Law of large numbers for a class of superdiffusions

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www.elsevier.com/locate/anihpb

Law of large numbers for a class of superdiffusions

János Engländer

^a,^∗

, Anita Winter

^b

aDepartment of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA bMathematisches Institut, Universität Erlangen–Nürnberg, Bismarckstraße 1 1/2, 91054 Erlangen, Germany

Received 1 March 2004; received in revised form 11 January 2005; accepted 15 March 2005 Available online 26 September 2005

Abstract

Pinsky [R.G. Pinsky, Transience, recurrence and local extinction properties of the support for supercritical finite measure- valued diffusions, Ann. Probab. 24 (1) 237–267] proved that the finite mass superdiffusionXcorresponding to the semilinear operatorLu+βu−αu²exhibits local extinction if and only ifλ_c0, whereλ_c:=λ_c(L+β)is the generalized principal eigenvalue ofL+βonR^d. For the case whenλc>0, it has been shown in Engländer and Turaev [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] that in law the superdiffusion locally behaves like exp[tλc]times a non-negative non-degenerate random variable, provided that the operatorL+β−λcsatisfies a certain spectral condition (‘product-criticality’), and thatαandµ=X₀are ‘not too large’.

In this article we will prove that the convergence in law used in the formulation in [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] can actually be replaced by convergence in probability.

Furthermore, instead ofR^dwe will consider a general Euclidean domainD⊆R^d.

As far as the proof of our main theorem is concerned, the heavy analytic method of [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] is replaced by a different, simpler and more probabilistic one. We introduce a space–time weighted superprocess (H-transformed superprocess) and use it in the proof along with some elementary probabilistic arguments.

Résumé

Pinsky [R.G. Pinsky, Transience, recurrence and local extinction properties of the support for supercritical finite measure- valued diffusions, Ann. Probab. 24 (1) 237–267] a prouvé que le processus de superdiffusion de masse finieXcorrespondant á l’operateur semilinéaireLu+βu−αu²possède la propriété d’extinction locale si, et seulement si,λ_c0, oùλ_c:=λ_c(L+β) est la valeur propre principale généralisée deL+βdansR^d. Dans le cas oùλc>0, et pour un opérateurL+β−λcsatisfaisant une condition spectrale (de ‘criticalité-produit’), et pourvu queαetµ=X₀ne soient pas trop grands, Engländer and Turaev [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] ont montré le processus se comporte localement et en loi comme exp[tλ_c]avec une constante multiplicative aléatoire non dégénérée.

* Corresponding author.

E-mail addresses: [email protected] (J. Engländer), [email protected] (A. Winter).

doi:10.1016/j.anihpb.2005.03.004

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Dans cet article, nous montrons que la convergence en loi de [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] peut être renforcée en la convergence en probabilité. De plus, l’espaceR^dpeut étre remplacé par un domaine euclidien quelconque.

S’agissant de la preuve du theorème principal la lourde methode analytique de [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] est remplacée par une approche probabiliste plus simple.

Nous introduisons une renormalisation spatio-temporelle du superprocessus (‘H-transformed superprocess’) que nous utilisons dans la preuve combinée a des arguments probabilistes élémentaires.

MSC: 60J60; 60J80

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

1. Introduction

1.1. Preparation

We consider a superprocess arising as the short life time and high density diffusion limit of a branching particle system, which can be described as follows: in thenth approximation step each particle has mass 1/nand lives a random time which is exponential with mean 1/n. While a particle is alive, its motion is described by a diffusion process corresponding to the operator L. At the end of its life, the particle dies and is replaced by a random number of particles situated at the parent particle’s final position. The distribution law of the number of descendants is spatially varying such that the mean number of descendants is 1+β(x)/n, while the variance is assumed to be 2α(x). All these mechanisms are independent of each other. The process is determined by the quadruple (L, β, α;D), whereLis a second order elliptic operator corresponding to the underlying diffusion process onD.

See Appendix A in Engländer and Pinsky [7] for a precise statement on the particle approximation.

We start by presenting a formal description of the model considered in this article. For convenience we first recall the basic notation: let D⊆R^d be a domain and let B(D)denote the Borel sets ofD. We writeMf(D) andMc(D)for the class of finite measures resp. the class of finite measures with compact support onB(D). For µ∈Mf(D), denoteµ :=µ(D)and letC_b⁺(D)andC_c⁺(D)be the class of non-negative bounded continuous resp. non-negative continuous functions D→Rhaving compact support. Write C^k,η(D)for the usual Hölder spaces of indexη∈(0,1]including derivatives of orderk, and setC^η(D):=C^0,η(D).

We continue with the definition of the (L, β, α;D)-superdiffusion, X. Let L be an elliptic operator on the domainD⊆R^dof the form

L:=1

2∇ ·a∇ +b· ∇, (1)

wherea_i,j, b_i ∈C^1,η(D),i, j=1, . . . , d, for someη∈(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.

