Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses

The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles moving in $R^d$ according to a symmetric $\alpha$-stable L\'evy process $(0<\alpha\leq 2)$, splitting with a critical $(1+\beta)$-branching law $(0<\beta\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_T\mu_\gamma$, occupation time fluctuation limits for the system as $T\to\infty$ have been obtained in two special cases: Lebesgue measure ($\gamma=0$, the homogeneous case), and finite measures $(\gamma>d)$. In some cases $H_T\equiv 1$ and in others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit processes are quite different for Lebesgue and for finite measures. Therefore the question arises of what kinds of limits can be obtained for Poisson intensity measures that are intermediate between Lebesgue measure and finite measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are used for investigating this question. Occupation time fluctuation limits are obtained which interpolate in some way between the two previous extreme cases. The limit processes depend on different arrangements of the parameters $d,\alpha,\beta,\gamma$. Related results for the corresponding $(d,\alpha,\beta,\gamma)$-superprocess are also given.


Introduction
Occupation time fluctuation limits have been proved for the so-called (d, α, β)-branching particle systems in R d with initial Poisson states in two special cases, namely, if the Poisson intensity measure is either Lebesgue measure, denoted by λ, or a finite measure [BGT1], [BGT2], [BGT3], [BGT4], [BGT6].Those cases are quite special, as explained below, and the limit processes are very different.Therefore the question arises of what happens with Poisson intensity measures that are intermediate between Lebesgue measure and finite measures.That is the main motivation for the present paper, and our aim is to obtain limit processes that interpolate in some way between those of the two special cases.One of our objectives is to find out when the limits have long-range dependence behavior and to describe it.Another motivation is to derive analogous results for the corresponding superprocesses.
In a (d, α, β)-branching particle system the particles move independently in R d according to a standard spherically symmetric α-stable Lévy process, 0 < α ≤ 2, the particle lifetime is exponentially distributed with parameter V , and the branching law is critical with generating function where 0 < β ≤ 1 (called (1 + β)-branching law), which is binary branching for β = 1.The parameter V is not particularly relevant, but it is convenient to use it.The empirical measure process N = (N t ) t≥0 is defined by N t (A) = number of particles in the Borel set A ⊂ R d at time t. (1.2) A common assumption for the initial distribution N 0 is to take a Poisson random measure with locally finite intensity measure µ.The corresponding (d, α, β)-superprocess Y = (Y t ) t≥0 is a measure-valued process, which is a high-density/short-life/small-particle limit of the particle system, with Y 0 = µ.See [D], [E], [P2] for background on those particle systems and superprocesses.In this paper we investigate (the limiting behavior of) the corresponding occupation time processes, i.e., t 0 N s ds, t ≥ 0, and t 0 Y s ds, t ≥ 0.
We recall that the distributions of these processes are characterized by their Laplace functionals as follows [GLM], [DP2]: where v ϕ (x, t) is the unique (mild) solution of the non-linear equation where u ϕ (x, t) is the unique (mild) solution of the non-linear equation u ϕ (x, 0) = 0, and ∆ α is the infinitesimal generator of the α-stable process.(See the end of the Introduction for the standard notations , , S(R d ).)For µ = λ, N 0 is homogeneous Poisson.This case is special (and technically simpler) because λ is invariant for the α-stable process (which implies in particular that EN t = λ for all t), and there is the following persistence/extinction dichotomy [GW], which heuristically explains the need for high density in some cases in order to obtain non-trivial occupation time fluctuation limits, and anticipates the situation we will encounter in this paper: (i) Persistence: If d > α/β, then N t converges in law to an equilibrium state N ∞ as t → ∞, such that EN ∞ = λ.(ii) Extinction: If d ≤ α/β, then N t becomes locally extinct in probability as t → ∞, i.e., for any bounded Borel set A, N t (A) → 0 in probability.An analogous persistence/extinction dichotomy holds for the corresponding superprocess (with Y 0 = λ) [DP1].For α = 2 (Brownian motion) and d < 2/β a stronger extinction holds: the superprocess becomes locally extinct in finite time a.s.[I2], and we shall see that so does the particle system.
The case of µ finite is special because the particle system goes to extinction globally in finite time a.s.for every dimension d, and so does the superprocess [P2].
The time-rescaled occupation time fluctuation process X T = (X T (t)) t≥0 of the particle system is defined by where N s is given by (1.2) and F T is a norming.The problem is to find F T such that X T converges in distribution (in some way) as T → ∞, and to identify the limit process and study its properties.This was done for µ = λ in the persistence case, d > α/β, [BGT3], [BGT4].
