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1 T t Y (t) := N ds, t 0, (2) T F s ≥ T Z0 and its fluctuations (XT (t))t≥0 by
1 T t X (t) := (N EN ) ds, t 0. (3) T F s − s ≥ T Z0 In both cases FT is a deterministic norming which may vary in different situations. We will now discuss the results obtained in the paper. The behaviour of both models is very similar hence will be presented together. Informally speaking, when FT = T the following law of large numbers holds - Y (t) µt, for a certain positive measure µ. Our aim is to estimate the speed of T →T this convergence. This will be done through the following:
Central limit theorems (CLT) The objectives of this part are to find suitable FT , such that XT converges in law as T + to a non-trivial limit and identify this limit. It is convenient to → ∞ regard X as a process with values in the space of tempered distributions ′(Rd) and prove T S convergence in this space. In the paper we prove a functional central limit theorem for the superprocess in Theorem 4.5 (the result for the BPS is already known [18, Theorem 2.1]). 1/2 The theorem is in a sense classical as FT = T and the limit is Gaussian, namely a Wiener process. The temporal structure of the limit is simple - the increments are independent, which contrasts sharply with the spatial structure being an ′(Rd)-valued Gaussian random field S with the law depending on the properties of the Markov family η. This result can be explained by the subcriticality of the branching law. Below we present a shortened version of a heuristic argument presented in [18]. It uses a particle picture so it refers directly only to the BPS, nev- ertheless by the approximation presented in Section 3 it is also applicable to the superprocess. The life-span of a family descending from one particle is short (its tail decays exponentially). Therefore, a particle hardly ever visits the same site multiple times. If we consider two dis- tant, disjoint time intervals, it is likely that distinct (independent) families contribute to the increases of the occupation time in them. This results in independent increments of the limit process. Consequently, under mild assumptions, the properties of the movement play a minor role in the temporal part of the limit. On the other hand, the life-span of a family is too short to “smooth out the grains in the space” which, in turn, gives rise to the complicated spatial structure.
2 Large and moderate deviation principles (LDP/MDP) They are standard ways of studying rare events (on an exponential scale) when a random object converges to a deterministic limit. Moderate deviations can be also regarded as a link between the central limit theorem and large deviations (see Remark 4.7 ). In the paper we prove version of large deviation principles for the rescaled occupation process YT for the BPS and the superprocess. They are contained in Theorem 4.2 and Theorem 4.6. The rate functions in these cases are quite complicated. Roughly speaking they are the Leg- endre transforms of functions expressed in terms of equations related to the systems. In both cases the results are not complete as the upper bounds and lower bounds are potentially not optimal and the upper bounds are derived only for a subclass of the compact sets. The reasons for this are to some extent fundamental. Exponential tightness is not likely to hold in this case, therefore a “strong” large deviation principle is impossible - see Remark 4.4. Moreover, the functional approach is technically demanding. In Theorem 4.3 and Theorem 4.7 we also present less powerful versions for the one-dimensional distribution which can be formulated more elegantly. The above limitations are not relevant to moderate deviation principles presented in Theo- rem 4.4 and Theorem 4.8. We were able to obtain so-called strong deviation principles in a functional setting (i.e. for random variables taking values in the space ([0, 1], R)). The rate C functions in both cases are “quite explicit”. What is more, the theorems closely resemble the Schilder theorem, which, together with the central limit theorems, strongly suggests that the large space-time scale behaviour is similar to the one of the Wiener process - see Remark 4.8.
The distinctive feature of all results presented in the paper is the fact that they were obtained for a large class of Markov processes η. This is uncommon for stochastic models of this kind; usually η is a well-known process (e.g. Brownian motion, α-stable process, Lévy process), which makes the analysis more tractable and explicit. The subcriticality of branching law suppresses the influence of the properties of η (as it was discussed for the CLT), which makes it possible to carry out the reasoning in our fairly general setting. We investigated the speed of convergence in the law of large numbers for the occupation time process YT using two complementary tools: central limit theorems and large deviations, which together pro- vide the full picture on various scales. The central limit theorems and moderate deviation principles indicate very close relation to the Brownian motion on large time scales which is slightly undermined by the large deviation principles. This phenomenon stems from the fact that in the “exponential scale” of the large deviation principles the properties of the Markov family η finally play a role. While the results on the moderate deviations, given in this paper, seem quite satisfactory, it remains unclear whether the large deviation principles could be refined or are already ultimate. We stress that the results were obtained in the functional setting. The paper is written as a self-contained reference hence we summarise the results obtained earlier and present one-dimensional versions. We will now present our results against the state-of-the-art in the field. Central limit theorems for similar models with critical branching were studied intensively by Bojdecki et al. and Milos. We just mention [8, 19], in which the reader finds further references. These systems (and related ones) were also studied using large deviation principles; [14, 11, 21, 15] with [15] describing the most recent developments. We also refer to [3] as an example of similar results for branching random walks. The results for critical branching systems are qualitatively different from the ones presented here. The
3 dependence on the properties of the particle movement is much stronger as the notion of transience and recurrence (for the movement itself and families of particles) plays vital role to the form of the limit - see also Remark 4.3. Systems with subcritical branching were largely neglected until recent works [18, 13]. While the studies of critical branching models concentrates mostly on the systems with “well-behaving” pro- cesses governing the particles movement (usually Brownian motion or α-stable processes), [18, 13] admit a large class of Markov processes. This paper extends and virtually completes their devel- opments. Firstly, we converted the functional central limit theorem for branching particle systems, [18, Theorem 2.1], to superprocesses, Theorem 4.5. Secondly, we extend the results of [13] where the authors showed a large and moderate deviation principle for one-dimensional distribution of the superprocess [13, Theorem 4.1 Theorem 5.1]. In the paper we present their functional counterparts and establish analogous results for the BPS. As it was mentioned above the only open issue left is the possibility of refining the large deviation principles. Not surprisingly the proof techniques bear resemblance to the ones in [18, 13]. However they had to be enhanced to handle new situations. Loosely speaking, the main technical difficulty was to com- bine the methods of [18] suitable for functional setting with the methods of [13] developed to deal with large and moderate deviation principles. This required some delicate estimations of solutions of partial differential equations. The proof of the exponential tightness, which was the technically most cumbersome part, required also estimations of the suprema of stochastic processes. The BPS and the superprocesses are similar models and the proofs in both cases are similar, though usually more difficult for BPS. The paper is organised as follows. In the next section we present the notation used throughout the paper. Section 3 is devoted to the detailed description of the superprocess. In Section 4 the results are presented. Finally, Section 5 contains the proofs.
2 Notation
In the whole paper we will use superscripts B, S to indicate the BPS and the superprocesses, respectively. We shall skip the superscripts when a quantity (equation) will apply to both models or when it is clear from the context which model we are dealing with. By (E) we will denote the B Borel sets on space E. By BV (E) we will denote the set of Borel measures on E of bounded total variation. ′(Rd) is a space of tempered distributions i.e. a nuclear space dual to the Schwartz space of rapidly S decreasing functions (Rd). The duality will be denoted by , . By ′(Rd) ′(Rd) we will S h· ·i S + ⊂ S denote the subspace of positive functions. In the whole paper Q := V (1 2q), (4) − which intuitively denotes the “intensity of dying”. Recall that V is the intensity of branching and 2q is the expected number of particles spawning from one particle. Clearly, the subcriticality of the branching law implies Q> 0. By ( t)t≥0 and A we will denote, respectively, the semigroup and the infinitesimal operator cor- T P responding to the Markov family (ηt, x)t≥0,x∈Rd presented in Introduction. Sometimes instead of E E x writing xf(ηt) we write f(ηt ).
4 For brevity of notation we also denote the semigroup Qf(x) := e−Qt f(x), Tt Tt and the potential operator corresponding to it +∞ Qf(x) := Qf(x)dt. (5) U Tt Z0 Three kinds of convergence are used. The convergence of finite-dimensional distributions is denoted by . For a continuous, ′(Rd)-valued process X = (X ) and any τ > 0 one can define an →fdd S t t≥0 ′(Rd+1)-valued random variable S τ τ X˜ , Φ := Xt, Φ( ,t) dt. (6) 0 h · i D E Z If for any τ > 0 X˜ X˜ in distribution, we say that the convergence in the space-time sense n → holds and denote this fact by . Finally, we consider the functional weak convergence denoted by →i X X. It holds if for any τ > 0 processes X = (X (t)) converge to X = (X(t)) n →c n n t∈[0,τ] t∈[0,τ] weakly in ([0,τ], ′(Rd)) (in the sequel without loss of generality we assume τ = 1 and skip the C S superscript). It is known that and do not imply each other, but either of them together →i →fdd with tightness implies . Conversely, implies both , . →c →c →i →fdd For a measure ν BV ([0, 1]) we write ∈ χν(s) := ν((s, 1]), χν,T (t) := χν(t/T ). (7) We will skip the subscript ν when the measure is obvious from the context. By H1 ([0, 1], R) we ′ t ′ ⊂C ′ denote the space of functions f for which there exists f such that f(t)= 0 f (s)ds and f is square integrable. We denote ′ R f 1 := f 2 . k kH k kL We also define := f ([0, 1], R) : f is differentiable and f ′(x) (a, b) . (8) Ca,b ∈C ∀x∈[0,1] ∈ For a function f ([0, 1], R) we denote its modulus of continuity ∈C w(f, δ):= sup f(s) f(t) . (9) s,t∈[0,1] | − | |s−t|<δ
By c, c1,...,C,C1,... we will denote generic constants.
3 Subcritical superprocess with immigration
n In this section we recall the construction of the superprocess. By (Nt )t≥0 we denote the n-th ap- proximation of superprocess i.e. the branching particle system in which particles live for exponential time with parameter Vn = 2nV q. The branching law is given by a generating function 2nq + 2q 1 F (s)= q s2 + (1 q ), q := − . (10) n n − n n 4nq
5 The system starts from the null measure (the starting measure does not affect the results, hence this assumption can be easily dropped) and the immigration is given by a space-time homogenous Poisson random field with intensity nH(λd+1). We also assume that each of the particles carries mass 1/n. The particular choice of Vn,qn is to some extent arbitral and was made to keep the intensity of dying fixed at level Q and for the sake of convenience (e.g. to have the same constants in forthcoming equations (28), (29)). It does not affect the generality of the results as one can easily reformulate theorems for any other choice. By N we denote a measure-valued homogenous Markov process with the following Laplace transform
t Eexp ( N , ϕ ) = exp H λ, H ϕ ds , − h t i − h s i Z0 where ϕ : Rd R is measurable and H is a semigroup given by equation 7→ + s t H ϕ(x)= Qϕ(x) Vq Q (H ϕ)2( ) (x)ds. t Tt − Tt−s s · Z0 The process N will be called the superprocess related to the BPS, which is justified by
Proposition 3.1. The following convergence holds
N n N. →c The proof is standard. For instance, one can follow the lines of [12, Section 1.4]).
