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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](x)=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. These are discrete analogues of KPO type equations with general (polynomial) nonlinear ties. 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.
Joint work with Xaver Kriechbaum and Lenya Ryzhik