For the extinction case, d ≤ α/β, in [BGT6] we introduced high density, meaning that the initial Poisson intensity measure was taken of the form H T λ, with H T → ∞ as T → ∞, so as to counteract the local extinction, and we obtained similarly high-density limits with µ finite.For µ = λ and d > α/β the same results hold with or without high density (with different normings) [BGT6].The limit processes are different for µ = λ and µ finite (some differences are mentioned below), and the results for any finite measure are essentially the same.
In order to study asymptotics of X T as T → ∞ with measures µ that are intermediate between the two previous cases, we consider Poisson intensity measures of the form µ γ (dx) = 1 1 + |x| γ dx, γ ≥ 0. (1.8) We call the model so defined a (d, α, β, γ)-branching particle system, and a (d, α, β, γ)-superprocess the corresponding measure-valued process.To obtain non-trivial limits we multiply µ γ by H T , which is suitably chosen in each case.Since µ 0 = λ and µ γ is finite for γ > d, by varying γ in the interval (0, d] we obtain limits that are between those of the two previous cases, which are extreme in this sense, in a way that interpolates between them.The substantial role of γ was already noted in the simpler model of particle systems without branching [BGT5]. The above mentioned behaviors of N t in the cases γ = 0 and γ > d raise the following questions on what happens for γ ∈ (0, d], and on its effect on asymptotics of X T : When does N t suffer a.s.local extinction in the sense that for each bounded Borel set A there is a finite random time τ A such that N t (A) = 0 for all t ≥ τ A a.s.? In this case the total occupation time ∞ 0 N t (A)dt is finite a.s., and therefore high density is needed in order to obtain non-trivial limits for X T .For γ > 0, N t (A) converges to 0 in probability as t → ∞ for any bounded Borel set A and every dimension d, so local extinction in probability occurs, but the total occupation time may or may not be finite.It turns out that the threshold between the need for high density or not is given by d = α/β + γ, and then a natural question is whether d = α/β + γ is also the border to a.s.local extinction of the particle system.We will come back to this question.
The limits for X T in [BGT6] are of three different kinds for both µ = λ and µ finite.In the first case there is a critical dimension, d c = α(1 + β)/β.For the "low" dimensions, d < d c , the limit has a simple spatial structure (the measure λ) and a complex temporal structure (with long-range dependence).For the "high" dimensions, d > d c , the limit has a complex spatial structure (distribution-valued) and a simple temporal structure (with stationary independent increments).For the "critical" dimension, d = d c , the spatial and the temporal structures are both simple, but the order of the fluctuations (F T ) is larger, as is typical in phase transitions.The limit processes are always continuous for d < d c , and for d ≥ d c they are continuous if and only if β = 1 (when the limits are Gaussian).For µ finite, an analogous trichotomy of results holds, with a new critical dimension, d c = α(2 + β)/(1 + β), another difference being that the limits for the critical and high dimensions are constant in time for t > 0.
In this paper we show analogous limits of X T for (d, α, β, γ)-branching particle system; the critical dimension changes between the ones above, α(1 + β)/β for γ = 0, and α(2 + β)/(1 + β) for γ > d, and they are linked with a unified formula, which interpolates between the two cases (see Remark 2.2(a) for a precise statement).There are several limit processes depending on different arrangements of d, α, β, γ.Some are analogous to those for µ = λ, and some are similar to those for µ finite (or even essentially the same).For γ < d there are six different cases that include the three ones recalled above for γ = 0.For γ > d there are the three cases obtained in [BGT6] (generally for finite µ).In the case γ < d and d < d c (γ), the temporal structure of the limit is a new real, stable, self-similar, continuous, long-range dependence process ξ, defined in (2.1) below, which has two different long-range dependence regimes if α < 2 (Theorem 1(a) and Proposition 2.3).This strange type of long-range dependence behavior already appears in the homogeneous case, γ = 0 [BGT3], [BGT6].An analogous phenomenon occurs with 0 < γ < d, the border between the two longrange dependence regimes changes continuously with γ, and it disappears in the limit γ ր d (see formula (2.13)).For γ > d there is only one long-range dependence regime, not depending on γ, β.
In [BGT1], [BGT2], [BGT3], [BGT4], [BGT5], [BGT6] we have given the convergence results for X T in a strong form (functional convergence when it holds), but in the present article our main objective is identifying the limits, so we have not attempted to prove the strongest form of convergence in each case, nevertheless we expect that convergence in law in a space of continuous functions holds in all cases where the limit is continuous.We prove functional convergence only in the case of the above mentioned long-range dependence process ξ because of its special properties.A technical difficulty for the tightness proof is the lack of moments if β < 1.
The time-rescaled occupation time fluctuation process for the (d, α, β, γ)-superprocess Y is defined analogously as (1.7), and the limits are obtained from (the proofs of) the results for the (d, α, β, γ)-branching particle systems, as a consequence of the fact that the log-Laplace equation of the occupation time of the superprocess is simpler than that of the particle system (see (1.3), (1.4), and (1.5), (1.6).) Our results on the fluctuation limits of superprocesses generalize those of Iscoe [I1], who considered the homogeneous case (γ = 0) only.