4 Results P Firstly we present the restrictions imposed on the Markov family (ηt, x)t≥0,x∈Rd . They are mild and easy to check in concrete cases. Let us denote the quadratic forms
T (ϕ) := Q ϕ Qϕ , ϕ (Rd), (11) 1 kU U k1 ∈ S T (ϕ) := Q ( Qϕ)2 , ϕ (Rd), (12) 2 kU U k1 ∈ S also, slightly abusing notation, we will also use T1 and T2 to denote the corresponding bilinear forms.
4.1 Assumptions P (A1) The Markov family (ηt, x)t≥0,x∈Rd is almost uniformly stochastically continuous i.e.
r>0 ǫ>0 lim inf inf Px(ηs,B(x,ǫ)) = 1, ∀ ∀ s→0 |x|≤r
where B(x,ǫ) denotes the ball of radius ǫ with the centre in x.
(A2) Let DA denotes the domain of the infinitesimal operator A. We have
(Rd) D . S ⊂ A
6 (A3) For any ϕ (Rd) the semigroup ( ϕ) given by ∈ S Tt t≥0 t ϕf(x) := E exp ϕ(η )ds f(η ), Tt x s t Z0 is a Feller semigroup.
(A4) For any ϕ (Rd) ∈ S T (ϕ) < + , T (ϕ) < + . (13) 1 ∞ 2 ∞ (A5) For any ϕ (Rd) ∈ S Q t3/2 ϕ 0. (14) kTt k1 → (A6) For any h 2 there exist C > 0 and Q′ > 0 such that ∈L ′ Qh Ce−Q t h , . kTt k2 ≤ k k2 ∀t≥0 (A7) For any h 1 there exist C > 0 and Q′ > 0 such that ∈L ′ Qh Ce−Q t h , . kTt k1 ≤ k k1 ∀t≥0 (A8) For any ϕ (Rd) there exist ǫ> 0,c> 0 such that ∈ S + Qϕ c 1 t−2−ǫ . kTt k1 ≤ ∧ (A9) For any ϕ (Rd) there exist ǫ> 0,c> 0 such that and for all h, l ∈ S + Q Qϕ( ) Qϕ( ) c 1 t−2−ǫ . kTt Th · Tl · k1 ≤ ∧ h i Remark 4.1. The assumptions above are used in various configurations and are not independent. E.g. (A7) implies (A5), (A8) and (A9). Remark 4.2. The conditions above can be easily checked for concrete processes. For example they hold for any Lévy process. Consider also the Ornstein-Uhlembeck process η given by the { t}t≥0 stochastic equation dη = θη dt + σdW , t − t t where θ> 0,σ > 0 and W is the Wiener process. X fulfils the above assumptions if
θ This condition has a clear interpretation. θ determines the speed at which particles arrive in prox- imity of 0 (this interpretation would be totally strict if σ = 0). The intensity of dying i.e. Q have to be large enough to prevent clumping particles near 0. 7 4.2 Branching process In this subsection N denotes the BPS described in Introduction. The processes (2) and (3) are defined with this N. Firstly we recall the central limit theorem [18, Theorem 2.1] Theorem 4.1. Let XT be the rescaled occupation time fluctuations process given by (3). Assume 1/2 that FT = T and assumptions (A1)-(A5) are fulfilled. Then X X, and X X, T →i T →fdd where X is a generalised ′(Rd)-valued Wiener process with covariance functional S Cov( X , ϕ , X , ϕ )= H (s t) (T (ϕ , ϕ )+ V qT (ϕ , ϕ )) , ϕ , ϕ (Rd), h t 1i h s 2i ∧ 1 1 2 2 1 2 1 2 ∈ S if, additionally, assumptions (A8)-(A9) are fulfilled then X X. T →c Remark 4.3. The limit is an ′(Rd)-valued Wiener process with a simple time structure and a S complicated temporal one (for any d). This resembles the result for the system with critical branching in large dimensions (e.g. [6], [17]). The main reason of this is a short (exponentially-tailed) life-span of a family descending from one particle. It leads to independent increments in the limit (as there are no “related” particles in the long term). On the other hand the movement is “not strong enough” to smooth out the spatial structure. Let us now recall (4) and denote QB = V (1 2 q(1 q)). (15) 0 − − A LDP contained in Theorem 4.2 is our next objective.p Before that we need Lemma 4.1. Let ϕ (Rd) and θ vϕ(x,θ) := lim vϕ(x,t,θ). (16) t→+∞ The proof is deferred to Section 5.2. We define now ∂ A := lim vϕ(x,θ). (17) B θ→Q0 /kϕk∞ ∂θ Fix ϕ (Rd) and let us recall (7). For ν BV ([0, 1]) such that χ 8 Theorem 4.2. Let ϕ (Rd) and Y be the rescaled occupation time process given by (2). Assume ∈ S + T that F = T and assumptions (A1)-(A3) are fulfilled. Then for any open set U ([0, 1], R), T ⊂C −1 ∗ lim inf T log P ( YT , ϕ U) inf Λ (f). T →+∞ h i∈ ≥− f∈U∩C0,A/H For any f and any δ > 0 there exists r> 0 such that ∈C0,A/H −1 ∗ lim sup T log P ( YT , ϕ B(f, r)) δ Λ (f), T →+∞ h i∈ ≤ − where B(f, r) is a ball in ([0, 1], R) of radius r centred at f. C Remark 4.4. We checked that for certain Markov families η exponential tightness does not hold. Consequently, in these cases a strong large deviation principle cannot hold either. We conjecture that this phenomenon is general and exponential tightness does not hold for any Markov family η. Remark 4.5. In the lower-bound formula the restriction to is fairly acceptable. The paths of C0,A/H YT are continuous and non-decreasing. Hence large class of open sets U can be “well-approximated” by in a sense that any function in U has to increase “very fast” on some intervals. C0,A/H \C0,A/H This requires a lot particles to gather in a small set which is not very likely in our system. Remark 4.6. The restriction in the lower-bound case is more awkward. We conjecture that −1 ∗ lim sup T log P ( YT , ϕ K) inf Λ (f). T →+∞ h i∈ ≤− f∈K∩C0,A/H is true for some class of compact sets K. To give full picture we also recall here a non-functional counterpart of the above theorem. Since Theorem 4.2 is a weak version of large deviations we cannot use the contraction principle [9, Theorem 4.2.1] and the theorem below requires a separate proof (which obviously is much simpler than the one of Theorem 4.2 and hence skipped). Theorem 4.3. Let ϕ (Rd) and Y be the rescaled occupation time process given by (2). Assume ∈ S + T that FT = T and assumptions (A1)-(A3) are fulfilled. Then there exists δ > 0 such that for any open set U (0, δ), closed set L (0, δ) ⊂ ⊂ −1 ∗ lim inf T log P ( YT (1), ϕ U) inf Λ (x), (18) T →+∞ h i∈ ≥− x∈U −1 ∗ lim sup T log P ( YT (1), ϕ L) inf Λ (x), T →+∞ h i∈ ≤− x∈L where ∗ Λ (x)= sup [xθ vϕ( ,θ), λ ] , B − h · i θ≤Q0 /kϕk∞ where vϕ is given by (16). Now we present a strong moderate deviation principle. 9 d Theorem 4.4. Let ϕ (R )+ and XT be the rescaled occupation time fluctuations process given ∈ S (1+α)/2 by (3). Assume that 0 <α< 1, FT = T and assumptions (A1)-(A7) are fulfilled. Then, for any open set U ([0, 1], R) and any closed set L ([0, 1], R) we have ⊂C ⊂C −α ∗ lim inf T log P ( XT , ϕ U) inf Λ (f), T →+∞ h i∈ ≥− f∈U −α ∗ lim sup T log P ( XT , ϕ L) inf Λ (f), T →+∞ h i∈ ≤− f∈L where 2 f 1 Λ∗(f)= k kH , 4H (T1(ϕ)+ V qT2(ϕ)) if f H1 and Λ∗(f)= if f / H1. ∈ ∞ ∈ Remark 4.7. This theorem “links” the central limit theorem and the large deviation principle. Roughly speaking α 0 corresponds to Theorem 4.1 and α 1 to Theorem 4.2. → → Remark 4.8. Let us also notice a close resemblance of this result to the Schilder theorem [9, Theorem 5.2.3] which is a strong large deviation principle for the Wiener process. This together with Theorem 4.1 imply that the process of fluctuations of the occupation time is much alike the Wiener process. While the CLT establishes this fact for “typical” paths the MDP complements it to “moderately rare” events. That also means that the properties of the movement of the particles merely influence the properties of converge. It should be noted however that in the LDP of Theorem 4.2 the analogy breaks, meaning that for “extremely rare” events the properties of the movement finally commence to play a significant role. We present also a moderate deviation principle for XT (1). d Corollary 4.1. Let ϕ (R )+ and XT be the rescaled occupation time fluctuations process given ∈ S (1+α)/2 by (3). Assume that 0 <α< 1, FT = T and assumptions (A1)-(A7) are fulfilled. Then for any open set U R and any closed set L R we have ⊂ ⊂ −α ∗ lim inf T log P ( XT (1), ϕ U) inf Λ (x), T →+∞ h i∈ ≥− x∈U −α ∗ lim sup T log P ( XT (1), ϕ L) inf Λ (x), T →+∞ h i∈ ≤− x∈L where x2 Λ∗(x)= . 