently,the limit appearing in Theorem1is mean-one.

We close this section with examples forD=R^d.

Example 1 (Supercritical SBM). The assumptions are satisfied for supercritical super-Brownian motion. Indeed, if β(·)≡β >0, thenλ_c=β, becauseλ_c(Δ,R^d)=0. Furthermore chooseh≡1,r(dx):=dxand the Brownian scaling factors_t :=t^d/2. Finally, as far as the spatial spread of the process is concerned,D_tcan be defined asD_t:=B₍^√_2β+ε)t, ε >0, whereB_r denotes the ball of radiusr >0 centered at the origin (see [6]); thusζ_t can be defined e.g. asζ_t=t^m withm >2. This setting satisfies conditions (A.1)–(A.3), as long as 0< αis bounded from above, and so Theorem 1 holds.

Consider now the simplest case of the previous example, the one whenαis a positive constant. Then the non- degenerate random variableW_∞can be thought of as the scaled limit of a one dimensional diffusion. Indeed,Y :=

Xis a diffusion corresponding to the operatorx(α ^∂2

∂x²+β_∂x^∂ )on[0,∞)withY0= μ, andW_∞=limt→∞e^−βtYt. Next, take a smooth, positive, but otherwise arbitrary functionh:D→Rand recall that the spatialh-transform is a particular case of the space–timeH-transform withH (t, x)=h(x), t≥0.

LetXbe the supercritical super-Brownian motion of Example 1 (β >0 is constant andα >0 is upper bounded).

Since the WLLN holds true forXstarting with any nonzero measure inMc(D), therefore it is also true forX^hstarting with any nonzero measure inMc(D).

Then, different particular choices ofhlead to different further examples as follows. Writex=(x1, x2, . . . , xd)and x²:= |x|²=d

i=1x_i². The supercritical SBM with drift, supercritical SBM with outward drift, supercritical Super Ornstein–Uhlenbeck process with quadraticβ and supercritical outward SOU with quadraticβ can all be treated by applyingh-transforms (whereh(x):=e^cx1,h(x):=e^c|x|, |x| 1,h(x):=e^−cx2 andh(x):=e^cx2, respectively and c >0). The WLLN for these cases follows fromh-transform invariance and the limiting random variable is always non-degenerate. Sinceα^h=αh, in each caseαhas to satisfyα(x)=O(h(x)),as|x| → ∞.

2. Proofs

Proof of Theorem 1. We first claim that(L+β)h=λ_ch. To see, this note that (A.2)(c) withμ=δ_xyields lim_t→∞s_t· S_t(f )(x)= r, f ·h(x)forx∈Dandf ∈C_c⁺(D). Using the notationufor time derivative and definingβ(t, x):=

β(x)+(logs_t), the equation(L+β)h=λ_chfollows from the fact thatu(t, x):=S_t(f )(x)solves(L+β−λ_c)u=u and thereforev(t, x):=st·u(t, x)solves(L+β−λc)v=v−(logst)v,that is(L+β−λc)v=v.By (A.2)(a),

sup_x∈D|β(t, x) −β(x)|tends to zero ast↑ ∞. Then a standard argument (see [2], p. 708) together with the second relation in (A.2)(a) gives thatw(x):=limt→∞v(t, x)= f, r(dy) ·h(x)belongs toC^2,η(D)and solves the steady state equation(L+β−λc)w=0. Then, also

(L+β−λc)h=0, (8)

and

L^h₀(u)=(L+β−λc)^h(u)=h⁻¹(L+β−λc)(hu)=H⁻¹(L+β+∂t)(H u).

By (8), (X^H,P) is the (L^h₀,0, αhe^−λct;D)-superdiffusion. Recalling the definition of S and defining ν :=

hμ, g:=f/ h, q:=hr,one can reformulate (A.2)(c):

(A*.2)(c) lim

t→∞ν(dx), st·St(g)(x)

= q, g · ν, g∈C_c⁺(D), ν∈Mf(D).

Similarly, (A.3)(d)–(A.3)(e) become (A*.³)(d) lim

t→∞P^ν W_t

D_t

>0

=0 and (A*.³)(e) lim

t→∞sup

x∈Dt

s_ζt ·S_ζt(g)(x)− q, g=0, g∈C_c⁺(D).