We now present our model.

Definition 1 (Time-homogeneous superdiffusion). Let (X,P^µ, µ∈Mf(D)) denote the(L, β, α;D)-superdiffu- sion. That is,Xis the uniqueMf(D)-valued continuous (time-homogeneous) Markov process which satisfies, for anyg∈C_b⁺(D),

E^µexpX_t,−g =exp

µ,−u(·, t )

, (2)

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whereuis the minimal non-negative solution to u_t=Lu+βu−αu² onD×(0,∞), lim_t_↓₀u(·, t )=g(·).

(3) As usual,ν, fdenotes the integral

Dν(dx)f (x).

(See Dynkin [4,5] or Dawson [2] for the definition of superprocesses in general; see Engländer and Pinsky [7]

for more on the definition in the particular setting above.)

Remark 1 (Time-inhomogeneous superdiffusion). The model under consideration is a time-homogeneous process.

However, it is important to point out that for the formulation of the main theorem and the proof, the introduction of certain time-inhomogeneous superdiffusions is required. The previous definition will therefore be generalized for time-inhomogeneous superdiffusions in Appendix B. (See Definition 2.)

Let

λ_c:=λ_c(L+β):=inf

λ∈R: ∃u >0 satisfying(L+β−λ)u=0 inD

(4) denote the generalized principal eigenvalue forL+βonD. Letξ^Lbe the diffusion process onDcorresponding toL, and denote byP^x the law ofξ^Lstarting atx∈D. Then from a probabilistic point of view, the generalized principal eigenvalue can be equivalently expressed as

λ_c= sup

{A:AD, ∂AisC^2,η} tlim→∞

1 t logE^x

exp

t 0

β ξ_s^L ds

; τ^A> t

, (5)

for anyx∈D, whereτ^A=inf{t0: ξ^L(t ) /∈A}, and theC^2,η-boundary is defined with the help ofC^2,η-maps in the usual way. (See Section 4.4 in Pinsky [15] on the subject). Hence, sinceβis bounded from above,λ_c<∞; and it is known from standard theory that for any λλ_c, there exists a function 0< f ∈C^2,η(D) such that (L+β)f =λf onD. (See Section 4.3 in Pinsky [15].)

Pinsky proved thatX exhibits local extinction (i.e., the support ofXleaves any given bounded set, P^µ-a.s. for eachµ∈Mc) if and only ifλc0. (See Theorem 6 and Remark 1 in Pinsky [16].)

From now on we are interested in the situation whereXdoes not exhibit local extinction. We therefore assume that λc>0. We get a first rough impression about the local growth rate by the following statement taken from Theorem 7(b) in Pinsky [16]:

Lemma 0 (Local behavior in expectation). Forµ∈Mc(D), andg∈C_c⁺(D), satisfyingµ =0 andg≡0, lim sup

t→∞ exp[−ρt]E^µ X_t, g

=

0 ifρ > λ_c,

∞ ifρ < λ_c. (6)

(See also Appendix A.)

We are therefore going to concentrate on scaling with the exponentρ=λ_c. In addition to the concept of the generalized principal eigenvalue we will then need some further ones from the so-called criticality theory of second order elliptic operators. In particular, we will use the concepts of critical and product-critical (or product-L¹ critical) operators. Recall that the operatorL+β−λ_cis called critical if there exists a positive functionf satisfying that(L+β−λ_c)f =0 but there is no (minimal positive) Green’s function for the operatorL+β−λ_c. In this case f is unique up to constant multiples and is called the ground state. The operatorL+β−λ_cis called product-critical if it is critical with ground state 0< φ_c, andφ_c andφ˜_c(i.e. the ground state for the formal adjoint ofL+β−λ_c) satisfydx, φ_cφ˜_c<∞. In this case we normalize them bydx, φ_cφ˜_c =1.

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IfL+β−λ_cpossesses a Green’s function, then it is called subcritical.

For the reader, it will be handy to have Appendix 2 of [9] at hand, where a review on criticality theory is given.

For a complete presentation of the theory, the reader is referred to Chapter 4 in [15].

1.2. Motivation

WhenD=R^d andL+β−λc is product-critical, it is known from Theorem 1 in [9] that ifαφc∞<∞and the initial stateµis such thatµ, φ_c<∞, then the following holds in the vague topology:

tlim→∞exp[−λ_ct]X_t(dx)=N^µφ˜_cdx in law, (7)

where the limiting non-negative non-degenerate random variable N^µ was identified with the help of a certain invariant curve.

It is important to point out that even though product-criticality is equivalent to the ergodicity of an auxiliary diffusion process (see next section), the original motion process corresponding toLdoes not have to be ergodic.

In fact it can even be transient – see Example 23 in [9].