4H (T1(ϕ)+ V qT2(ϕ)) The proof is an easy application of the contraction principle [9, Theorem 4.2.1] to Theorem 4.4. 4.3 Superprocess In this subsection N denotes the superprocess described in Section 3. The processes (2) and (3) are defined with this N. Firstly we present a central limit theorem 10 Theorem 4.5. Let XT be the rescaled occupation time fluctuations process given by (3). Assume 1/2 that FT = T and assumptions (A1)-(A5) are fulfilled. Then X X, and X X, T →i T →fdd where X is a generalised ′(Rd)-valued Wiener process with covariance functional S Cov( X , ϕ , X , ϕ )= VHq (s t) T (ϕ , ϕ ) ϕ , ϕ (Rd), h t 1i h s 2i ∧ 2 1 2 1 2 ∈ S if, additionally, assumptions (A8)-(A9) are fulfilled then X X. T →c Let us recall (4) and denote Q2 QS := . (19) 0 4Vq The LDP contained in Theorem 4.6 is our next aim. To this end we need to formulate the following lemma Lemma 4.2. Let ϕ (Rd) and θ vϕ(x,θ) := lim vθϕ(t,x,θ). (20) t→+∞ The proof follows the lines of the proof of Lemma 4.1 and is skipped. We define now ∂ A := lim vϕ(x,θ). (21) S θ→Q0 /kϕk∞ ∂θ Fix ϕ (Rd) and let us recall (7). For ν such that χ 11 To give full picture we also recall here a non-functional counterpart of the above theorem. It is a slightly modified version of [13, Theorem 4.1]. Theorem 4.7. Let ϕ (Rd) and Y be the rescaled occupation time process given by (2). Assume ∈ S + T that FT = T and assumptions (A1)-(A3) are fulfilled. Then there exists δ > 0 for any open set U (0, δ) and any closed set L (0, δ) we have ⊂ ⊂ −1 ∗ lim inf T log P ( YT (1), ϕ U) inf Λ (x), T →+∞ h i∈ ≥− x∈U −1 ∗ lim sup T log P ( YT (1), ϕ L) inf Λ (x), T →+∞ h i∈ ≤− x∈L where ∗ Λ (x)= sup [xθ vϕ( ,θ), λ ] , S − h · i θ≤Q0 /kϕk∞ where vϕ(θ) is given by (20). Now we present a strong moderate deviation principle d Theorem 4.8. Let ϕ (R )+ and XT be the rescaled occupation time fluctuations process given ∈ S (1+α)/2 by (3). Assume that FT = T , 0 <α< 1 and assumptions (A1)-(A7) are fulfilled. Then, for any open set U ([0, 1], R) and any closed set L ([0, 1], R) we have ⊂C ⊂C −α ∗ lim inf T log P ( XT , ϕ U) inf Λ (f), T →+∞ h i∈ ≥− f∈U −α ∗ lim sup T log P ( XT , ϕ L) inf Λ (f), T →+∞ h i∈ ≤− f∈L where 2 f 1 Λ∗(f)= k kH , 4HVqT2(ϕ) if f H1 and Λ∗(f)= if f / H1. ∈ ∞ ∈ The contraction principle [9, Theorem 4.2.1] can be applied here to obtain a moderate deviation principle for XT (1). This result was already known [13, Theorem 5.1], we put it here for readers’ convenience d Corollary 4.2. Let ϕ (R )+ and XT be the rescaled occupation time fluctuations process given ∈ S (1+α)/2 by (3). Assume that 0 <α< 1, FT = T and assumptions (A1)-(A7) are fulfilled. Then, for any open set U R and any closed set L R we have ⊂ ⊂ −α ∗ lim inf T log P ( XT (1), ϕ U) inf Λ (x), T →+∞ h i∈ ≥− x∈U −α ∗ lim sup T log P ( XT (1), ϕ L) inf Λ (x), T →+∞ h i∈ ≤− x∈L where x2 Λ∗(x)= . 4HV qT2(ϕ) Remark 4.9. Remarks concerning the BPS contained in the previous subsection are also valid for the superprocess. 12 5 Proofs The proofs for the branching particle system and the superprocess are similar and are presented together. 5.1 Notation In the proofs we use the following notation. Firstly, Φ is always of either of two forms Φ(x,s)= ϕ(x)ψ(s), ϕ (Rd), ψ (R). (24) ∈ S ∈ S Φ(x, ds)= ϕ(x)ν(ds), ϕ (Rd),ν is a signed measure of finite variation ∈ S For the Φ of the first form we define 1 1 t ϕ (x) := ϕ (x) , χ(s) := ψ(u)du, χ := χ . (25) T F T T T Zs and for the Φ of the second form we use χ given by (7). Note that it is a cádlág function. Secondly, throughout the paper Ψ always is Ψ(x,s)= ϕ(x)χ(s), 1 s ΨT (x,s)= Ψ x, = ϕT (x)χT (s). (26) FT T In the following the notation is always assumed unless stated otherwise. 5.2 One-particle equation In this section we present an equation describing the behaviour of the occupation time for the branching system starting from a single particle. Subsequently we also acquire its analogue for the superprocess. These equations play key role in the rest of the proofs. Recall (1) and define G(s) := F (1 s) (1 s) − − − G(s)= qs2 + (1 2q)s. − We denote the Laplace transform of the occupation time for the system starting off from a single particle at x t vB (x,r,t) := E exp N x, Ψ ( , r + s) ds 1, Ψ (Rd+1), (27) Ψ h s · i − ∈ S Z0 x x where Ns s≥0 denotes the empirical measure of the particle system with the initial condition N0 = x{ } δx. N is a system in which particles evolve according to the dynamics described in Introduction but without immigration. B B Lemma 5.1. Let Q0 be given by (15). Assume that Ψ < Q0 and assumptions (A1)-(A3) are fulfilled then t vB (x,r,t)= Q Ψ ( , r + t s)(1+ vB ( , r + t s,s)) + Vq(vB ( , r + t s,s))2 (x) ds. Ψ Tt−s · − Ψ · − Ψ · − Z0 (28) 13 The proof of this lemma is a rather obvious modification of the proof [18, Lemma 3.1] though one needs to be careful as the equation may blow up for some Ψ. The proof of [13, Lemma 3.2] can be modified to exclude this possibility and to show that vB admits a unique solution. S Now we move to the analogue for the superprocess. Let vΨ be analogue of (27) for the superprocess x i.e. this time by N we understand the superprocess with the initial condition δx again without immigration. S S Lemma 5.2. Let Q0 be given by (19). Assume that Ψ t 2 v¯S (x,r,t)= Q Ψ ( , r + t s) Vq v¯S ( , r + t s,s) (x) ds. (30) Ψ Tt−s · − − Ψ · − Z0 h i This equation is much much simpler in analysis but can be used only in the proof of the CLT. In the proof of the deviation principles we will need (27). Proof. Recall that Fn denotes the generating function of the branching law (10). The one particle equation from Lemma 5.1 for the n-th approximation is t 2 vB (x,r,t)= Qn Ψ ( , r + t s) 1+ vB ( , r + t s,s) + V q vB ( , r + t s,s) (x) ds, n,Ψ Tt−s · − n,Ψ · − n n n,Ψ · − Z0 h i B where Q := V (1 2q ). Denote now h = nv Ψ (which reflects the fact that the n-th approx- n n n n,Ψ n, − n imation consists of the particles of size 1/n and the initial number of particles is n times greater). It is easy to check that Qn = Q hence t 1 V q h (x,r,t)= Q Ψ ( , r + t s) 1+ h ( , r + t s,s) + n n h2 ( , r + t s,s) (x) ds. n,Ψ Tt−s · − n n,Ψ · − n n,Ψ · − Z0 We have Vnqn Vq. By the convergence from Proposition 3.1 (we use a different starting condition n → but it does not influence the convergence) h vS (31) n,Ψ → Ψ S It is easy to check that vΨ fulfils (29). Equation (29) has a useful series representation. Firstly, for f, g (Rd+1) we define the ∈ B convolution operator t (g f)(x,t)= Q [g( ,s)f( ,s)] (x)ds. ∗ Tt−s · · Z0 14 Let us fix t > 0 and denote vS (x,s) := vS (x,t t,t). By (29) it fulfils 0 Ψ Ψ 0 − t 2 vS (x,t)= Q Ψ( ,t s)+ Vq vS ( ,s) (x) ds, t [0,t ]. Ψ Tt−s · 0 − Ψ · ∈ 0 Z0 h i Following the reasoning of [13, Section 3] and assuming that Ψ t0 n−1 t0 t ∗n Q ∗k ∗(n−k) F ( ,s) 1ds t−s F ( ,s) F ( ,s) 1dsdt. 0 k · k ≤ 0 0 kT · · k Z Xk=1 Z Z h i For n 3 by (A7) and Lemma 5.3 we get ≥ t0 ⌈n/2⌉ t0 t ∗n 1−2(n−k) n−k −Q′(t−s) ∗k F ( ,s) 1ds 2C Bn−kQ VqΨ ∞ e F ( ,s) 1dsdt. 0 k · k ≤ k k 0 0 k · k Z Xk=1 Z Z h i 15 Changing the order of integration we get t0 ⌈n/2⌉ t0 ∗n 2C 1−2(n−k) n−k ∗k F ( ,s) 1ds ′ Bn−kQ VqΨ ∞ F ( ,s) 1ds. 0 k · k ≤ Q k k 0 k · k Z Xk=1 Z h i Assume now that we already know that (33) is true for k t0 ⌈n/2⌉ t0 ∗n 2C 1−2(n−k) n−k n−k −k F ( ,s) 1ds ′ Bn−kQ (C1Vq) ϕ ∞ a ϕ 1 χ(s)ds . 