A Note on the Diffusive Scaling Limit for a Class of Linear Systems

We consider a class of continuous-time stochastic growth models on $d$-dimensional lattice with non-negative real numbers as possible values per site. We remark that the diffusive scaling limit proven in our previous work [Nagahata, Y., Yoshida, N.: Central Limit Theorem for a Class of Linear Systems, Electron. J. Probab. Vol. 14, No. 34, 960--977. (2009)] can be extended to wider class of models so that it covers the cases of potlatch/smoothing processes.


The model
We go directly into the formal definition of the model, referring the reader to [NY09a,NY09b] for relevant backgrounds. The class of growth models considered here is a reasonably ample subclass of the one considered in [Lig85, Chapter IX] as "linear systems". We introduce a random vector K = (K x ) x∈Z d such that 0 ≤ K x ≤ b K 1 {|x|≤r K } a.s. for some constants b K , r K ∈ [0, ∞), (1.1) the set {x ∈ Z d ; P [K x ] = 0} contains a linear basis of R d . (1. 2) The first condition (1.1) amounts to the standard boundedness and the finite range assumptions for the transition rate of interacting particle systems. The second condition (1.2) makes the model "truly d-dimensional". Let τ z,i , (z ∈ Z d , i ∈ N * ) be i.i.d. mean-one exponential random variables and T z,i = τ z,1 + ... + τ z,i . Let also K z,i = (K z,i x ) x∈Z d (z ∈ Z d , i ∈ N * ) be i.i.d. random vectors with the same distributions as K, independent of {τ z,i } z∈Z d ,i∈N * . We suppose that the process (η t ) starts from a deterministic configuration η 0 = (η 0,x ) x∈Z d ∈ N Z d with |η 0 | < ∞. At time t = T z,i , η t− is replaced by η t , where (1.3) We also consider the dual process ζ t ∈ [0, ∞) Z d , t ≥ 0 which evolves in the same way as (η t ) t≥0 except that (1.3) is replaced by its transpose: Here are some typical examples which fall into the above set-up: • The binary contact path process (BCPP): The binary contact path process (BCPP), originally introduced by D. Griffeath [Gri83] is a special case the model, where K = (δ x,0 + δ x,e ) x∈Z d with probability λ 2dλ+1 , for each 2d neighbor e of 0 0 with probability 1 2dλ+1 . (1.5) The process is interpreted as the spread of an infection, with η t,x infected individuals at time t at the site x. The first line of (1.5) says that, with probability λ 2dλ+1 for each |e| = 1, all the infected individuals at site x − e are duplicated and added to those on the site x. On the other hand, the second line of (1.5) says that, all the infected individuals at a site become healthy with probability 1 2dλ+1 . A motivation to study the BCPP comes from the fact that the projected process (η t,x ∧ 1) x∈Z d , t ≥ 0 is the basic contact process [Gri83].
• The potlatch/smoothing processes: The potlatch process discussed in e.g. [HL81] and [Lig85,Chapter IX] is also a special case of the above set-up, in which (1.6) Here, k = (k x ) x∈Z d ∈ [0, ∞) Z d is a non-random vector and W is a non-negative, bounded, mean-one random variable such that P (W = 1) < 1 (so that the notation k here is consistent with the definition (1.7) below). The smoothing process is the dual process of the potlatch process. The potlatch/smoothing processes were first introduced in [Spi81] for the case W ≡ 1 and discussed further in [LS81]. It was in [HL81] where case with W ≡ 1 was introduced and discussed. Note that we do not assume that k x is a transition probability of an irreducible random walk, unlike in the literatures mentioned above.
We now recall the following facts from [Lig85, page 433, Theorems 2.2 and 2.3]. Let F t be the σ-field generated by η s , s ≤ t. Let (η x t ) t≥0 be the process (η t ) t≥0 starting from one particle at the site x: η x 0 = δ x . Similarly, let (ζ x t ) t≥0 be the dual process starting from one particle at the site x: ζ x 0 = δ x .
c) The above a)-b), with η t replaced by ζ t are true for the dual process.

Results
We are now in position to state our main result in this article (Theorem 1.2.1). It extends our previous result [NY09a, Theorem 1.2.1] to wider class of models so that it covers the cases of potlatch/smoothing processes, cf. Remarks 1)-2) after Theorem 1.2.1. We first introduce some more notation. For η, ζ ∈ R Z d , the inner product and the discrete convolution are defined respectively by provided the summations converge. We define for x, y ∈ Z d , If we simply write β in the sequel, it stands for the function x → β x . Note then that (1.13) We also introduce: where ((S t ) t≥0 , P x S ) is the continuous-time random walk on Z d starting from x ∈ Z d , with the generator Theorem 1.2.1 Suppose d ≥ 3. Then, the following conditions are equivalent: The main point of Theorem 1.2.1 is that a) implies d) and d'), while the equivalences between the other conditions are byproducts.