(On the other hand, it follows from the discussion in Appendix A that whenL+β−λ_c onD⊆R^d is not product-critical, then forµ∈Mc, andg∈C_c⁺(D),

tlim→∞exp[−λct]Xt, g =0 inL¹.) (8)

There are two disadvantages of the method used in [9]. First, the assumption thatD=R^dis essential. Second, the proof does not yield any probabilistic insight.

In this paper our goal is to improve the statement in (7) and to provide a proof that is probabilistic in nature.

We will show that the ‘Law of Large Numbers’ holds, that is, that one can replace the convergence in law by convergence in probability.¹Furthermore we will drop the assumption thatD=R^d. In the proof we will replace the analytic reasoning given in [9] (which relies on dynamical systems) by a more probabilistic one using space–

time weighted superprocesses (H-transforms).

We suspect that in fact the Strong Law of Large Numbers holds, that is that the convergence in probability can be replaced by almost sure convergence. However we could not upgrade the proof of this paper to give the Strong Law.

In the recent paper [11] the authors study a supercritical superprocess taking values in the space of finite measures on[0,1], whose underlying motion is the Wright–Fisher diffusion corresponding to the operator

L=1

2x(1−x) d² dx²,

and whose branching mechanism isγ u(1−u)withγ >0 (that is,α=β=γ). They establish a dichotomy in the long-time behavior of this superprocess. Forγ1, the mass in the interior(0,1)dies out after a finite random time, while forγ >1, the mass in(0,1)grows exponentially with rateγ−1 (ast→ ∞and with positive probability) and is approximately uniformly distributed over(0,1).

This result is in line with that of [9] if one considers the restriction of the superprocess on the (open) domain (0,1). In fact it is easy to show thatλc:=γ −1 is the principal eigenvalue of the linearized elliptic operator L+γ. Here is a possible argument: the operator L+γ −λc=L+1 can be h-transformed (h=v, wherev is an explicitly given function in the paper) into a diffusion operator, which – according to their Lemma 20 – corresponds to a (positive) recurrent diffusion. Consequently, thish-transformed operator is critical, and thus its principal eigenvalue is zero. Byh-transform invariance, the same is then true for the original operatorL+γ−λ_c. (See again Chapter 4 in [15]). Furthermore, the product-criticality and boundedness assumptions are automatically satisfied by the boundedness ofD=(0,1).

1 Since the limit is not constant, therefore, unlike in classical probability theory, one has to distinguish between convergence in law (WLLN) and convergence in probability (LLN).

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Finally the fact that the limiting measure is the Lebesgue measure, is also in line with [9]. Indeed, according to [9], the limiting density is a harmonic function with respect to the adjoint ofL+γ−λ_c=L+1, that is with respect toL˜+1, whereL˜ is the adjoint ofL. An easy computation reveals that

L˜+1=1

2x(1−x) d²

dx² +(1−2x) d dx.

Since the adjoint of a critical operator is also critical, and since positive harmonic functions for a critical operator are unique up to constant multiples, the limiting density must be a properly normalized constant on the unit interval, that is, the limiting density is one.

However, as the authors point out referring to [9], ‘their methods use in an essential way the fact that their underlying space isR^d(and not an open subset ofR^d, like(0,1)), and therefore their results are not applicable to our situation.’

In the present article, as already mentioned, we manage to overcome this difficulty, so the result of [11] will fit our main result. (In [11] the result is somewhat stronger as they prove convergence inL².)

2. Main result

Recall from (4) the definition of the principal eigenvalueλ_c of L+β on D and the corresponding ground stateφ_c, and that throughout the paper we assume thatλ_c>0. Also,{Ss}s0will denote the semigroup (‘expectation semigroup’) corresponding to the operatorL+βonD. So far we have recalled (7). In order to replace in (7) the convergence in law by convergence in probability, we will assume the same conditions as in [9], except that we work with a generic domainD.

Assumption 1. In addition to the assumption thatλ_c>0, also assume thatL+β−λ_cis product-critical, thatαφ_c is bounded and thatXstarts in a stateµwithµ, φ_c<∞.

Before reading the remainder of this section, it is recommended that the reader consults Appendix B regarding the definition of time-inhomogeneous superdiffusions as well as the space–timeH-transform (weighted superdiffusion) introduced there.

LetXbe a(L, β, α;D)-superdiffusion withX0=µ. LetH (x, t ):=exp(−λct )φc(x),x∈D,t0. It turns out (see Lemma 3 in Appendix B) that the (time-inhomogeneous) processX^Hdefined by

X_t^H:=H (·, t )Xt

that is, dX^H_t

dX_t =H (·, t )

, t0 (9)

is an(L+a^∇_φ^φ^c

c · ∇,0, αφ_ce⁻^λ^c^t;D)-superdiffusion. In the sequelE andVar will denote expectation and variance with respect to the law ofX^H.