Finally, the theorem itself transforms into the following statement: if 0≡g∈C_c⁺(D)and0≡ν∈Mc(D),then in P^ν-probability,

tlim→∞

X^H_t , g

E^νX^H_t , g=W_∞

ν or, equivalently, lim

t→∞stWt, g = q, gW_∞. (9)

(Recall (A*.2)(c) and note thatSis theexpectationsemigroup.)

In order to show (9), the main idea is to use the comparison with the deterministic flow as in [5], however, there is an essential difference. In [5] we argued that by considering some large timet+T (botht andT are large), the changes ofX^H are negligible aftert, while the remaining timeT is still long enough to distribute the produced mass according to the ergodic flow given by theH-transformed migration. We then letT ↑ ∞and thent↑ ∞.

Reading carefully the proof in [5] one can see that this method relied heavily on the ergodicity of the flow and would break down here. Hence, instead of letting first T and then t go to infinity, we now define T :=T_t =ζ_t. Similarly to [5], the strategy is to first show that the total mass more or less stabilizes by timeT_t, then to identify the limit of thescaled flow(starting from the state of the process atTt) at timet+Tt, and finally to show that it agrees with the scaling limit of the process itself. Of course, the first part is simple: being a supermartingale, thetotal mass converges:

tlim→∞W_t =W_∞, P^ν-a.s. (10)

Unlike in [5], however, we do not know a priori, that the limit is non-zero, and moreover, one cannot proceed further without exploiting what is known about the radial speed of the process. Therefore we continue as fol- lows. Let {Z_Wt(s)}s≥0 denote the deterministic flow starting from the (random) measure W_t. Since given W_t, Z_Wt(ζ_t), g = W_t(dx),S_ζt(g)(x),one has

P^νs_ζt Z_Wt(ζ_t), g

− W_tq, g> ε

≤P^ν

Wtsup

x∈Dt

sζ_t

Sζ_t(g)

(x)− q, g> ε +P^ν

Wt

D_t

>0 .

Let us call the two terms on the righthand sideAt andBt. By (A*.3)(d and e), one has limt→∞At=limt→∞Bt =0.

Hence,

tlim→∞P^νs_ζt Z_Wt(ζ_t), g

− W_tq, g> ε

=0. (11)

From this, (A.3)(c) and (10), one obtains thescaling limit of the flow:

t→∞limP^νs_t+ζt Z_Wt(ζ_t), g

−W_∞q, g> ε

=0. (12)

Our goal is therefore to show thatthe scaling limit of the flow agrees with the scaling limit of the measure-valued process. To achieve this, recall (A.2)(b). A computation using Chebysev and the supermartingale property (essentially the same computation as the one giving formula (28) in [5]) yields:

P^νs_t+ζt Z_Wt(ζ_t), g

−s_t+ζtW_t+ζt,g> ε

≤ CE^νW_ts_t+ζ²

t

ε²λ_ce^λct ≤Cνs²_t+ζ

t

ε²λ_ce^λct ,

whereC=C(g,αh):=18αh · g². Recall the abbreviationT :=ζ_t. To finish the estimate it is enough to show that limt→∞e^−λcts_t²_+T =0.By the first condition in (A.2)(a) and by (A.3)(b), there is ak >0 such that

log

e^−λcts_t²_+T

= −λ_ct+2 logs_t+T ≤ −λ_ct+2klog(t+T ) and

tlim→∞

−λ_ct+2klog(t+T )

= lim

t→∞t

−λ_c+2klog(t+T ) t

= −∞.

Proof of Theorem 2. Suppose thatXpossesses the compact support property but the diffusion corresponding toL^h₀ is not conservative. Since the support of the superprocess is invariant underh-transforms,X^hpossesses the compact support property too. However, by Theorem 3 in [3], if the diffusion process corresponding toLonDis not conser- vative and sup_x∈Dα(x) <∞ and infx∈Dβ(x) >−∞, then the compact support property does not hold. (In [3] the domain isD=R^d, but the proof goes through for generalD.) SinceX^h is the(L^h₀, λ_c, αh;D)-superprocess,L^h₀ is

not conservative, andαhis bounded from above, we got a contradiction.

Acknowledgment

The author owes thanks to J. Biggins for a helpful discussion.

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