Remarks
For example, BCPP satisfies (1.19), while the potlatch/smoothing processes do not.
2) Let π d be the return probability for the simple random walk on Z d . We then have that 2dλ + 1 δ x,y , and G S (0) = 2dλ + 1 2dλ To see (1.20) for the potlatch/smoothing processes, we note that 1 2 (k +ǩ) * G S = |k|G S − δ 0 , withǩ x = k −x and that from which (1.20) for the potlatch/smoothing processes follows.
3) It will be seen from the proof that the inequalities in (1.16) and (1.18) can be replaced by the equality, keeping the other statement of Theorem 1.2.1.
As an immediate consequence of Theorem 1.2.1, we have the following Proof: The case of (η · ) follows from Theorem 1.2.1d). Note also that if P (|η ∞ | > 0) > 0, then, up to a null set, which follows from [NY09b, Lemma 2.1.2]. The proof for the case of (ζ · ) is the same. We first show the Feynman-Kac formula for two-point function, which is the basis of the proof of Theorem 1.2.1. To state it, we introduce Markov chains (X, X) and (Y, Y ) which are also exploited in [NY09a]. Let (X, ) be the continuous-time Markov chains on Z d × Z d starting from (x, x), with the generators It is useful to note that y, y L X, X (x, x, y, y) = 2(|k| − 1) + β x− x , (2.3) y, y L Y, Y (x, x, y, y) = 2(|k| − 1) + β, 1 δ x, x . (2.4) Recall also the notation (η x t ) t≥0 and (ζ x t ) t≥0 introduced before Lemma 1.1.1.
Proof: By the time-reversal argument as in [Lig85, Theorem 1.25], we see that (η x t,y , η x t, y ) and (ζ y t,x , ζ y t, x ) have the same law. This implies the first equality. In [NY09a, Lemma 2.1.1], we showed the second equality, using (2.4). Finally, we see from (2.2) -(2.4) that the operators: are transpose to each other, and hence are the semi-groups generated by the above operators. This proves the last equality of the lemma. 2 ) are Markov chains with the generators: (2.5) respectively (cf. (1.15)). Moreover, these Markov chains are transient for d ≥ 3.
Proof: Let (Z, Z) = (X, X) or (Y, Y ). Since (Z, Z) is shift-invariant, in the sense that Markov chain. Moreover, the jump rates L Z− Z (x, y), x = y are computed as follows: These prove (2.5). By (1.2), the random walk S · is transient for d ≥ 3. Thus, Z − Z is transient d ≥ 3, since L Z− Z (x, ·) = 2L S (x, ·) except for finitely many x.
In particular it solves (1.16) with equality. b) ⇒ c): By Lemma 2.1.1, we have that where e Y, Y ,t = exp β, 1 t 0 δ Ys, Ys ds . By Lemma 2.1.2, (1.16) reads: and thus, Since h takes its values in [1, sup h] with sup h < ∞, we have By this and 1), we obtain that On the other hand, we have by 1) that for any x, x ∈ Z d , where the last inequality comes from Schwarz inequality and the shift-invariance. Thus, (2.9) Therefore, we can define h : .
Plugging this into (2.8), we have a). 2 Remark: The function h defined by (2.10) solves (2.7) with equality, as can be seen by the way it is defined. This proves c) ⇒ b) directly. It is also easy to see from (2.8) that the function h defined by (2.10) and by (2.6) coincide.

The equivalence of c) and d)
To proceed from c) to the diffusive scaling limit d), we will use the following variant of [NY09a, Lemma 2.2.2]: Lemma 2.2.1 Let ((Z t ) t≥0 , P x ) be a continuous-time random walk on Z d starting from x, with the generator: where we assume that: On the other hand, let Z = (( Z t ) t≥0 , P x ) be the continuous-time Markov chain on Z d starting from x, with the generator: We assume that z ∈ Z d , D ⊂ Z d and a function v : Then, for where m = x∈Z d xL Z (0, x) and ν is the Gaussian measure with: Since c) implies that lim t→∞ |η t | = |η ∞ | in L 2 (P ), it is enough to prove that Note that by (2.9) and c), Since |η 0 | < ∞, it is enough to prove that for each x, To prove this, we apply Lemma 2.2.1 to the Markov chain Z t def.
= (Y t , Y t ) and the random walk (Z t ) on Z d × Z d with the generator: Moreover, by Lemma 2.1.2, 3) D is transient both for (Z t ) and for ( Z t ).
Finally, the Gaussian measure ν ⊗ ν is the limit law in the central limit theorem for the random walk (Z t ). Therefore, by 1)-3) and Lemma 2.2.1, Since h(0) = 2 and lim |x|→∞ h(x) = 2 − (β * G S )(0) ∈ (0, ∞), h is bounded away from both 0 and ∞. Therefore, a constant multiple of the above h satisfies the conditions in b'). b') ⇔ c'): This can be seen similarly as b) ⇔ c) (cf. the remark at the end of section 2.1). c') ⇒ a) : We first note that 1) lim |x|→∞ (β * G S )(x) = 0, since G S vanishes at infinity and β is of finite support. We then set: Then, there exists positive constant M such that 1 M ≤ h 0 ≤ M and By 1), h 2 is also bounded and This implies that there exists a constant c such that h 2 ≡ c on the subgroup H of Z d generated by the set {x ∈ Z d ; k x + k −x > 0}, i.e., By setting x = 0 in 2), we have On the other hand, we see from 1)-2) that These imply β, G S < 2. 2 Lemma 2.3.1 For d ≥ 3, Proof: The function β x can be either positive or negative. To control this inconvenience, we introduce: β x = y∈Z d P [K y K x+y ]. Since β x ≥ 0 and G S (x + y)G S (0) ≥ G S (x)G S (y) for all x, y ∈ Z d , we have 1) G S (0)(G S * β)(x) ≥ G S (x)(G S * β)(0).
Then, we have 0 < 1 2M ≤ h 0 ≤ M and This implies, as in the proof of b) ⇒ c) that sup t≥0 P x, x X, X e p X, X,t ≤ 2M 2 < ∞ for x, x ∈ Z d , which guarantees the uniform integrability of e X, X,t , t ≥ 0 required to apply Lemma 2.2.1. The rest of the proof is the same as in c) ⇒ d). 2