0 k · k ≤ Q k k k k 0 Z Xk=1 Z w Using the fact that B 5−n | 0 we can find C small{z enough to have w} < a−n. The cases n = 1, 2 n → 1 can be checked directly, finally by appealing to the induction we finish the proof. Let us also define t v˜ (x,r,t)= Q Ψ( , r + t s)ds, Ψ (Rd+1). (34) Ψ Tt−s · − ∈ S Z0 uB := vB v˜ , uS := vS v˜ . (35) Ψ Ψ − Ψ Ψ Ψ − Ψ We have B S Lemma 5.5. uΨ, uΨ satisfy the equations t uB(x,r,t)= Q Ψ( , r + t s)vB( , r + t s,s)+ Vq(vB( , r + t s,s))2 (x)ds. (36) Ψ Tt−s · − Ψ · − Ψ · − Z0 t uS (x,r,t)= Q Vq(vS ( , r + t s,s))2 (x)ds. (37) Ψ Tt−s Ψ · − Z0 B For proof see [18, Lemma 3.2]. We will now present two lemmas for estimation of vΨ . They will have a common proof. Lemma 5.6. There exists C ,C > 0 such that for any Ψ (Rd+1) such that Ψ Let now Ψ be of the form Ψ(x,t)= ϕ(t)1[s,s+δ](t) for some s,δ > 0. Lemma 5.7. There exists C ,C > 0 such that for any ϕ (Rd) such that ϕ δ 16 Proof. We notice that it is enough to show the claim for v(x,t) := vB(x, T t,t), for any T > 0 Ψ − (with constants independent of T ). Moreover it is upper-bounded by L, being the solution of t L(x,t)= Q Ψ ( , T s) (1 + L ( ,s)) + Vq(L ( ,s))2 (x) ds. Tt−s k · − k∞ · · Z0 Obviously the space parameter is now superfluous, skipping it we obtain yet simpler equations t L(t)= e−Q(t−s) Ψ ( , T s) (1 + L (s)) + Vq(L (s))2 ds, k · − k∞ Z0 which in a differential form writes as L′(t)= QL(t)+ Ψ ( , T t) (1 + L(t)) + Vq(L(t))2, L(0) = 0. − k · − k∞ It is upper-bounded by the solution of K′(t)= (Q/2)K(t)+ Ψ ( , T t) (1 + K(t)), K(0) = 0, − k · − k∞ as long as K Q . One checks that once we assume Ψ Q/4 we have k k∞ ≤ 2Vq k k∞ ≤ Ψ 4 Ψ K(t) k k∞ e(kΨk∞−Q/2)t 1 k k∞ . ≤ Ψ ∞ Q/2 − ≤ Q k k − Now one can easily choose C1 such that Lemma 5.6 holds. To see Lemma 5.7 we notice that without loss of generality we may assume that K′(t)= (Q/2)K(t)+ ϕ 1 (t)(1 + K(t)), K(0) = 0. − k k∞ [0,δ] The sup is attained for K(δ) hence we have ϕ K(t) K(δ)= k k∞ e(kϕk∞−Q/2)δ 1 ekϕk∞δ 1. ≤ ϕ ∞ Q/2 − ≤ − k k − The final step is to choose C small enough to have eC1 1 Q . 1 − ≤ 2Vq S B Due to the representation presented above vΨ is easier to handle. It is useful to know that vΨ is S comparable with vΨ. This property allows to convert some proofs for the superprocess into proofs for the BPS semi-automatically. Lemma 5.8. There exists C > 0 such that for any Ψ (Rd+1) which fulfils the assumptions of ∈B + Lemma 5.6 or Lemma 5.7 we have vB (x,r,t) vS (x,r,t) vB(x,r,t). (39) CΨ ≤ Ψ ≤ Ψ Proof. The second inequality is easy and is left to the reader. We may choose C > 0 such that C(1 + vB ) 1. Therefore k CΨk ≤ t vB (x,r,t)= Q CΨ ( , r + t s)(1+ vB ( , r + t s,s)) + Vq(vB ( , r + t s,s))2 (x) ds CΨ Tt−s · − CΨ · − CΨ · − Z0 t Q Ψ ( , r + t s,s)+ Vq(vB ( , r + t s,s))2 (x) ds. ≤ Tt−s · − CΨ · − Z0 Easy application of the Banach contraction principle concludes the proof. 17 Proof of Lemma 4.1(sketch). Let us define operator t t (F (v))(x,r,t) = Q θϕ(x)+ Q θϕ( ,s)v( ,s,θ)+ V qv2( ,s,θ) (x)ds. Tt−s Tt−s · · · Z0 Z0 For θ > 0 the sequence (0, F (0), F (F (0)),...) is non-decreasing and by the proof of the previous lemma bounded. Therefore it converges to a solution of the equation. One can also check that for t small enough the operator is contraction. The unicity can be proven easily by subtractions of two distinct solutions. To keep the proofs comprehensive we utilise the following notation B B B B vT (x,r,t) := vΨT (x,r,t) and vT (x) := vT (x, 0, T ), (40) The same also applies to uB , vS , uS . ΨT ΨT ΨT 5.3 Laplace transforms This section we are going to compute the Laplace transforms of space time variables Y˜T and X˜T defined by (6) for (2) and (3). Let us recall notation (24). We start with the branching particle system N B. Proposition 5.1. Let Φ (Rd+1) such that Ψ QB then for Y˜ B, X˜ B defined for the BPS N B ∈ B ≤ 0 T T we have T E ˜ B B exp XT , Φ = exp H uT (x, T s,s)ds , 0 Rd − D E Z Z T E ˜ B B exp YT , Φ = exp H vT (x, T s,s)ds . 0 Rd − D E Z Z The proof is analogous to the proof in [18, Section 3.3]. For the superprocess N S we have Proposition 5.2. Let Φ (Rd+1) such that Ψ QS then for Y˜ S, X˜ S defined for the superprocess ∈B ≤ 0 T T N S we have T E ˜ S S exp XT , Φ = exp H uT (x, T s,s)ds , 0 Rd − D E Z Z T E ˜ S S exp YT , Φ = exp H vT (x, T s,s)ds . 0 Rd − D E Z Z Proof. Let us fix Ψ fulfilling the assumptions. In Section 3 we defined the sequence N n approxi- S ˜ n { } n mating the superprocess N . We denote by YT the space time variable for (2) defined for N . We have 1 T ˜ n T n n YT , Φ = NTs, Ψ( ,s) ds = Ns , ΨT ( ,s) ds. (41) FT 0 h · i 0 h · i D E Z Z Let us recall notation (26) and denote T n E ˜ n E n KT (Φ) := exp YT , Φ = exp Ns , ΨT ( ,s) ds . 0 h · i D E Z 18 Conditioning with respect to Immn (Poisson random field describing the immigration of the n-th approximation), using independence of evolution of particles (branching Markov property) and (27) for the BPS starting from one particle we obtain T E exp N n, Ψ ( ,s) ds Immn = h s T · i Z0 T E x,t,n B exp N , ΨT ( ,s) ds = v Ψ (x,t,T t) 1 , (42) s−t · n, T − − \n t \n n (t,x)Y∈Imm Z D E (t,x)Y∈Imm n [ n n where Imm is a (random) set such that (t,x)∈Imm\ δ(t,x) = Imm a.s. viz. δ(t,x) corresponds to x,t,n a particle which immigrate to the systemP at time t to location x. By N we denote the BPS starting from location x at time t adhering to the dynamics of the n-the approximation. Following the notation of the proof of Lemma 5.2 we have T n n B Eexp N , ΨT ( ,s) ds = Eexp Imm , log(v Ψ ( , ⋆, T ⋆) 1 , − h s · i n, T · − − Z0 n where ,⋆ denote integration with respect to space and time, respectively. Taking into account · distribution of Immn we obtain T T n E n B KT (Φ) = exp Ns , ΨT ( ,s) ds = exp H nv ΨT (x, T t,t)dxdt . h · i Rd n, − Z0 Z0 Z n By the convergence (31) and Proposition 3.1 we get T n S KT (Φ) KT (Φ) = exp H vΨT (x, T t,t)dxdt , (43) → Rd − Z0 Z E ˜ S where KT (Φ) = exp YT , Φ . Simple calculations show that D E T E ˜ S S LT (Φ) := exp XT , Φ = exp H uT (x, T t,t)dxdt . (44) 0 Rd − nD Eo Z Z 5.4 Central limit theorem In this section we present the proof of Theorem 4.1. We follow closely the lines of the proof of [18, Theorem 2.1]. To make it clear we present a general scheme first. Although the processes XT are signed-measure-valued it is convenient to regard them as ′(Rd)-valued. In this space one may S employ a space-time method introduced in [4] which together with Mitoma’s theorem constitute a powerful technique in proving weak functional convergence. 