Let us come back to the question of high density and local extinction for the (d, α, β, γ)branching particle system.From Theorems 2.1, 2.5 and 2.6 it follows immediately (see Corollary 2.10) that in all cases where high density is not necessary (i.e., we may take H T ≡ 1) there is no a.s.local extinction (in spite of the fact that local extinction in probability occurs if γ > 0).For instance, condition (2.6) in Theorem 2.1(a) holds automatically if d > α/β + γ (with γ < d), hence high density is not necessary for a non-trivial limit of X T in this case.On the other hand, high density is indispensable to obtain a non-trivial limit if either d ≤ α/β + γ or α < γ ≤ d.In the latter case the total occupation time of any bounded Borel set by the process N is finite a.s.(it has finite mean).We prove a.s.local extinction for α = 2 and d < 2/β + γ, and we conjecture that a.s.local extinction holds generally for d < α/β + γ also if α < 2. This conjecture is supported by the fact that for d = α/β + γ, γ < α, there is an ergodic result (Proposition 2.9).For α = 2 and d < 2/β + γ it follows from [I2] (Theorem 3 β ) that the (d, 2, β, γ)-superprocess suffers a.s.local extinction.The method of [Z] can also be used to prove this (private communication).The proof of a.s.local extinction of the particle system consists in showing that a.s.local extinction of the (d, 2, β, γ)-superprocess implies a.s.local extinction of the (d, 2, β, γ)-branching particle system (Theorem 2.8).This implication is not as simple as it might seem because the well-known Cox relationship between the particle system and the superprocess (i.e., for each t, N t is a doubly stochastic Poisson random measure with random intensity measure given by Y t ) is not enough to relate the long time behaviors of the two processes.But the argument does not work for α < 2. In this case the superprocess Y has the instantaneous propagation of support property, i.e., with probability 1 for each t > 0 if the closed support of Y t is not empty, then it is all of R d .This follows from the result proved in [P1] for finite initial measure and finite variance branching (β = 1 in our model), which is extended in [LZ] for more general superprocesses and branching mechanisms (including β < 1 in our case).It follows that for the (d, α, β, γ)-superprocess with α < 2, a.s.local extinction and global extinction are equivalent, and it is known that if the initial measure has infinite total mass, the probability of global extinction in finite time is 0. Nevertheless, the total occupation time of a bounded set for the superprocess with α < 2 may or may not be finite, and this is what is directly relevant for us (see the proof of Theorem 2.8).Iscoe [I1] showed that for initial Lebesgue measure and α = 2 the total occupation time of a bounded set is finite if and only if d < 2/β, and he conjectured that an analogous result holds for α < 2 and d < α/β.So far as we know, this conjecture has not been proved.
Summarizing, if γ < d and γ < α, there are two thresholds for the asymptotics of X T , namely, α/β + γ, and d c (γ) given by (1.9).The first one, which is smaller than the second one, appears to be the border to a.s.local extinction (we know that it is for α = 2), and it determines the need for high density.The second one is the critical dimension between changes of behavior of the limit processes, in particular the change from long-range dependence to independent increments, and from continuity to discontinuity if β < 1.An interpretation of d c (γ) in terms of the model seems rather mysterious, even in the case γ = 0 (see [BGT4], Section 4, for several questions on the meaning of results).
The general methods of proof developed in [BGT3], [BGT4], [BGT6], and the special cases for β = 1, where the limits are Gaussian [BGT1], [BGT2], can be used for the proofs involving µ γ .However, a considerable amount of technical work is unavoidable in order to deal with γ > 0.Moreover, each case requires different calculations.We will abbreviate the proofs as much as possible.
We have given special attention to the long-range dependence stable process ξ and its properties because long-range dependence is currently a subject of much interest (see e.g.[DOT], [H1], [H2], [S], [T] for discussions and literature), hence it is worthwhile to study different types of stochastic models where it appears.Other types of long-range dependence processes have been found recently (e.g.[CS], [GNR], [HJ], [HV], [KT], [MY], [LT], [PTL]), in particular in models involving heavy-tailed distributions.
The following notation is used in the paper.
), in particular, integral of a function with respect to a tempered measure.
⇒ C : weak convergence on the space of continuous functions C([0, τ ], S ′ (R d )) for each τ > 0. ⇒ f : weak convergence of finite-dimensional distributions of S ′ (R d )-valued processes.⇒ i : integral convergence of S ′ (R d )-valued processes, i.e., X T ⇒ i X if, for any τ > 0, the S ′ (R d+1 )-valued random variables X T converge in law to X as T → ∞, where X (and, analogously X T ) is defined as a space-time random field by (1.11) ⇒ f,i : ⇒ f and ⇒ i together.