Lemma 1 (Bounded variance).

tlim→∞Var^φ^c^µ X_t^H= ∞

0

dse⁻^2λ^c^s µ,Ss

αφ²_c

<∞, (10)

andX^His a uniformly integrableP^φ^c^µ-martingale.

Proof. LetX^H denote the total mass process, i.e.,

X^H:=X^H. (11)

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Abbreviate

L^φ₀^c:=L+a∇φc

φc · ∇ (12)

and note that in fact

L^φ₀^c(u)=φ⁻_c¹(L+β−λc)(φcu)=H⁻¹(L+β+∂t)(H u).

(Here∂_t denotes differentiation with respect to time.) LetS^φ^c denote theh-transformed semigroup withh=φ_c, that isSs^φ^c(·)=(φc)⁻¹Ss(φc·).

DefineSs^H :=e⁻^λ^c^sSs^φ^c; then the semigroup{Ss^H}s0corresponds to the operatorL^φ₀^c that has no zeroth order part. In particular then

Ss^H11. (13)

Finally, the product-criticality assumption onL+β−λ_cguarantees that the diffusion process corresponding toL^φ₀^c onDis positive recurrent (ergodic) (see Section 4.4 in Pinsky [15]). (Since ergodicity implies conservativeness, thus in factS_s^H1=1; nonetheless, for us it will be enough to know (13).)

By Lemma 3(b) of Appendix B along with Theorem A2 in [7], we have that for all f ∈C_const² (D):= {f ∈ C²(D): ∃ΩDsuch thatf=const onD\Ω},

d X^H_t , f

=

X^H_t , L^φ₀^cf

dt+dM_t(f ), (14)

where{M_t(f )}t0is a square-integrableP^φ^c^µ-martingale, and its quadratic variation (i.e. the increasing process in the Doob–Meyer decomposition)M(f )is given by

M(f )

t= t

0

dse⁻^λ^c^s

X^H_s , αφ_cf²

, t0. (15)

(One can take the function classC_const² (D)instead of justC²_c(D), because the diffusion process corresponding to L^φ₀^c onDis conservative, that is, it never leaves the domainDwith probability one.)

Applying (14) to the functionf ≡1, it follows thatX^H is aP^φ^c^µ-martingale. Furthermore, by (15), E^φ^c^µ

X_t^H,12

= µ, φ_c²+ t

0

dse⁻^λ^c^s

φ_cµ,S_s^H[αφ_c]

. (16)

That is

Var^φ^c^µ X_t^H= t

0

dse⁻^λ^c^s

φ_cµ,Ss^H[αφ_c]

= t

0

dse⁻^2λ^c^s µ,Ss

αφ_c²

. (17)

Lettingt→ ∞we obtain the first statement of the lemma.

Replacingtby∞in the first of the integrals in (17), we have from (13) and from our assumptions that Var^φ^c^µ X_t^H

∞

0

dse⁻^λ^c^s

φcµ,Ss^H[αφc]

λc−1αφc∞µ, φc<∞.

Hence, by (16), sup_t₀E^µφ^c

X^H_t ,12

<∞,

and consequentlyX^His uniformly integrable. This completes the proof of the second statement of the lemma. 2

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Remark 2. Our proof of LLN will indeed use the condition thatαφ_c is bounded, however it is quite possible that this condition is not necessary and that assuming the finiteness of the integral in (10) (along withµ, φ_c<∞) would suffice.

An immediate consequence of uniform integrability is thatE^φ^c^µ[X^H_∞] = µ, φ_c, which is finite by assumption, and positive forµ=0. This yields thatP^φ^c^µ[X^H_∞=0]<1 forµ=0. We record this in a lemma.

Lemma 2 (Limit of the total mass). The martingaleX^Hhas aP^φ^c^µ-a.s. limitX^H_∞:=lim_t_→∞X^H_t which is positive with positive probability.

2.1. Heuristics for the Law of Large Numbers

Before stating the Law of Large Numbers for the class of superdiffusions under consideration, in this subsection we give some heuristic computations. These will justify why we call our main result ‘the Law of Large Numbers’.

Working with theH-transformed superprocess and at the same time, having the particle approximation in mind, consider particles with underlying motionY corresponding to the elliptic operatorL^φ₀^c (the probabilities forY will be denoted by{P^x,x∈D}) and with critical binary branching at rate exp[−λct]α(x)at positionx∈Dand time t0. Furthermore let the system be started with initial discrete measure being “close” toν=0.

LetItⁿ denote the collection of particles alive at timet in the nth approximation step. Finally, the event of

‘survival’ is{|Itⁿ|>0∀t >0}.

