Voting models and tightness for a family of recursion equations

Abstract We consider recursion equations of the form u n +1 ( x ) = Q [ u n ]( x ), n ≥ 1, x ∈ ℝ, with a non-local operator Q [ u ] = g ( u * q ), where g is a polynomial, satisfying g (0) = 0, g (1) = 1, g ((0, 1)) ⊆ (0, 1), and q is a (compactly supported) probability density with * denoting convolution. Motivated by a line of works for nonlinear PDEs initiated by Etheridge, Freeman and Penington (2017), we show that for general g , a probabilistic model based on branching random walk can be given to the solution of the recursion, while in case g is also strictly monotone, a probabilistic threshold-based model can be given. In the latter case, we provide a conditional tightness result. We analyze in detail the bistable case and prove for it convergence of the solution shifted around a linear in n centering.