Recall that in general ⇒ f and ⇒ i do not imply each other, and either one of them, together with tightness of The transition probability density, the semigroup, and the potential operator of the standard symmetric α-stable Lévy process on R d are denoted respectively by p t (x), T t (i.e., T t ϕ = p t * ϕ) and (for d > α) (1.12) where .
(1.13) Generic constants are written, C, C i , with possible dependencies in parenthesis.
Section 2 contains the results, and Section 3 the proofs.
For α ∈ (0, 2], γ ≥ 0, we define the process which is well defined provided that (see [ST]).For γ = 0, ξ is the same as the process ξ defined by (2.1) in [BGT6].We also recall the following process defined by (2.2) in [BGT6], We consider the (d, α, β, γ)-branching particle system described in the Introduction with X T defined by (1.7).Recall that the initial Poisson intensity measure is H T µ γ .We formulate the results for low, critical and high dimensions separately, since, as mentioned in the Introduction, the qualitative behaviors of the limit processes are different in each one of these cases.In the theorems below K is a positive number depending on d, α, β, γ, V , which may vary from case to case and it is possible to compute it explicitly.
The results for the low dimensions are contained in the following theorem.
Theorem 2.1 (a) Assume γ < d and (2.4) Then the process ξ given by (2.1) is well defined, and for we have and put (2.8) Then for with lim we have X T ⇒ f,i Kλζ, where ζ is defined by (2.3).
(b) For d satisfying (2.4) and additionally d > α/β + γ, condition (2.6) holds with H T = 1, so in this case high density is not needed, and the limit of X T is the same as for the high-density model.
(c) The case γ > d is included for completeness only, since it is contained in Theorem 2.7 of [BGT6], where a general finite intensity measure was considered.The same remark applies also to the theorems for critical and high dimensions (Theorems 2.5 and 2.6 below).
(d) Note that the limit process is the same (up to constant) for the infinite intensity measure as for finite measures.(e) In Theorem 2.1(b) we consider convergence ⇒ f,i only.We are sure that functional convergence holds (in fact, for the case γ > d this was proved in [BGT6]), but, as stated in the Introduction, we are mainly interested in the identification of limits and we do not attempt to give convergence results in the strongest forms.The same applies to the theorems that follow.
In the next proposition we gather some basic properties of the process ξ defined by (2.1), in particular its long-range dependence property.(The process ζ was discussed in [BGT6]).In [BGT3] we introduced a way of measuring long-range dependence in terms of the dependence exponent, defined by where (2.12) (see also [RZ]).
(c) ξ has continuous paths.(d) ξ has the long-range dependence property with dependence exponent (2.13) Remark 2.4 (a) Here, as in the case γ = 0 (Theorem 2.7 of [BGT3]), the intriguing phenomenon of two long-range dependence regimes occurs for α < 2. It seems also interesting to note that putting formally γ ≥ d in (2.13) we obtain κ = d/α (with no change of regime), which is indeed the dependence exponent of the process ζ (Proposition 2.9 of [BGT6]).On the other hand, the process ζ itself is not obtained from ξ by putting γ ≥ d.
We now turn to the critical dimensions, i.e., the cases where the inequalities in (2.4) and (2.7) are replaced by equalities.It turns out that in spite of different conditions on the normings, the limits have always the same form as for finite intensity measure, with the only exception of the case given in Theorem 2.5(a) below. and where η is a (1 + β)-stable process with independent, non-stationary increments (for γ > 0) whose laws are determined by (b) In all the remaining critical cases, i.e., (i) γ = α, γ < d with d satisfying (2.14), ) ) with F T satisfying (2.16) and where ϑ is a real process such that ϑ 0 = 0 and for t > 0, ϑ t = ϑ 1 = (1 + β)-stable random variable totally skewed to the right, i.e., It remains to consider the high dimensions. and -stable process with independent, non-stationary increments (for γ > 0) determined by and G is defined by (1.12).(b) Assume γ < d, γ = α, and d satisfying (2.23) with ) where X is an S ′ (R d )-valued process such that X 0 = 0, and for t > 0, X t = X 1 = (1 + β)-stable random variable determined by (2.28) and lim Then X T ⇒ f,i X, where X is an S ′ (R d )-valued process such that X 0 = 0, and for t > 0, X t = X 1 = (1 + β)-stable random variable determined by with c β given by (2.26).
Remark 2.7 (a) As in all the cases studied previously [BGT3], [BGT4], [BGT6], we observe the same phenomenon that in low dimensions the limit processes are continuous with a simple spatial structure and a complicated temporal structure (with long-range dependence), while in high dimensions they are truly S ′ (R d )-valued with independent increments, and not necessarily continuous.
