An asymptotic shape theorem for additive random linear growth models

In this paper, we define a class of additive random growth models whose growth is at least and at most linear and prove an asymptotic shape theorem for these models. This proof generalizes already known proofs for the classical contact process [T.E. Harris, Ann. Probab. 2 (1974) 969–988; R. Durrett and D. Griffeath, Z. Wahrsch. Verw. Gebiete 59 (1982) 535–552] or some of its variants (contact process on supercritical random environment [O. Garet and R. Marchand, Ann. Appl. Probab. 22 (2012) 1362–1410] or contact process with aging [A. Deshayes, ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 845–883]) and allows us to obtain conjectured asymptotic shape theorems for Richardson’s model with stirring and the contact process with stirring [R. Marchand et al., arXiv 2504.03627 (2025)].