19 ˜ S Convergence From now on we will denote by XT a space-time variable (recall (6) with τ = 1) S ˜ S defined for XT for the superprocess. To prove convergence of XT we will use the Laplace functional L (Φ) = Eexp X˜ S , Φ , Φ (Rd+1) . T − T ∈ S + D E For the limit process X denote L(Φ) = Eexp X˜ S, Φ , Φ (Rd+1) . − ∈ S + D E Once we have established convergence L (Φ) L(Φ), as T + , Φ (Rd+1) . (45) T → → ∞ ∈ S + we will obtain weak convergence X˜ X˜ and consequently X X. Two technical remarks T ⇒ T →i should be made here. We consider only non-negative Φ of the first form described in Section 5.1. The procedure how to extend the convergence to any Φ is explained in [5, Section 3.2]. Another ˜ S issue is the fact that XT , Φ is not non-negative. The usage of the Laplace transform in this paper is justified by the specialD (Gaussian)E form of the limit. For more detailed explanation one can check also [5, Section 3.2]. As explained in [7] due to the special form of the Laplace transform convergence (45) implies also finite-dimensional convergence. We have T S Rd+1 LT (Φ) = exp H u¯T (x, T s,s)ds , Φ ( )+, − Rd − ∈ S Z0 Z ! A(T ) where u¯S :=v ˜ v¯S is an analogue of| (37) defined{z by } T T − T t u¯S (x,r,t)= Q V (¯vS ( , r + t s,s))2 ds. (46) T Tt−s T · − Z0 S and v¯T is an analogue of (29) given by (30). The formula for L(T ) can be proved in the same as S S Proposition 5.2. Note here that v¯T and u¯T are much alike vT and uT in [18]. It is worthwhile to S S mention that v¯T is far easier to analyse than vT because we have obvious inequalities S CΦ 0 v¯T v˜T . (47) ≤ ≤ ≤ FT We can also use the following simple estimation S CΦ u¯T 2 . (48) ≤ FT S Our aim now is to calculate the limit of A(T ). To this end we replace vT with v˜T in (46) and calculate the limit for such changed expression. T t ˜ Q 2 A(T )= V t−s v˜T ( , T s,s) (x)dsdtdx. (49) Rd T · − Z Z0 Z0 20 A˜(T ) is the same as A˜4(T ) in [18, Section 3.3]. Therefore we have 1 +∞ ˜ 2 Q Q Q Q lim A(T ) = 2V χ(1 v) dv s ϕ( ) s ϕ( ) (x)dxds. T →+∞ − Rd U T · T U · Z0 Z0 Z Note that by assumptions (A5) the integral above is finite. We are left with estimation of A˜(T ) S − A(T ). By the definition of u¯T and inequality (47) we have T t ˜ Q S A(T ) A(T ) 2V t−s u¯T ( , T s,s)v ˜T ( , T s,s) (x)dsdtdx. | − |≤ Rd T · − · − Z Z0 Z0 Using (48) and (34) we write T t s ˜ 2V Q Q A(T ) A(T ) 2 t−s s−uΨT ( , T s) (x)dudsdtdx. | − |≤ F Rd T T · − T Z Z0 Z0 Z0 Using (26), after simple calculations, we get T t ˜ 2V Q A(T ) A(T ) 3 u u ϕ(x)dudtdx. | − |≤ F Rd T T Z Z0 Z0 Now, by using d’Hospital rule, it follows easily from assumption (A5) that A˜(T ) A(T ) 0, as T + . | − |→ → ∞ Tightness Using additional assumptions (A8),(A9) the tightness can be proved utilising the Mit- oma theorem [20]. It states that tightness of X with trajectories in C([0, 1], ′(Rd)) is equivalent { T }T S to tightness of X , ϕ , in ([0,τ], R) for every ϕ (Rd). We adopt a technique introduced in [6]. h T i C ∈ S Recall a classical criterion [1, Theorem 12.3], i.e. a process X (t), ϕ is tight if for any t,s 0 and h T i ≥ constant C > 0 E( X (t), ϕ X (s), ϕ )4 C(t s)2. (50) h T i − h T i ≤ − Following the scheme in [6] we define a sequence (ψ ) in (R), and χ (u)= 1 ψ (s)ds in a such n n S n u n way that R ψ δ δ , 0 χ 1 . n → t − s ≤ n ≤ [s,t] Denote Φ = ϕ ψ . We have n ⊗ n lim XT , Φn = XT (t), ϕ XT (s), ϕ n→+∞ h i h i − h i thus by the Fatou lemma and the definition of ψn we will obtain (50) if we prove that 4 E X˜ , Φ C(t s)2, T n ≤ − D E where C is a constant independent of n and T . From now on we fix n and denote Φ := Φn and χ := χn. By properties of the Laplace transform we have 4 d4 E X˜ , Φ = Eexp θ X˜ , Φ T dθ4 − T θ=0 D E D E 21 Hence the proof of tightness will be completed if we show d4 Eexp θ X˜ , Φ C(t s)2. dθ4 − T ≤ − θ=0 D E For the sake of brevity the detailed calculation are left for the reader. 5.5 Large deviation principle In this section we present the proof of Theorem 4.6. The proof of Theorem 4.2 is very similar (some parts can be transformed directly and some with a help of Lemma 5.8). Let us recall definition (22) and denote 1 −1 E S Λϕ(T,ν) := T log exp T YT (t), ϕ ν(dt) . Z0 Lemma 5.9. Let ϕ (Rd) and ν BV (R) such that ϕ χ x Proof. Let us recall notation (25) and denote ΨT (x,s) := ϕ(x)χT (s). Let N denote the superpro- x cess starting from N0 = δx without immigration. Definition (27) yields T t vS (x, T (1 t),Tt)= E exp N x, ϕ χ ((1 t)+ s/T )ds 1. ΨT − h s i ν − − Z0 By [13, Lemma 4.1] it is finite. Without loss of generality we may assume that χν is a càdlàg. Let us denote now T t T t A(T ) := N x, ϕ χ((1 t)+ s/T )ds B(T ) := χ(1 t) N x, ϕ ds. h s i − − h s i Z0 Z0 We have T t A(T ) B(T ) N x, ϕ χ((1 t)+ s/T ) χ(1 t) ds. | − |≤ h s i | − − − | Z0 By the dominated Lebesgue convergence theorem (by the sub-criticality of the branching law there is ∞ N x, ϕ ds is finite a.s. ) we have 0 h s i R A(T ) B(T ) 0, a.s. | − |→ A next usage of the dominated convergence theorem yields S vϕ(x,χ(1 t)) = lim vΨ (x, T (1 t),Tt), (51) − T →+∞ T − where vϕ is defined by (20). By Proposition 5.2 we get T 1 −1 S Λϕ(T,ν)= T H vΨT (x, T t,t)dxdt = H vΨT (x, T (1 t),Tt)dx. Rd − Rd − Z0 Z Z0 Z Appealing to (51) and the dominated Lebesgue theorem concludes. 22 Proof of Theorem 4.6. Upper bound We follow a standard route of showing functional large devi- ation principle via studying multidimensional case. This is also emphasised by the use of the same notation as in the infinite dimensional case. Let us consider θ = (θ ,θ , ,θ ) Rn and set 1 2 · · · n ∈ n χθ(s) := (θi + θi+1 + . . . + θn)1 i−1 i (s). (52) [ n , n ) Xi=1 n It is straightforward to check that according to (7) χθ = χν for ν = i=1 θiδi/n. We also denote n P −1 Λϕ(T,θ)= T log Eexp T θi YT (i/n), ϕ h i! Xi=1 By Proposition 5.1 and Lemma 5.9 for any θ Rn such that sup χ (t) QS/ ϕ we have ∈ t θ ≤ 0 k k∞ n 1 H Λϕ(θ) := lim inf Λ(T,θ)= H vϕ(x,χθ(t))dxdt = vϕ(x,θi + θi+1 + + θn)dx. T →+∞ 0 Rd n Rd · · · Z Z Xi=1 Z We check that Λ is a convex function with respect to each θ . Let us recall (21) and take f Rn i ∈ f = (f ,f ,...,f ) such that 0 Indeed, let Vϕ(θ) := Rd vϕ(x,θ)dx. Using standard calculus we know that sup is attained at the solution of the following set of equations: R j ∂ H ′ 0= ( f,θ Λϕ(θ)) = fj Vϕ(x,θi + + θn), j 1, ,n . ∂θj h i− − n · · · ∈ { · · · } Xi=1 The vector f was chosen in such a way that the solution of the above set of equations θ = (θ ,θ ,...,θ ) is such that θ + . . . + θ < QS/ ϕ (more details can be found in the proof 1 2 n ∀i i n 0 k k∞ of [13, Theorem 4.1]). In other words, for this θ we have sup χ (t) QS/ ϕ ). This consid- t θ ≤ 0 k k∞ erations together with [9, Lemma 2.3.9] entitle us to use the Gärtner-Ellis theorem (see e.g. [9, Theorem 2.3.6]) establishing the result in the finite-dimensional setting. Now we are ready to prove the upper bound. It is suff‘icient to show that for any f and ∈ C0,A/H any δ > 0 −1P ∗ lim inf T ( YT , ϕ B(f, δ)) Λϕ(f), T h i∈ ≥− where