(b) For low dimensions the forms of the limits depend on the relation between d and γ only, whereas for critical and high dimensions only the relationship between α and γ is relevant.More precisely, in critical dimensions we have different forms of the limits for γ < α and γ ≥ α, and in high dimensions the forms are different for γ < α, γ = α and γ > α.In the case γ > α even the normings are the same, depending only on β.
(c) For β = 1 the limits are centered Gaussian.In high dimensions there is no continuous transition between the cases β < 1 and β = 1; in the latter case an additional term appears.
The coefficient c β defined in (2.26) was introduced in order to present the results in unified forms.
(d) We have assumed that the initial intensity measure is determined by µ γ of the form (1.8).
It will be clear that the same results are obtained for the measure µ γ (dx Analogously as in the non-branching case [BGT5], other generalizations are also possible. Let us look further into the need for high density (i.e., to assume H T → ∞) and the question of a.s.local extinction.For γ > d the Poisson intensity measure is finite, so there is only a finite number of particles at time t = 0 and the system becomes globally extinct in finite time a.s.due to the criticality of the branching.Also, for γ ∧ d > α it is not difficult to see that the total occupation time ∞ 0 N s ds is finite a.s. on bounded sets (see [BGT5], Proposition 2.1, because EN s is the same for the systems with and without branching), so high density is also necessary.We have a more delicate situation in the remaining cases where the threshold is d = α/β + γ.Concerning extinction, the situation is completely clear for α = 2.In Theorem 2.8 below we state that for α = 2 and d < 2/β + γ there is a.s.local extinction, hence the total occupation time of any bounded set is finite a.s.We conjecture that the same is true for d < α/β + γ if α < 2, but we have not been able to prove it.
The proof of this theorem relies on Iscoe's a.s.local extinction result for the superprocess [I2], by showing that in general (i.e., for 0 < α ≤ 2) a.s.local finiteness of the total occupation time of the (d, α, β, γ)-superprocess implies a.s.local extinction of the (d, α, β, γ)-branching particle system.On the other hand, as explained in the Introduction, for α < 2 the a.s.local extinction for the superprocess cannot occur, and we do not know how to prove directly the a.s.local finiteness of its total occupation time.
The next ergodic-type result, which is a direct generalization of [Ta], shows that α/β + γ is indeed a natural threshold.
Proposition 2.9 Assume and denote (2.34) where ξ is a real non-negative process with Laplace transform for any τ > 0, where θ 1 , . . ., θ n ≥ 0, 0 ≤ t 1 < • • • < t n ≤ τ , and v(x, t) is the unique nonnegative solution of the equation (2.37) To complete the discussion on a.s.local extinction we formulate a corollary which follows immediately from our results but which, nevertheless, seems worth stating explicitly.
We end with the results for the superprocess.
Theorem 2.11 Let Y be the (d, α, β, γ)-superprocess and X T its occupation time fluctuation process defined by (1.10).Then the limit results for X T as T → ∞ are the same as those in Theorems 2.1, 2.5 and 2.6, with the same normings, and c β = 0 in all cases in Theorem 2.6.

Scheme of proofs
The proofs of Theorems 2.1, 2.5 and 2.6 follow the general scheme presented in [BGT6].For completeness we recall the main steps.
As explained in [BGT3,BGT4,BGT6], in order to prove convergence ⇒ i it suffices to show lim for each Φ ∈ S(R d+1 ), Φ ≥ 0, where X is the corresponding limit process and X T , X are defined by (1.11).To prove convergence ⇒ C according to the space-time approach [BGR] it is enough to show additionally that the family { X T , ϕ Without loss of generality we may fix τ = 1 (see (1.11)).To simplify slightly the calculations we consider Φ of the form We define where N x is the empirical process of the branching system started from a single particle at x.
The following equation for v T was derived in [BGT3] (formula (3.8), see also [BGT1]) by means of the Feynman-Kac formula: (3.5) For brevity we denote (3.6) By the Poisson property and (3.4) we have where (3.9) (3.10) In the proofs of Theorems 2.1, 2.5 and in Theorem 2.6 for β < 1 we show lim and lim where (3.13) is obtained from J 2 (T ) = 0, (3.17) where We remark that the proof of (3.15) is the only place where the high density (with specific conditions on H T ) is required in some cases.Finally, the ⇒ f convergence is obtained by an analogous argument as explained in the proof of Theorem 2.1 in [BGT4].

Auxiliary estimates
Recall that the transition density p t of the standard α-stable process has the self-similarity property and it satisfies where the lower bound holds for α < 2. Denote The following estimate can be easily deduced from (3.20) and (3.21): (3.24) We will also use the following elementary estimates: Let (3.28) where f is defined in (3.22).From the estimates above we obtain and (3.31)

Proof of Theorem 2.1(a)
According to the scheme sketched above, in order to prove ⇒ f,i convergence we show (3.1).By (3.7)-(3.11)and(2.1) it is enough to prove (3.12), which amounts to lim (3.32) (see (3.9)) and, additionally, (3.13) and (3.15).To simplify the notation we will carry out the proof for µ γ of the form µ γ (dx) = |x| −γ dx instead of (1.8).It will be clear that in the present case (d < γ) this will not affect the result.
