The paper under review builds a renewal structure for the front propagation process through a microscopic model of infection or epidemic reaction on the integer lattice. The initial condition is that at each integer there are a random number of particles whose distribution is Poisson with parameter \(\rho\), and the random numbers at different integers are independent. Particles at negative integers are red, particles at positive integers are blue. The front is the asymptotic behavior of the rightmost site \(r_t\) in the positive integers occupied by red particles at time \(t\).
The first main result (Theorem 1) states that there exists a nonrandom number \(0<\sigma_*^2<+\infty\) such that \(B_t^{\varepsilon} = {\varepsilon}^{1/2}(r_{{\varepsilon}^{-1}t}- {\varepsilon}^{-1}v_* t)\) converges to a Brownian motion with variance \(\sigma_*^2\) in law in the Skorokhod space, provided that the total jump rate \(D_R= D_B\) is the same for both red and blue particles (referring to the single-rate KS infection model). Then Theorem 2 and Theorem 3 show that the same results hold for the remanent KS infection model (\(0<D_B \leq D_R\)). The proofs of those results are based on a renewal structure originally defined by F. Comets et al. [Trans. Am. Math. Soc. 361, No. 11, 6165–6189 (2009; Zbl 1177.82081)], and the core lies in finding an appropriate definition for the renewal structure and then proving the required tail estimates.
The idea is to find random times \(k_n\) with: (1) the history of the front after \(k_n\) does not depend on the future trajectories of particles located below \(r_{k_n}\), and (2) the distribution of particles located above \(r_{k_n}\) is fixed up to transition. This will shape a renewal structure. Then tail estimates of the random variables \(k_1\), \(r_{k_1}\), \(k_{n+1}-k_n\), \(r_{k_{n+1}}-r_{k_n}\) for \(n\geq 1\) are found. Mainly adapting techniques from [loc. cit.], the authors achieve (2) by extending the trajectories of random walks infinitely far into the past. The invariance properties of the Poisson distribution of particles play a key role in the arguments.
Section 2 sets up the single-rate process with reference spaces and reference probability, and introduces the infection dynamics; Section 3 gives a detailed construction of the random time \(k_n\) as a stopping time, and Theorem 4 characterizes when the process under the reference probability is indeed a renewal structure. The authors first state Proposition 3 and Proposition 4 and deduce Theorem 4 from them. The rest of Section 3 is devoted to the proof of Proposition 3.
Section 4 is devoted to the proof of Proposition 4 by tail estimates. One has to construct another sequence of stopping times \(S_n\leq D_n\leq S_{n+1} \leq \cdots\) for the \(n\)-th attempt at obtaining an \(\alpha\)-separation time. In the case that this attempt fails, \(D_n\) is the time at which the failure is detected, in order to control the tail of \(k_1\) and \(r_{k_1}\). The authors provide the precise definition of the random variables \(S_n\) and \(D_n\) in Subsection 4.2, and then list all parameters, assumptions and conventions in Subsection 4.3. Elementary results on the hitting times and hitting probability of a straight line by a system of independent random walks are proved in Subsection 4.4. Subsections 4.5 and 4.6 analyze an extension of the quantitative ballisticity (among others). The final ingredients to finish the proof of Proposition 4 are given in subsection 4.7.
Section 5 briefly explains how to extend Theorem 1 to Theorems 2 and 3. Note that it is very different from the case \(D_R<D_B\). The only available results for a model of this kind were obtained in [H. Kesten and V. Sidoravicius, Ann. Probab. 36, No. 5, 1838–1879 (2008; Zbl 1154.60075)], where it was conjectured that a positive asymptotic velocity is obtained for sufficiently large \(\rho\). It would be certainly interesting to see if the main results in this paper can also be extended to higher dimensional models.