B(f, δ) denotes ball in ([0, 1], R). Let us denote C O = g ([0, 1], R) : g(i/n) (f(i/n) ǫ,f(i/n)+ ǫ), i 0, 1,...,n , n N,ǫ> 0 n,ǫ { ∈C ∈ − ∈ { }} ∈ One can check that there exists n and ǫ such that O B(f, δ) contains only functions which are n,ǫ \ not increasing (i.e. for any such function g we can find s P ( Y , ϕ B(f, δ)) P ( Y , ϕ O ) ( ) h T i∈ ≥ h T i∈ n,ǫ ≥ ∗ 23 This reduced the problem to finite number of dimensions therefore −1 ( ) = lim inf T P ( YT (i/n), ϕ (f(i/n) ǫ,f(i/n)+ ǫ), i 1,...,n ) ∗ T h i∈ − ∈ { } Λ∗ ((f(0),f(1/n),...,f(1 1/n))). ≥− ϕ − Notice that the last quantity is the same as (23) if we restrict B in its definition to the set of point measures with the support in the set 1/n, 2/n,..., 1 . { } Lower bound Let us take function in f , for any δ > 0 we can find ν such that ∈ C0,A/H 0 f,ν Λ (ν ) Λ∗ (f) δ , and sup χ (t) < QS/ ϕ . ν can be approximated by a point h 0i− ϕ 0 ≥ ϕ − 4 t ν0 0 k k∞ 0 measure as in the previous section such that δ f,ν Λ (ν) Λ∗ (f) . h i− ϕ ≥ ϕ − 2 Following the notation of the previous section we write its total variation ν = n θ < + | | i=1 | i| ∞ (as each θ is bounded; for details see the proof of [13, Theorem 4.1]). We define r := δ/(2 ν ) | i| P | | and consider a ball B(f, r). For any g B(f, r) we have ν,f g δ/2. Using the Chebyshev ∈ h − i ≤ inequality we obtain 1 log P Y , ϕ B(f, r) T h T i∈ 1 1 1 1 log P ( Y (s), ϕ f(s)) ν(ds) δ/2 = log P Y (s), ϕ ν(ds) ν,f δ/2 ≤ T h T i− ≥− T h T i ≥ h i− Z0 Z0 1 1 log P exp T Y (s), ϕ ν(ds) exp Tν,f T δ/2 Λ (T,ν) ν,f + δ/2 ≤ T h T i ≥ {h i− } ≤ ϕ − h i Z0 Now we easily conclude that 1 ∗ lim sup log P YT B(f, r) Λ (f)+ δ. T →+∞ T ∈ ≤− 5.6 Functional moderate deviation principle In this section we prove Theorem 4.4 (we skip the proof of Theorem 4.8 which is simpler). Through- out the whole proof F = T (1+α)/2, 0 <α< 1 and ϕ (Rd) is fixed. We will skip the superscript T ∈ S + B. Lower bound Firstly we will prove the lower estimate for compact set. Let ν BV (R) and define ∈ 1 Λ (T,ν) := T −α log E exp θT α X (t), ϕ ν(dt) . ϕ h T i Z0 24 We introduce an additional parameter θ; in this part of proof it is always θ = 1. By Proposition 5.1 we have T −α Λϕ(T,ν)= T H uΨ(T,θ)(x, T t,t)dxdt. (53) Rd − Z0 Z where Ψ(T,θ) = θT (α−1)/2ϕ χ (see notation in Section 5.1). Since Ψ(T,θ) 0 we know that ⊗ T →T Λϕ(T,ν) is well-defined for T ’s large enough. T −α Λϕ(T,ν)= HT uΨ(T,θ)(x, T t,t)dxdt. Rd − Z0 Z Using equation (36) we obtain Λϕ(T,ν) = Λ1(T,ν) + Λ2(T,ν), where T t −α Q Λ1(T,ν) := HT t−s Ψ(T,θ)( , T s)vΨ(T,θ)( , T s,s) (x)dsdxdt, Rd T · − · − Z0 Z Z0 T t −α Q 2 Λ2(T,ν) := HVqT t−s vΨ(T,θ)( , T s,s) (x)dsdxdt. (54) 0 Rd 0 T · − Z Z Z h i Using the Fubini theorem we get T T −s −α Q Λ1(T,ν)= HT u Ψ(T,θ)( , T s)vΨ(T,θ)( , T s,s) (x)dudxds, Rd T · − · − Z0 Z Z0 T T −s −α Q 2 Λ2(T,ν)= HVqT u vΨ(T,θ)( , T s,s) (x)dudxds. 0 Rd 0 T · − Z Z Z h i This is approximated by (we recall (5)) T −α Q Λ1a(T,ν) := HT Ψ(T,θ)( , T s)vΨ(T,θ)( , T s,s) (x)dxds, Rd U · − · − Z0 Z T −α Q 2 Λ2a(T,ν) := HVqT vΨ(T,θ)( , T s,s) (x)dxds. (55) 0 Rd U · − Z Z h i In the next step we approximate v (x, T s,s) with s Q Ψ(T,θ)du = θT (α−1)/2 s Q ϕ(x)χ (T Ψ(T,θ) − 0 Ts−u 0 Ts−u T − u)du, namely R R T s 2 −1 Q Q Λ1b(T,ν) := Hθ T ϕ( )χT (T s) s−uϕ( )χT (T u)du (x)dxds, Rd U · − T · − Z0 Z Z0 T s 2 2 −1 Q Q Λ2b(T,ν) := HVqθ T s−uϕ( )χT (T u)du (x)dxds. (56) Rd U T · − Z0 Z "Z0 # 25 Now we substitute s Ts, use the Fubini theorem and (25) → 1 Ts 2 Q Q Λ1b(T,ν)= θ H ϕ( ) Ts−uϕ( )χ(1 u/T )χ(1 s) (x)dxduds, 0 0 Rd U · T · − − Z Z Z h i 1 Ts 2 2 Q Q Λ2b(T,ν)= θ HVq Ts−uϕ( )χ(1 u/T )du (x)dxds. Rd U T · − Z0 Z "Z0 # Substituting u Ts u we get → − 1 Ts 2 Q Q Λ1b(T,ν)= θ H ϕ( ) u ϕ( ) (x)χ(1 s + u/T )χ(1 s)dxduds, Rd U · T · − − Z0 Z0 Z 1 Ts 2 2 Q Q Λ2b(T,ν)= θ HVq u ϕ( )χ(1 s + u/T )du (x)dxds. Rd U T · − Z0 Z "Z0 # We define also 1 Ts 2 Q Q 2 Λ1c(T,ν) := θ H ϕ( ) u ϕ( ) (x)χ (1 s)dxduds, Rd U · T · − Z0 Z0 Z 1 Ts 2 2 Q Q Λ2c(T,ν) := θ HVq u ϕ( )χ(1 s)du (x)dxds. (57) Rd U T · − Z0 Z "Z0 # Finally we notice that both function are increasing in T and recall assumption (A4) to get (we denote T (χ)= 1 χ(1 s)2ds) 3 0 − R 2 2 lim Λ1c(T,ν) = (θ H)T1(ϕ)T3(χ), lim Λ2c(T,ν) = (θ HVq)T2(ϕ)T3(χ). T →+∞ T →+∞ We will show that in the limits of Λ1c, Λ2c are the same as the ones of Λ1, Λ2. Firstly, using (A7) we easily get 1 Ts 2 Q Λ2c(T,ν) Λ2b(T,ν) θ C u ϕ(x) χ(1 s)+ χ(1 s + u/T ) du | − |≤ Rd T | − − | Z0 Z Z0 Ts Qϕ(x) χ(1 s) χ(1 s + u/T ) du dxds. Tu | − − − | Z0 Using (A7) the first integral is finite, hence 1 Ts 2 Q Λ2c(T,ν) Λ2b(T,ν) θ C1 u ϕ(x) χ(1 s) χ(1 s + u/T ) du dxds 0. | − |≤ Rd T | − − − | → Z0 Z Z0 since we can observe that χ < + and one can use the Lebesgue dominated convergence k k∞ ∞ theorem. Analogously Λ (T,ν) Λ (T,ν) 0. | 1c − 1b |→ 26 Using assumption (A7) again we have T −α Λ1a(T,ν) Λ1b(T,ν) C1T Ψ (T,θ)(x, T s)u|Ψ|(T,θ)(x, T s,s)dxds. | − |≤ Rd | | − − Z0 Z By (38) we know that for T ’s large enough u (x, T s,s) CT −1+α hence |Ψ|(T,θ) − ≤ T −1 3 (α−1)/2 Λ1a(T,ν) Λ1b(T,ν) C2T Ψ (T,θ)(x, T s)dsdx C2θ T 0. | − |≤ Rd | | − ≤ → Z0 Z Similarly using assumption (A7) and Lemma 5.6 we get T −α Λ2a(T,ν) Λ2b(T,ν) C1T u|Ψ|(T,θ)(x, T s,s)v|Ψ|(T,θ)(x, T s,s)dxds, | − |≤ Rd − − Z0 Z T −1 3 (α−1)/2 Λ2a(T,ν) Λ2b(T,ν) C2T v|Ψ|(T,θ)(x, T s,s)dxds C2θ T 0. | − |≤ Rd − ≤ → Z0 Z Once again we utilise (A7) to obtain T −α −(T −s)Q′ Λ1a(T,ν) Λ1(T,ν) C1T e Ψ (T,θ)(x, T s)v|Ψ|(T,θ)(x, T s,s)dsdx. | − |≤ Rd | | − − Z0 Z By Lemma 5.6 we get T −1 −(T −s)Q′ −1 Λ1a(T,ν) Λ1(T,ν) C2T e Ψ (T,θ)(x, T s)dsdx C3T 0. | − |≤ Rd | | − ≤ → Z0 Z Analogously using Lemma 5.6 once more and then Lemma 5.4 we have T −1 −(T −s)Q′ (α−1)/2 Λ2a(T,ν) Λ2(T,ν) HT e v|Ψ|(T,θ)(x, T s,s)dsdx CT 0. | − |≤ Rd − ≤ → Z0 Z Finally, we put all the calculation together 1 2 2 Λϕ(ν) := lim Λϕ(T,ν)= θ H χ(1 s) ds (T1(ϕ)+ V qT2(ϕ)) . (58) T →+∞ − Z0 Once we prove that the exponential tightness holds - Section 5.6.1 - by [10, Theorem 2.2.4] and [10, Lemma 1.3.8] lower bound in Theorem 4.4 will be established. Upper bound We start with the multidimensional case. We utilise the notation introduced in the proof of the large deviation principle. Namely, consider a vector θ = (θ ,θ , ,θ ) and recall 1 2 · · · n (52). Further we denote n −α α Λϕ(T,θ) := T log Eexp T θi XT (i/n), ϕ . h i! Xi=1 27 This is in fact a special case of (58) hence we already know that 1 2 Λϕ(θ) = lim Λϕ(T,θ)= H χθ(1 s) ds (T1(ϕ)+ V qT2(ϕ)) . T →+∞ − Z0 ∗ Denote its Legendre transform by Λϕ. [9, Lemma 2.3.9] entitle us to use the Gärtner-Ellis theorem (see e.g. [9, Theorem 2.3.6]) hence for any open set O Rn we get the following deviation principle ⊂ −α P ∗ lim inf T log (( XT (1/n), ϕ ,..., XT (1), ϕ ) O) inf Λϕ(x), T →+∞ h i h i ∈ ≥− x∈O In order to prove (18) it suffices to show that for any f H1 and ǫ> 0 ∈ −α P ∗ lim inf T log ( XT , ϕ B(f,ǫ)) Λϕ(f). (59) T →+∞ h i∈ ≥− where B(f,ǫ) denotes a ball in ([0, 1], R). Consider now C O := g ([0, 1], R) : g(i/n) (f(i/n) ǫ/2,f(i/n)+ ǫ/2), i 1,...n , . n { ∈C ∈ − ∈ { }} O˜ := Πn (f(i/n) ǫ/2,f(i/n)+ ǫ/2) Rn n i=1 − ⊂ It is easy to check that for any f O one have Λ∗ (f) Λ∗ (f(1/n),f(2/n),...,f(1)) hence ∈ n ϕ ≥ ϕ −α P ∗ ∗ ∗ lim inf T log ( XT , ϕ On) inf Λϕ(x) inf Λϕ(g) Λϕ(f), T →+∞ h i∈ ≥− x∈O˜n ≥− g∈On ≥− To finish the proof we show that for any C > 0 there exists n such that −α lim sup T log P ( XT , ϕ On B(f,ǫ)) C. T →+∞ h i∈ \ ≤− ∗ Using the upper bound estimate we have only to prove that inff∈cl(On\B(f,ǫ)) Λϕ(f) > C. To this end we choose n such that w(f, 1/n) < ǫ/10 (w being defined by (9)). If f cl (O B(f,ǫ)) then ∈ n\ there exist s,t such that s t 1/n and f(s) f(t) > ǫ/5. Using the Jensen inequality it is easy | −2 |≤ | − | to show that t f ′(u)2du ǫ n . s ≥ 10 R 5.6.1 Exponential tightness Let us fix ϕ (Rd) and recall that F = T (1+α)/2. In this section we will prove exponential ∈ S + T tightness of X , ϕ . To keep notation short we write {h T i}T x (t) := X (t), ϕ , t [0, 1]. T h T i ∈ Remark 5.2. If the exponential tightness holds for x defined with some ϕ it is also true for xT defined with Cϕ for any C > 0. Therefore we are entitled to decrease ϕ (finitely many times) if necessary. By ST (xT ) we denote stopping times relative to the natural filtration of xT . In our context 28 Lemma 5.10. Assume that for all λ> 0 we have −α lim lim sup T log P sup xT (t) η = , (60) η→+∞ T →+∞ t∈[0,1] | |≥ ! −∞ −α lim lim sup sup T log P sup xT ((τ + t) 1) xT (τ) λ = . (61) δ→0 T →+∞ τ∈ST (xT ) t≤δ | ∧ − |≥ ! −∞ then sequence x is exponentially tight. { T }T This follows easily from [16, Theorem 3.1]. Let us denote now total fluctuation of occupation time by x (i.e. we can take x(t) := xt(1) with FT = 1 as a definition). We will need the following estimate Lemma 5.11. Let ǫ (0, 1/2) then there exist c > 0 and T > 0 such that for T > T and ∈ 0 0 0 θ′ T −ǫ ≤ ≤ Eexp θ′ [x(t) x(s)] exp c(t s)(θ′)2 , 0 s We claim that Λ (T,ν) cH(T, θ, b). To be more precise| the{z lemma} will| be shown{z once} we prove ϕ ≈ Λ (T,ν) H(T, Θ, b) lim sup sup sup ϕ − = C < + . T →+∞ Θ∈(0,1) 0 Λ2(T,ν)−H2(T,Θ,b) We start with considering H(T,Θ,b) . In this direction we going to use the chain of approxi- mations of (53) from Section 5.6. That is Λ2, Λ2a, Λ2b, Λ2c given by (54)-(57). Obviously, by the fact that ϕ 0 and χ = χν 0 (recall (7)), we have Λ2(T ) Λ2a(T ), so we have only check the rest ≥ ≥ (α−1)/2 ≤ of the terms. Let us recall that Ψ(T, Θ)=ΘT ϕχT . Using definitions (34), (35), assumption (A7), inequalities (38), v˜ v , (38) and finally Lemma 5.4 we prove Ψ(T,θ) ≤ Ψ(T,θ) −α T Q Λ (T,ν) Λ (T,ν) C1T d u (x, T s,s)v (x, T s,s) dsdx 2a 2b 0 R U Ψ(T,θ) − Ψ(T,θ) − − 2 −1 H(T, Θ, b) ≤ R R CΘ T b ≤ 2 T T C2Θ Rd vΨ(T,θ)(x, T s,s)dsdx 0 − C T (α−1)/2b−1 χ (T s)ds = C T (α−1)/2 0. Θ2b ≤ 3 T − 3 → R R Z0 29 Further using assumption (A7) again we derive 2 2 2 1 Ts Q Ts Q C Θ d u ϕ(x)χ(1 s)du u ϕ(x)χ(1 s + u/T )du dxds Λ (T,ν) Λ (T,ν) 1 0 R 0 T − − 0 T − | 2b − 2c | R R R 2 −1R H(T, Θ,t) ≤ CΘ T b 1 Ts Q Ts Q C2 0 Rd 0 u ϕ(x)(χ(1 s)+ χ(1 s + u/T ))du 0 u ϕ(x) χ(1 s) χ(1 s + u/T ) du dxds T − − T | − − − | . ≤ R R R bT −1 R · Q It is easy to check that u ϕ(x) χ(1 s) χ(1 s + u/T ) du C . Next we substitute u T u, 0 T | − − − | ≤ 3 → use definition of ν and assumption (A7) to obtain R 1 s Λ2b(T,ν) Λ2c(T,ν) −1 2 Q | − | C4b T Tuϕ(x)(χ(1 s)+ χ(1 s + u))dudxds = H(T, Θ, b) ≤ 0 Rd 0 T − − 1 1 Z Z Z 1 −1 2 Q 2 Q C5b T Tuϕ(x) (χ(1 s)+ χ(1 s + u))dsdxdu C5T Tuϕ(x)dxdu 0. Rd T − − ≤ Rd T → Z0 Z Zu Z0 Z Λ1(T,ν)−H1(T,Θ,b) In similar way one can upper-bound H(T,Θ,b) which concludes the proof. We need also a method of estimating suprema of processes. Let us denote the set of dyadic rationals D = i/2k : i 0, 1,..., 2k . k ∈ n n oo Lemma 5.12. Let x : [0, 1] R be a cádl´gfunction then → ∞ sup x(t) 2 Lk(x)+ x(1) , t∈[0,1] | |≤ | | Xk=1 where Lk(x)= max mrst(x), r,s,t∈Dk s−r=t−s=2−k and m (x)= x(s) x(r) x(t) x(s) . rst | − | ∧ | − | The proof is standard; the reader is referred to [2, Section 10]. Now we proceed to the proof of exponential tightness. We start with (60). Using Lemma 5.12 we write ∞ P sup xT (t) η P 2 Lk(xT ) η/2 + P ( xT (1) η/2) . t∈[0,1] | |≥ ! ≤ ≥ ! | |≥ Xk=1 Therefore (60) will be shown once we obtain ∞ −α −α lim lim sup T log P ( xT (1) η)= , lim lim sup T log P Lk(xT ) η = . η→+∞ T →+∞ | |≥ −∞ η→+∞ T →+∞ ≥ ! −∞ Xk=1 30 The first one follows from [13, Theorem 5.1]. To prove the second one we set ǫ := (1 α)/50, ǫ := (1 α)/70, θ := 2−ǫ1 , (63) 1 − 2 − and write ∞ ∞ k P Lk(xT ) η P Lk(xT ) θ ηθ ≥ ! ≤ ≥ k=1 k=1 X X ∞ k −k −k k 2 max P xT (i2 ) xT ((i 1)2 ) θ ηθ , ≤ i∈ 1,2,...,2k | − − |≥ Xk=1 { } where η = η(1 θ)/θ. We denote also K := 1−α log T and k := log T and split the sum θ − T 8 log(2θ) T ∞ KT kT ∞ P Lk(xT ) η . . . + . . . + . . . =: I(T )+ II(T )+ III(T ). ≥ ! ≤ Xk=1 Xk=1 kX=KT kX=kT Estimation of III(T ) First we will estimate the probability in the sum above. For k N we denote δ := 2−k, l := 2k ∈ k k and take any u , u R such that u u = δ and by x total fluctuation (as in Lemma 5.11). We 1 2 ∈ + 2 − 1 k have u2 E B Ak := exp (lk(x(u2) x(u1))) = exp uΨk (x, u2 t,t)dxdt − Rd − Z0 Z u2 t Q B 2 = exp t−s vΨk ( , u2 s,s) (x)dsdxdt , Rd T · − Z0 Z Z0 1 where Ψk(x,t) = lkϕ(x) [u1,u2](t). One must be aware that in the above equation we go slightly beyond the scope of Proposition 5.2 and Lemma 5.5. However let us notice that all functions above are analytic as functions of complex parameter l . We understand uS and vS as the analytic k Ψk Ψk extension of the definitions in Section 5.2. Using assumption (A7) and the Fubini theorem we get u2 B 2 log Ak c vΨk (x, u2 s,s) dxds. ≤ Rd − Z0 Z We are going to estimate the right-hand side. Let us notice that by Lemma 5.8 it is sufficient to u2 S 2 B prove the estimation of Rd v −1 (x, u2 s,s) dxds. We denote vk(x,s) := v −1 (x, u2 s,s). 0 C Ψk − C Ψk − Equation (29) writes as R R t v (x,t)= Q l ϕ( )1 (s)+ v2( ,s) (x)ds. k Tt−s k · [0,δk] k · Z0 We are going to estimate v ( ,t) . Using the representation (32) (we use analytic extensions again k k · k2 and skip Vq to make calculations trackable) we have ∞ ∗n vk(x,t)= Fk (x,t), Xk=0 31 where F ∗1(x,t) = t Q l ϕ( )1 (s) (x)ds. Let us recall Q′ from assumption (A6); we will k 0 Tt−s k · [0,δk ] prove that R ′ F ∗n( ,t) 2−ne−(Q t)/2. k k · k2 ≤ For n = 1 we have and t > δk using assumption (A6) and the generalised Minkowski inequality t ′ δk ∗1 Q Q 1 −Q (t−δk ) Q F ( ,t) lkϕ( ) [0,δ ](s) (x)ds lke ϕ(x)ds k k · k2 ≤ kTt−δk Tt−s · k k2 ≤ k Ts k2 ≤ Z0 Z0 ′ δk ′ δk ′ ′ l e−Q (t−δk) Qϕ(x) ds l e−Q (t−δk) ϕ ds = e−Q (t−δk ) ϕ 2−1e−(Q t)/2. k kTs k2 ≤ k k k2 k k2 ≤ Z0 Z0 We decrease ϕ is necessary - see Remark 5.2. For t < δ it is easy to check that F ( ,t) is k k k · k2 even smaller. Using Lemma 5.3 we have F ∗n C−n for any constant C (possibly decreasing ϕ k k k∞ ≤ once more). For n > 1 we estimate using the induction argument together with the (generalised) Minkowski inequality n−1 F ∗n( ,t) = F ∗j F ∗(n−j) ( ,t) k k · k2 k k ∗ k · k2 Xj=1 ⌈n/2⌉ t ⌈n/2⌉ t Q ∗j ∗(n−j) Q ∗j ∗(n−j) 2 t−s Fk ( ,s)Fk ( ,s) (x)ds 2 2 t−sFk ( ,s)ds 2 Fk ∞ ≤ k 0 T · · k ≤ k 0 T · k k k Xj=1 Z h i Xj=1 Z ⌈n/2⌉ t ⌈n/2⌉ t Q ∗j −(n−j) −(n−j) Q ∗j 2 t−s Fk ( ,s) ds 2C 2 C t−s Fk ( ,s) 2ds. ≤ k 0 T · k ≤ 0 kT · k Xj=1 Z h i Xj=1 Z h i We