By (3.9), (3.2), (3.6), the definition of T t , substituting s ′ = 1 − s/T, u ′ = 1 − u/T , we have and where f is defined by (3.22).By (3.20) and (2.5), making obvious spatial substitutions in (3.33), we obtain Note that if we consider the measure µ γ of the form (1.8), then in (3.37) instead of |x| −γ we have ( dz almost everywhere in y.Hence, to prove (3.32) it remains to justify the passage to the limit under the integrals in (3.37).Denote First we prove pointwise convergence of h T , which amounts to showing that the integrand is majorized by an integrable function independent of T .Fix y = 0. We use (3.36) and observe that by the unimodal property of the α-stable density and since ϕ ∈ S(R d ).We conclude by noting that to prove convergence of I 1 (T ) it suffices to show that the right-hand side of (3.40) (denoted by If γ < α, then f γ is bounded by (3.31), and the assumption (2.4) implies that Next assume γ ≥ α.It is easily seen that h * T 1 1 {|y|≥1} converges in L 1 (R d ), by (3.41) and (3.30).To prove that h * T (y)1 1 {|y|<1} converges in L 1 (R d ) too, it suffices to find p, q > 1, 1/p + 1/q = 1, such that (3.43) If γ = α, then (3.31) implies that (3.42) holds for any p > 1, and by (2.4) it is clear that (3.43) is satisfied for q sufficiently close to 1.
Then by (3.45) and (2.5) we obtain that I ′′ 2 (T ) → 0 as T → ∞.Thus, we have proved (3.13).According to the general scheme, in order to obtain (3.15) it suffices to show (3.16) and (3.17).The proofs are quite similar to the argument presented above, therefore we omit the proof of (3.16) and we give an outline of the proof of (3.17), since this is the only place where the condition (2.6) is needed.
By (3.19), (2.5) and the usual substitutions we have , where 1/p + 1/q = 1.We already know that there exist such p and q that this expression is finite (see (3.42) and (3.43)).
We have thus established the convergence In order to obtain ⇒ C convergence it suffices to show that the family { X T , ϕ } T ≥1 is tight in C([0, 1], R) for any ϕ ∈ S(R d ) ( [Mi]).One may additionally assume that ϕ ≥ 0. We apply the method presented in [BGT3] and [BGT6].We start with the inequality valid for any ψ ∈ S(R), δ > 0. Fix 0 ≤ t 1 < t 2 ≤ 1 and take ψ approximating Then the left hand side of (3.54) approximates So, in order to show tightness one should prove that the right hand side of (3.54) is estimated by C(t h 2 − t h 1 ) 1+σ for some h, σ > 0. By the argument in [BGT6] this reduces to showing that and where I 1 is defined by (3.9), and The proofs of (3.56) and (3.57) are quite involved and lengthy, therefore, as an example we present only the argument for the case γ < α, which, together with (2.4) implies We start with (3.57).By self-similarity of p s we have Using this, (3.37), (3.35) and (3.55) we obtain where Fix any ρ such that (see (3.59).For any fixed s ∈ [0, t 1 ] we apply the Jensen inequality to the measure We have which, after the substitution s ′ = s/u, by (3.64) and γ < α, yields (3.66)where h = 2 − γ/α + ρβ − (d/α)β > 0 by assumptions.Next, by (3.63) we have dsdy.

Proof of Theorem 2.1(b)
We prove the theorem for Recall that in this case k(T ) occurring in (2.9) and (2.10) is log T .
According to the discussion in Section 3.1 it suffices to prove (3.12), (3.13) and (3.15).By the form of the limit process (see (2.By (3.9), (3.2), (3.6), using similar substitutions as in the previous section, we obtain where ϕ T is given by (3.34).We write where Passing to polar coordinates in the integral with respect to x we have where g is defined by (3.35).The crucial step is the substitution which gives It is now clear that if one could pass to the limit under the integrals as T → ∞, then I ′ 1 (T ) would converge to the right hand side of (3.69).This procedure is indeed justified by the fact that for f defined by (3.22) we have is estimated as follows: by (3.76).This and (3.71) prove (3.69).
For J 1 (T ) we obtain the estimate (log T and H T appear with negative powers only), whereas → 0 by assumption (2.10).The proof of Theorem 2.1 is complete 2

Proof of Proposition 2.3
Properties (a)-(c) are clear, following from (2.1) and Theorem 2.1(a).Recall that the index of self-similarity is defined as a ∈ R such that the process (ξ ct ) t∈R + has the same distribution as (c a ξ t ) t∈R + for any c > 0.