The Law of Large Numbers would mean that if 0≡f ∈C⁺_c(D), then ast→ ∞, (and without further specifying what “≈” means),

ν(1/|Itⁿ|)

x∈Itⁿf (x)

ν,E^x[f (Y_t)] ≈1 onItⁿ>0∀t >0

. (18)

Now recalling that in thenth approximating step the individual particle mass is scaled down bynand recalling also Lemma 2, one has that (fornlarge),|Itⁿ| ≈nX_∞^H ast→ ∞. Putting this together with (18), one gets formally, that for largen,

ν(1/n)

x∈Itⁿf (x)

ν,E^x[f (Yt)] ≈ X^H_∞, ast→ ∞. (19) Note that in fact

ν,E^x f (Y_t)

=E^ν X^H_t , f

=e⁻^λ^c^tE^µX_t, f φ_c

(ν=φ_cµ). (The first equality can be shown for instance by taking first the particular caseν=δ_x,x∈D, and using that the two expectations satisfy the same parabolic problem; then integrating with respect toν(dx).) Furthermore, passing to the limit (asn→ ∞) formally, the numerator of the fraction on the left hand side of (19) becomes

X^H_t , f

=e⁻^λ^c^tX_t, f φ_c.

Hence, for the new test function 0≡ ˆf :=f φ_c∈C_c⁺(D), X_t,fˆ

E^µXt,fˆ ≈X^H_∞ ν. 2.2. Main theorem

Making the intuition of the previous subsection precise, we now state our main result:

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Theorem 1 (Law of Large Numbers). Letf∈C_c⁺(D). Iff ≡0 andµ =0, then

tlim→∞

X_t, f

E^µX_t, f= X_∞^H

µ, φ_c, in P^µ-probability. (20)

Comparing our theorem with (7), we can now identify the limiting distribution:N^µ= X_∞^H in law.

Remark 3. One has to be a bit careful though when making heuristic inferences using the particle picture as in the previous subsection.

Obviously, the discrete system in thenth approximation step is so that lim inf_t_→∞|I_tⁿ|1 under survival. That is, lim inf_t_→∞_n¹|I_tⁿ|>0 under survival. Recall that, heuristically (fornlarge), ¹_n|I_tⁿ| ≈ X^H_∞ast→ ∞.

On the other hand, in the recent paper [6] an example of a superprocess is given that satisfies the conditions of our previous theorem and for which

P^µ X_∞^H =0|S

>0,

whereSis the event of survival,S:= {X_t>0,∀t0}.

Conjecture 1. We conjecture that convergence in probability in (20) can be replaced by almost sure convergence.

We close this section with a remark concerning an old result.

Remark 4. A simple case of a superdiffusion is when D=R^d,d 1, L= ¹₂∆, with α, β positive constants (supercritical super-Brownian motion). Hereλc=βand

1

2∆+β−λ_c=1 2∆.

Sinceφ_c= ˜φ_c≡1, d1, the operator ¹₂∆is either critical but not product-critical (d2), or subcritical (d3).

Therefore this case is not included in our setup. On the other hand, the corresponding (Strong) Law of large Numbers is well known – at least for the discrete particle systems. Using techniques from Fourier transform theory, Watanabe [18] proved SLLN for branching-Brownian motion inR^dand in certain subdomains of it. It is not clear however if his method can be generalized for more general branching diffusions.

3. Proof of the result

The proof is based on two observations. The first one is that the problem can be formulated in terms ofX^H, that is, one can reduce the problem to the investigation of a critical superdiffusion with ergodic motion component and exponentially decaying branching rate (again, recall thatX^His an(L+a^∇_φ^φ^c

c ·∇,0, αφ_ce⁻^λ^c^t;D)-superdiffusion).

The second one is that by considering some large timet+T (where botht andT are large), the changes of

X^H are negligible after time t, while the remaining timeT is still long enough to distribute the produced mass according to the ergodic flow given by theH-transformed migration.

To simplify notation, we will writeW:=X^H (and, accordingly,W_∞:= X^H_∞). Denoteν:=W₀. By assumption, ν = µ, φ_c<∞. We need to show that for all >0,

tlim→∞P^µ

X_t, f

E^µX_t, f−W_∞ ν

>

=0. (21)

Recall thatP denotes the probabilities with respect to the law of W. Denotingφ_c⁻¹f =:g∈C_c⁺(D), we rewrite (21) as

tlim→∞P^ν

Wt, g

E^νW_t, g−W_∞ ν

>

=0.

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It is easy to check that the limiting invariant density for the diffusion corresponding toL^φ₀^c is φ_cφ˜_c (recall the normalization

Ddx φ_c(x)φ˜_c(x)=1). SinceZ:=E^νWis just the deterministicL^φ₀^c-flow starting fromν, therefore

tlim→∞E^νW_t, g = ν · φ_cφ˜_c, g,

and consequently our statement is tantamount to saying that for all >0,

tlim→∞P^ν

W_t, g − φ_cφ˜_c, g W_∞ >

=0.