can now use assumption (A6), the induction hypothesis and choose suitable C > 0 to get ⌈n/2⌉ t ∗n −(n−j) −Q′(t−s) −j −(Q′s)/2 Fk( ,t) 2 2 C e 2 e ds k · k ≤ 0 Xj=1 Z ⌈n/2⌉ ′ ′ e−(Q t)/2 Q′C−(n−j)2−j 2−ne−(Q t)/2. ≤ ≤ Xj=1 ′ It is now obvious that v( ,t) = ∞ F ( ,t)∗n e−(Q t)/2 and the estimate does not depend k · k2 k n=1 · k2 ≤ on k, so neither does A . The Chebyshev inequality yields k P Ak k P (x(u2) x(u1) λ) C2 exp 2 λ . − ≥ ≤ exp(lkλ) ≤ − It is easy to derive an analogous estimate for P (x(u ) x(u ) λ). Employing these to III(T ) we 1 − 2 ≥ get ∞ ∞ III(T ) 2C 2k exp 2k−kT T (1+α)/2θkη = 2C exp k ln 2 (2θ)k−kT θkT T (1+α)/2η ≤ 2 − θ 2 − θ kX=kT kX=kT 32 k −ǫ1 The choice of θ yields θ T = T . For T ’s large enough (depending on ηθ) and certain C > 0 we have k ln 2 (2θ)k−kT T −ǫ1+(1+α)/2η kCT αη , hence − θ ≤− θ ∞ α α exp ( kT CT ηθ) III(T ) exp ( kCT ηθ) − α . ≤ − ≤ 1 exp ( kT CT ηθ) kX=kT − − It is now straightforward to check that lim lim sup T −α log III(T )= . η→+∞ T →+∞ −∞ 1−α ′ Estimation of I(T ) We will use Lemma 5.11 with ǫ := 4 and θ = lT given by η l := θ (2θ)kT (α−1)/2, T 2c where c is the same as in the lemma. It is straightforward to check that for T ’s large enough (depending on η ,θ and c) and for k < K we have l T (α−1)/4. Consequently, by Lemma 5.11 θ T T ≤ and the Chebyshev inequality we have η2 P x (i2−k) x ((i 1)2−k) θkη exp l2 cT 2−k l T (1+α)/2θkη = exp θ (2θ2)kT α . T − T − ≥ θ ≤ T − T θ − 4c An analogous inequality for P x ((i 1)2−k) x (i2−k) θkη also holds. Consequently T − − T ≥ θ K K T η2 T η2 I(T ) 2 2k exp θ (2θ2)kT α = 2 exp k ln 2 θ (2θ2)kT α . ≤ − 4c − 4c Xk=1 Xk=1 2 k Recalling (63) it is easy to check that θ> 1/√2 hence (2θ ) > c1k. Finally we get η2 KT θ α η2 exp ln 2 c1 4c T I(T ) exp k ln 2 c k θ T α − 1 η2 ≤ − 4c ≤ 1 exp ln 2 c θ T α Xk=1 − − 1 4c It is now straightforward to check that lim lim sup T −α log I(T )= . η→+∞ T →+∞ −∞ Estimation of II(T ) By (63) one checks that (7α + 1)/8 ǫ ǫ α. We will use Lemma 5.11 − 2 − 1 ≥ with some ǫ< (3 3α)/8 ǫ and θ′ = l given by − − 2 T η l = θ T (3α−3)/8−ǫ2 , T 2c where c is the same as in the lemma. For large T (depending on η and c) we have l T −ǫ. By θ T ≤ Lemma 5.11 and the Chebyshev inequality we get P x (i2−k) x ((i 1)2−k) θkη exp l2 cT 2−k l T (1+α)/2θkη = T − T − ≥ θ ≤ T − T θ η2 η2 η2 η2 exp θ T (3α+1)/4−2ǫ2 2−k θ T (7α+1)/8−ǫ2 θk exp θ T (7α+1)/8−2ǫ2 θ T (7α+1)/8−ǫ2−ǫ1 , 4c − 2c ≤ 4c − 2c 33 where the last estimate follows by θkT T −ǫ1 and 2−k 2−KT T (α−1)/8. An analogous estimate ≥ ≤ ≤ holds also for P x ((i 1)2−k) X (i2−k) θkη . Putting these together we write T − − T ≥ θ η2 η2 II(T ) C T exp θ T (7α+1)/8−2ǫ2 θ T (7α+1)/8−ǫ2−ǫ1 . ≤ 1 4c − 2c It is now straightforward to check that lim lim sup T −α log I(T )= . η→+∞ T →+∞ −∞ This end the proof of (60). Now we turn to (61). For any τ ST (x ) we have ∈ T sup xT ((τ + t) 1) xT (τ) w(xT , δ), t≤δ | ∧ − |≤ where w is the modulus of continuity (9). Using this fact together with [2, Theorem 7.4] we get −1 sup P sup xT ((τ + t) 1) xT (τ) λ sup δ P sup xT (s + t) xT (s) λ/3 . τ∈ST (xT ) t∈[0,δ] | ∧ − |≥ ! ≤ s∈[0,1−δ] t∈[0,δ] | − |≥ ! To prove (61) it is enough to prove that for any λ> 0 there is −α lim lim sup T sup log P sup xT (s + t) xT (s) λ = . (64) δ→0 T →+∞ s∈[0,1−δ] t∈[0,δ] | − |≥ ! −∞ The following proof mimics the proof of (60) but is slightly more technically elaborated. By Lemma 5.12 we have ∞ P P δ,s P sup xT (t + s) xT (s) λ 2 Lk (xT ) λ/2 + ( xT (s + δ) xT (s) λ/2) t∈[0,δ] | − |≥ ! ≤ ≥ ! | − |≥ Xk=1 δ,s where Lk is defined analogously to Lk but on the interval [s,s + δ]. Finally, (64) will be shown once we have proved that −α lim lim sup T sup log P ( xT (s + δ) xT (s) λ)= , δ→0 T →+∞ s∈[0,1−δ] | − |≥ −∞ ∞ −α P δ,s lim lim sup T sup log Lk (xT ) λ = . δ→0 T →+∞ s∈[0,1−δ] ≥ ! −∞ Xk=1 ′ λ (α−1)/2 The first convergence can be obtained by application of Lemma 5.11 with θ = 2cδ T and the Chebyshev inequality, namely P ( x (s + δ) x (s) λ) exp λ T α . To prove the second we | T − T |≥ ≤ − 4cδ recall (63) and write ∞ ∞ P δ,s P δ,s k Lk (xT ) η Lk (xT ) cθθ η ≥ ! ≤ ≥ k=1 k=1 X X ∞ k −k −k k 2 max P xT (i2 ) xT ((i 1)2 ) cθθ η ≤ i∈ 1,2,...,2k | − − |≥ Xk=1 { } where λ = λ(1 θ)/θ. We denote also K := 1−α log T and k := log(δT ) then θ − T 8 log(2θ) T ∞ KT kT ∞ P δ,s Lk (xT ) η . . . + . . . + . . . =: I(T )+ II(T )+ III(T ). ≥ ! ≤ Xk=1 Xk=1 kX=KT kX=kT 34 Estimation of III(T ) Following the same lines of reasoning as in the previous section we arrive at ∞ III(T ) C 2k exp 2k−kT δ−1T (1+α)/2θkλ ≤ − θ kX=kT ∞ C exp k ln 2 (2θ)k−kT θkT δ−1T (1+α)/2λ . ≤ − θ kX=kT k −ǫ1 The choice of θ implies θ T = (T δ) . For T ’s large enough (depending on λθ and δ) and certain C > 0 we have k ln 2 (2θ)k−kT θkT δ−1T (1+α)/2λ kCδ−1−ǫ1 T αλ , hence − θ ≤− θ ∞ −1−ǫ1 α exp kT Cδ T λθ III(T ) exp kCδ−1−ǫ1 T αλ − . θ −1−ǫ1 α ≤ − ≤ 1 exp ( kT Cδ T λθ) k=kT − − X −α It is now straightforward to check that for any λ> 0 we have limδ→0 lim supT →+∞ T log III(T )= . −∞ 1−α ′ Estimation of I(T ) We use Lemma 5.11 with ǫ = 4 and θ = lT (c is given by the lemma) λ l = θ (2θ)kT (α−1)/2. T 2δc It is straightforward to check that for T ’s large enough (depending on λθ,δ,θ and c) for any k < KT we have l T (α−1)/4. Consequently, by Lemma 5.11 and the Chebyshev inequality we have T ≤ P x (s + iδ2−k) x (s + (i 1)δ2−k) θkλ T − T − ≥ θ λ2 exp l2 cδT 2−k l T (1+α)/2θkλ = exp θ (2θ2)kT α . ≤ T − T θ −4δc An analogous inequality holds also for P x (s + (i 1)δ2−k) x (s + iδ2−k) θkλ . Conse- T − − T ≥ θ quently K K T λ2 T λ2 I(T ) 2 2k exp θ (2θ2)kT α = exp k ln 2 θ (2θ2)kT α . ≤ −4δc − 4δc Xk=1 Xk=1 2 k We know that θ > 1/√2 hence (2θ ) > c1k for certain c1 > 0. Finally we get λ2 KT θ α λ2 exp ln 2 c1 4δc T I(T ) exp k ln 2 c k θ T α − . 1 λ2 ≤ − 4δc ≤ 1 exp ln 2 c θ T α Xk=1 − − 1 4δc −α It is now straightforward to check that for any λ> 0 we have limδ→0 lim supT →+∞ T log I(T ) = . −∞ 35 Estimation of II(T ) We apply Lemma 5.11 with some ǫ< (3 3α)/8 ǫ and θ′ = l given by − − 2 T λ l = θ T (3α−3)/8−ǫ2 , T 2δc where c is given by the lemma. For T ’s large enough (depending on λ ,θ,δ and c) l T −ǫ hence θ T ≤ Lemma 5.11 and the Chebyshev inequality yield P x (s + iδ2−k) x (s + (i 1)δ2−k) θkλ exp l2 cT δ2−k l T (1+α)/2θkλ = T − T − ≥ θ ≤ T − T θ λ2 λ2 exp θ T (3α+1)/4−2ǫ2 2−k θ T (7α+1)/8−ǫ2 θk 4δc − 2δc 2 2 λ 7α+1 λ exp θ T 8 −2ǫ2 θ T (7α+1)/8−ǫ2−ǫ1 , ≤ 4δc − 2δ1+ǫ1 c where in the last estimation we used the fact that θkT (δT )−ǫ1 and 2−k 2−KT T (α−1)/8. An ≈ ≤ ≤ analogous estimate holds also for P x (s + (i 1)δ2−k) x (s + iδ2−k) θkλ . Hence T − − T ≥ θ λ2 λ2c II(T ) CδT exp θ T (7α+1)/8−2ǫ2 θ θ T (7α+1)/8−ǫ2−ǫ1 ≤ 4δc − 2δ1+ǫ1 c −α It is now straightforward to check that for any λ> 0 we have limδ→0 lim supT →+∞ T log II(T )= . −∞ References [1] P. Billingsley. Convergence of Probability Measures. John Wiley, New York, 1968. [2] P. Billingsley. Convergence of Probability Measures. John Wiley, New York, 1999. [3] M. Birkner and I. Zähle. A functional CLT for the occupation time of state-dependent branching random walk. Ann. 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