To calculate the dependence exponent of ξ (see (2.11), (2.12)) first note that by (2.1) and Proposition 3.4.2 of [ST] the finite-dimensional distributions of ξ are given by Eexp{i(z The argument goes along the lines of the proof of Theorem 2.7 of [BGT3].For fixed z > 0 and 0 ≤ u < v < s < t we define (the formulas for D + , D − are obtained from (2.12) and (3.78)), and we prove and for T sufficiently large, The upper estimates are obtained similarly as (4.3), (4.4) in [BGT3] and (3.108) in [BGT6].The only difference is that in formulas (4.9) and (4.10) in [BGT3] a new factor, R d p r (x − y)|y| −γ dy, appears (which corresponds to p r (x) in (3.101) in [BGT6]).This factor is responsible for the new long-range dependence threshold and the form of the dependence exponent (2.13).
In the estimates we use (3.30).
The first of the lower estimates is obtained exactly as (4.18) in [BGT3].The new expression, that appears at the right-hand side is finite by (3.31).
To derive (3.79) we argue as in (4.22), (4.24) of [BGT3] and we apply estimates (4.21) (which holds for |x| ≤ T 1/α ) and (4.23) therein, obtaining where ε > 0 is sufficiently small.For 1 ≤ |x| ≤ T d/(d+α)α−ε we have Putting this into (3.80)we obtain (3.79). 2 3.6 Proof of Theorem 2.5 Each of the cases requires a different proof, and none of them is straightforward.We will present a detailed proof of the part (a) only.In the remaining cases we will confine ourselves to explaining why the limit processes have the forms given in the theorem (recall that part (b)(iv) has been proved in [BGT6]).It seems instructive to compare the proofs for this theorem to the argument given in the proof of Theorem 2.1(b) for γ = d.Although the critical cases are of different kinds, some of the technical tricks repeat in all cases, nevertheless they are applied in a slightly different way and are far from being identical.

Proof of case (a)
To simplify calculations we again consider the measure µ γ of the form µ γ (dx) = |x| −γ dx instead of (1.8).
In (3.9) we substitute u ′ = s − u and then s ′ = (T − s)/T , obtaining Using (2.15), (3.20) and substitution x ′ = x(sT ) −1/α we have where Passing to polar coordinates in the integral with respect to y and making substitution (3.75) we obtain We substitute z ′ = T −r z, u ′ = uT −rα , use (3.20) and (2.14), arriving at where It is clear that on the set of integration one should have lim where C d,α is given by (1.13), which should yield lim However, (3.87) and (3.88) need a justification.It is easy to see that the first integral in (3.86) can be replaced by it is clear that in order to prove (3.87) it suffices to show that sup T >2 sup is integrable in u.This is clear for u ≥ 1 since d > α, and for u < 1 we argue similarly as in (3.39) obtaining an integrable bound C 1 p u (w/2) + C 2 .In the same way one shows that h T (r, s, w) ≤ C.This together with (3.87) easily implies (3.88).
To prove (3.17 ).Again, it can be shown that the only significant term is I ′ 1 (T ), i.e., lim T →∞ I 1 (T ) = lim T →∞ I ′ 1 (T ).In order to derive this limit, in (3.85) we substitute s ′ = sT 1−rα , obtaining It is easy to see that the limit remains the same if the integral . . .ds is replaced by T ) after this change.Next, we substitute s ′ = log s/ log T and we have By (3.86), it is clear that on the set of integration lim This shows that we should have lim ), (3.95) and this passage to the limit can be indeed justified.The right-hand side of (3.95) is equal to log Eexp{−C X, ϕ ⊗ ψ }, where X(= Kλϑ) is the limit process defined in the theorem.We skip the remaining parts of the proof.
(iii) As d = γ, we must keep the measure µ γ in its original form (1.8).
Arguing as in the proof of (ii) and taking into account (2.16), instead of (3.96) we obtain it can be shown, with some effort, that lim Again, we omit the remaining parts of the proof.

Proof of Theorem 2.6
We only give an outline of the proof.The following lemma is constantly used.
Proof of part (b) of the theorem.Following the general scheme one can show lim Here, again, we use repeatedly the Lemma above together with the easily checked fact that sup for any integrable and bounded function h (recall that d > α).We omit details.
Proof of part (c) of the theorem.Recall that the case γ > d has been proved in [BGT6].For α < γ ≤ d we use the Lemma (part (c) is particularly important).We show lim The superprocesses Y ≡ Y 0 and Y ψ are obtained as (high-density/short-life/small-particle) limits of the same process N, removing first the killed particles in the case of Y ψ .Hence for any bounded set A.