LetT >0 and letZ_W_t denote the deterministicL^φ₀^c-flow starting from the (random) measureW_t. Then P^ν W_t₊_T, g − φ_cφ˜_c, g W_∞>

S₁(t )+S₂(t, T )+S₃(t, T ), (22) where

S₁(t ):=P^ν φcφ˜c, g W_∞− φcφ˜c, gWt> /3 , S2(t, T ):=P^ν φcφ˜c, gWt −

ZW_t(T ), g> /3 S₃(t, T ):=P^ν Z_W_t(T ), g

− W_t₊_T, g> /3 . Take lim sup_t_→∞lim sup_T_→∞on both sides of (22). We have

lim sup

t→∞ P^ν W_t, g − φ_cφ˜_c, g W_∞>

I+II+III, where

I:=lim sup

t→∞ S1(t ), (23)

II:=lim sup

t→∞ lim sup

T→∞ S2(t, T ), (24)

III:=lim sup

t→∞ lim sup

T→∞ S₃(t, T ). (25)

Now, sinceWt → W_∞ast→ ∞a.s., I= lim

t→∞S₁(t )=0.

Also II=0, because for all fixedt0, limT→∞S₂(t, T )=0. (Indeed, for allω∈Ω, limT→∞Z_W_t_(ω)(T ), g = φ_cφ˜_c, gW_t(ω).) Therefore, if we show that

III=0, (26)

then we are done.

In order to do this, use at timet thatW is a time-inhomogeneous Markov-process, and then apply Chebysev’s inequality:

S₃(t, T )=E^νP^W^t Z_W_t(T ), g

− W_t₊_T, g> /3

9⁻²E^νσˆ_W²

tW_T, g, (27)

whereP is the law of the(L+a^∇_φ^φ^c

c · ∇,0, αφ_ce⁻^λ^c^(t⁺^s);D)-superdiffusion (heretis fixed ands >0 is time) and ˆ

σ²denotes variance.

Let us now recall how the formulae for the first two moments ofW_T, gare obtained: by writingu_θ(t, x)for the solution of the semilinear parabolic evolution equation (corresponding to the superprocess) with initial valueθg, one differentiates (repeatedly) with respect toθand setsθ=0.

(10)

For time-homogeneous processes with constant branching rate this is written down in detail in [10], p. 37. Since the derivation of these ‘moment formulae’ only uses differentiation with respect toθ(and nottorx), therefore the proof goes through for the more general setting where coefficients are space–time-dependent.

In our case, from these moment formulae and from (27), one obtains (recall also (13)) that for allT >0,

S₃(t, T ) 9 ²·E^ν

T

0

ds2 e⁻^λ^c^(t⁺^s)

W_t,S_s^H

αφ_c S_T^H₋_sg₂

C·E^νWt

²·λ_ce^λ^c^t , (28)

with

C=C g,αφ_c

:=18αφ_c · g².

(Note that we have an extra factor 2 relative to [10] – indeed in [10] the non-linear term in the semilinear parabolic evolution equation is ¹₂γ u².) Recall thatWis aP^ν-martingale with meanνand continue (28) with

= C· ν ²·λ_ce^λ^c^t.

Since this holds for allT >0, thus lim sup

T→∞ S₃(t, T ) C· ν ²·λce^λ^c^t.

Lettingt→ ∞, one obtains (26), completing our proof. 2

Remark 5. Personal communication with T. Kurtz revealed the possibility of another, alternative way of completing the argument. Namely, once we know that the total mass ofX^H converges, we can use a (yet unpublished) spatial version of the so-called lookdown construction given in [3] to show that the ‘angular part’ of X^H converges in distribution. (Note that, conditional on the limit of the total mass, the limit of the angular component is constant.)

Appendix A. The behavior of the process in expectation

For the cases whenL+β−λcis subcritical, or critical butdx, φcφ˜c = ∞, Theorem 7(b)(ii) in [16] states that forg∈C_c⁺,

tlim→∞e⁻^λ^c^tE^µ X_t, g

=0. (29)

Note, however, that in the proof there is a glitch: the proof simply refers to Theorem 4.9.9 in [15] which deals with the product-critical case only, and is therefore not applicable for the cases mentioned.

Nevertheless, for the subcritical case, and for µ∈Mc(D), the statement can be verified by a very simple argument as follows (cf. Theorem 4.9.1 in [15]). First, note that by the first moment formula,

e⁻^λ^c^tE^µ X_t, g

=

µ,[Ttg](x) ,

where{Tt}t0= {e⁻^λ^c^tSt}t0denotes the semigroup corresponding to the operatorL+β−λ_conD.

