2023/12/06 by Xaver Kriechbaum, Kriechbaum, Xaver, Lenya Ryzhik +3
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2312.03944
openalex publication_date 2023/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider recursion equations of the form un+1(x)=Q[un](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. 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.