Proof of Lemma B. Let B R be a closed ball in R d with radius R centered at the origin.Let (t i , x i , τ i ), i = 1, 2, . . ., be a sequence of random vectors defined as follows for any realization of the branching particle system.First we exclude all the particles which start inside B R at time 0 and their progenies.Let t 1 be the first time any of the remaining particles enters B R , x 1 is the entry point, and τ 1 is the occupation time of the closed ball B 1 (x 1 ) of radius 1 centered at x 1 by the tree generated by the entered particle.We exclude this tree from further consideration.Let t 2 be the first time after t 1 that any of the remaining particles enters B R , with x 2 and τ 2 defined analogously as above; and so on.Let η denote the total number of first entries (t i , x i , τ i ), i = 1, . . ., η.We will show that η < ∞ a.s.. Suppose to the contrary that P and this is a contradiction since, as observed above, P [ ∞ i=1 τ i = ∞] = 0. Going back to the particles that start inside B R , there are only finitely many of them since µ(B R ) < ∞.
In conclusion, with probability 1 only finitely many initial particles generate trees that contribute to the occupation time of any given bounded set, and all those trees become extinct a.s. in finite time by criticality of the branching.So (3.110) is proved. 2 Now, to prove Theorem 2.8 it suffices to observe that under its assumptions the corresponding superprocess Y suffers local extinction by Theorem 3 β of [I2], hence (3.100) is clearly satisfied and the theorem follows immediately from the lemmas.
3.9 Proof of Proposition 2.9 First observe that it suffices to prove convergence of finite-dimensional distributions.Indeed, in the proof of Theorem 2.1(a) we have shown tightness of X T = Z T − EZ T , and the presence of high density was not relevant in that proof.On the other hand, from Proposition 2.1 of [BGT5] it follows easily that the family of deterministic processes (E Z T , ϕ Without loss of generality we assume that τ = 1.Fix 0 ≤ t 1 < t 2 < . . .< t n ≤ 1, ϕ 1 , . . ., ϕ n ∈ S(R d ), and we may additionally assume that ϕ 1 , . . ., ϕ n ≥ 0. In order to show ⇒ f convergence we prove that lim (3.111)where v satisfies (2.36) with ψ given by (2.37) for θ k = R d ϕ k (y)dy (as explained in [Ta], the solution of (2.36) is unique.)For simplicity we consider µ γ (dx) = |x| −γ dx (it will be clear that the limit is the same as for µ γ given by (1.8)).Also, to simplify the notation we take ϕ 1 = . . .= ϕ n = ϕ.Essentially the same argument can be carried out in the general case.
As in [Ta] and [BGT4] (the possibility to pass from space-time random variable to the (compare with the first formula on page 851 of [Ta]).We will show that lim , where f is defined by (3.22), and p, q > 1 are such that γp < d, (d − α)q < d, 1/p + 1/q = 1 (such p and q exist since γ < α < d).Hence (3.118) easily follows from (3. 23) and (3.24).
Next, we will show that lim where lim This can be derived in a similar way as in [Ta] (see (2.21) and subsequent estimates therein).
The only difference is that the term corresponding to I 2 (T ) in [Ta] requires a slightly more delicate treatment; in our case it has the form  3.122), it is seen that one can pass to the limit in (3.115) letting T → ∞, thus obtaining that the limit of h T satisfies (2.36).This proves (3.114) and completes the proof of the Proposition. 2 Proof of Theorem 2.11 The proof is similar to those for the particle system, starting from an equation analogous to (3.where the term I 2 (T ) in equation (3.8), given by (3.10), does not appear.This reflects the fact that comparing the log-Laplace equations (1.4) for the particle system and (1.6) for the superprocess, the term −ϕv ϕ is missing in (1.6).(Equation (3.123) can be obtained from (3.4) by the same limiting procedure that yields the superprocess from the branching particle system.An equation analogous to (3.7) for the superprocess can be derived from continuous dependence of the occupation time process with respect to the superprocess, and continuity of the mapping C([0, τ ], S ′ (R d ) ∋ x → x ∈ S ′ (R d+1 ) in (1.11) [BGR].)It follows that the results for the superprocess are the same as those for the particle system, except in the cases where I 2 (T ) has a non-zero limit, and to obtain the results in those cases it suffices to delete those non-zero limits.Therefore the limits in Theorems 2.1 and 2.5 are the same for the superprocess, and for those in Theorem 2.6, c β = 0 in all cases. 2
(3.103) and (3.105) coincide for ϕ = ψ, hence, from (3.102), (3.104), 3.101) is satisfied for any bounded set A ⊂ R d and the lemma is proved.2 Lemma B. Let N be the empirical process of the (d, α, β)-branching particle system with locally finite initial intensity measure µ.If (3.101) is satisfied for any bounded set A ⊂ R d , then