Make anh-transform now:

[Ttg](x)=h(x) Tt^h(g)ˆ

(x),

wheregˆ:=gh⁻¹, and pick anh >0 satisfying(L+β−λ_c)h=0, to reduce the problem to the proof of

tlim→∞

hµ,E^x ˆ g ξ_t^h

=0, (30)

(11)

whereξ^hdenotes the diffusion corresponding toT^h, that is, to the operatorL^h₀(defined analogously toL^φ₀^cin (12) withφ_creplaced byh) andEdenotes the corresponding expectation. (Of course,hµ∈Mc(D).)

Furthermore, it is enough to show the statement withµ:=δ_x, x∈D, because once we know that, the general statement follows by bounded convergence:E^x[ ˆg(ξ_t^h)] ˆgfor allx∈Dandt0.

In [15], Chapter 4 it is shown that subcriticality is invariant underh-transforms, and that the transience of a diffusion is equivalent to the subcriticality of the corresponding elliptic operator. Therefore, in our case, it follows thatξ^his transient. Sincegis compactly supported, by an obvious comparison argument, it is enough to show (30) withgreplaced by the indicator 1_B, whereBis a ball. By transience 1_B(ξ_t^h)→0 ast→ ∞a.s., and the statement follows from this and bounded convergence.

Similarly, the critical but non-product critical case can be reduced to the analogous (but much subtler) problem of showing (30) for a null-recurrentξ^h. (Cf. the well known analogous limit theorem for countable state space Markov chains – see e.g. Proposition 5.3 and Corollary 6.39 in [12].) In fact, as mentioned in the notes at the end of Chapter 4 in [15], this result is known in the case whenLis symmetric with respect to some reference measure ρdx (see [1] or [17]). (Recall thatLis symmetric if and only ifb=a∇Qfor someQ∈C^2,η(D),η∈(0,1], and in this caseLpossesses a self-adjoint extension due to the Friedrichs extension theorem – see Chapter 4 in [15].) Recently Pinchover completed the result by proving it for the general (non-selfadjoint) setting (see [14]).

Consequently, theρ > λc (over-scaling) part of (6) is immediate. One does not need however the above deep result for the ρ > λ_c part. Here is a simple alternative proof: using an h-transform with an h >0 satisfying (L+β−λ_c)h=0, the statement is equivalent to

tlim→∞e^(λ^c⁻^ρ)tE^x g ξ_t^h

=0, (31)

for eachx∈D, which is true in virtue of the boundedness ofg.

The ρ < λc (under-scaling) part of (6) is harder, and we are only able to provide the rigorous proof of the somewhat weaker assertion:

lim sup

t→∞ exp[−ρt]E^µ X_t, g

= ∞. (32)

To this end, denote byp^L⁺^β(t;x,·), x∈D, the kernel corresponding to the operatorL+βand note that since gis compactly supported, by an obvious comparison argument, it is enough to prove that

lim sup

t→∞

1 t log

B

µ(dx)p^L⁺^β(t;x, B)=λ_c, (33)

for eachx∈D, and Borel setBD. Clearly, we may assume that µ =1. To verify (33), make again anh- transform with anh >0 satisfying(L+β−λc)h=0 onD. ThenL+β transforms intoL^h₀+λc. Moreover, since the generalized principal eigenvalue is invariant underh-transforms and sinceλc(L^h₀+λc)=λc(L^h₀)+λc, it follows thatλ_c(L^h₀)=0. Letp₀^h(t, x,·)denote the transition measures corresponding toL^h₀. Fixx∈DandBD.

Sinceh(·)h⁻¹(x)p^L⁺^β(t, x,·)=e^λ^c^tp^h₀(t, x,·)(see Theorem 4.1.1 in [15]), and sincehis bounded between two positive constants onB, we have

lim sup

t→∞

1 t log

B

µ(dx)p₀^h(t;x, B)=lim sup

t→∞

1 t log

B

µ(dx)p^L⁺^β(t;x, B)−λc. (34) Sinceµ =1 andp^h₀(t;x, B)1, the left hand side of (34) is non-positive, giving immediately

λ₊(x, B):=lim sup

t→∞

1 t log

B

µ(dx)p^L⁺^β(t;x, B)λc. (35)

Suppose now thatλ₊(x, B) < λ_cand pick c∈ −λ_c,−λ₊(x, B)

. (36)

(12)

Then by (34) and (36), along with Fubini’s theorem, one obtains

B

µ(dx) ∞

0

e^ctp^L⁺^β(t, x, B)dt <∞. (37)

Hence, for almost everyx∈B, G^(L⁺^β⁺^c)(x, B):=

∞

0

e^ctp^L⁺^β(t, x, B)dt <∞. (38)

It follows from general theory then that e^ctp^L⁺^β(t, x, B)is in fact integrable for all x∈DandBD, that is, that the operator L+β+cpossesses a (minimal positive) Green’s function onD; however this contradicts to the well known fact that an operator with positive generalized principal eigenvalue does not possess a Green’s function. (In our caseλ_c(L+β+c)=λ_c+c >0.)

Appendix B. TheH-transform of superdiffusions

This section treats a generalization of theh-transform for superdiffusions introduced in [7]. Theh-transform was used in the proofs in [9]. The method used in the present paper however requires the spatial function 0< h=h(x) to be replaced by a space–time function 0< H =H (x, t ). (The reader should not confuse with the space–time harmonic transformation yielding a Girsanov-type change of measure – see e.g. [13].)

We start with the more general definition of a time-inhomogeneous superdiffusion. LetL˜ be an elliptic operator onD×R⁺of the form

L˜ :=1

2∇ · ˜a∇ + ˜b· ∇ (39)

where the functionsa˜_i,j,b˜_i:D×R⁺→R,i, j=1, . . . , d, areC^1,η(D)(for someη∈(0,1]) in the space, and continuously differentiable in the time coordinate. Moreover assume that the symmetric matrixa(x, t )˜ :=(a_i,j(x, t )) is positive definite for allx∈Dandt∈R⁺.

In addition, letα,˜ β˜:D×R⁺→R, beC^η(D)in the space, and continuously differentiable in the time coordinate. Finally assume thatα˜ is positive, andβ˜is bounded from above.

Definition 2 (Time-inhomogeneous(L,˜ β,˜ α˜;D)-superdiffusion).

(i) The (L,˜ β,˜ α˜;D)-superdiffusion is a measure-valued (inhomogeneous) Markov process, (X,P^µ,r; µ ∈ Mf(D), r0), that is, a family{P^µ,r}of probability measures where P^µ,r is a probability onC([r,∞)) and the family is indexed by Mf(D))× [0,∞), such that the following holds: for each g∈C_b⁺(D)and µ∈Mf(D),

E^µ,r

exp−X_t, g

=exp−

µ, u(·, r;t, g)

, (40)

whereu=u(·,·;t, g)is a particular non-negative solution to the backward equation

−∂_ru= ˜Lu+ ˜βu− ˜αu² inD×(0, t ), lim

r↑tu(·, r;t, g)=g(·). (41)

(13)

(ii) To determine the solutionuuniquely, use the equivalent forward equation along with the minimality of the solution: fixt >0 and introduce the ‘time-reversed’ operatorLˆ onD×(0, t )by

Lˆ:=1

2∇ · ˆa∇ + ˆb· ∇, (42)

where, forr∈ [0, t], ˆ

a(·, r):= ˜a(·, t−r) and b(ˆ ·, r):= ˜b(·, t−r); furthermore let

β(ˆ ·, r):= ˜β(·, t−r) and α(ˆ ·, r):= ˜α(·, t−r).

Consider nowv, the minimal non-negative solution to the forward equation

∂_rv= ˆLv+ ˆβv− ˆαv² inD×(0, t ), lim

r↓0v(·, r;t, g)=g(·). (43)

Then

u(·, r;t, g)=v(·, t−r;t, g).

(See also the remark following this definition.)

Remark 6 (Minimal non-negative solutions for forward equations). In the time-homogeneous case, minimal non- negative solutions for forward equations have been constructed in Appendix A in [7], and in Section 2 in [8] – the construction uses the approximation ofDby compactly embedded subdomains with Dirichlet condition on their boundaries. The construction goes through for the time-inhomogeneous setting.

In [7] and [8] the time interval is[0,∞)rather than[0, t]. However that does not make any difference – in fact the solution on the infinite time interval can be defined by first working on finite time intervals and then showing that they can be taken arbitrarily large without having the solution blown up.

As we will see in Lemma 3(b), one way of defining a time-inhomogeneous superdiffusion is to start with a time- homogeneous one, and then to apply an ‘H-transform’. In general, theH-transform of a time-inhomogeneous superdiffusion is defined as follows. Let 0< H∈C^2,η(D)×C^1,η(R⁺)and letXbe a(L,˜ β,˜ α˜;D)-superdiffusion.

We define a new processX^Hby X_t^H:=H (·, t )X_t

that is, dX^H_t

dX_t =H (·, t )

, t0. (44)

This way one obtains a new superdiffusion, which, in general, is not finite measure-valued but only σ-finite measure-valued. That is, ifM(D)denotes the family of all (finite or infinite) measures onD, then

X_t^H∈M^{(t )}_H(D):=

ν∈M(D)|H (·, t )⁻¹ν∈Mf(D) (cf. [7], p. 688).

In [7], Section 2, it was shown, that, from an analytical point of view, the (spatial)h-transform of the superdif- fusion is given by a certain transformation of the corresponding semilinear operator. This remains the case for the space–timeH-transform.

Lemma 3 (H-transform). LetX^H be defined by (44). Then

(a) X^H is a(L˜+ ˜a^∇_H^H · ∇,β˜+^LH^˜_H +^∂^r_H^H,αH˜ ;D)-superdiffusion.