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<p>We consider recursion equations of the form <i>u</i><sub><i>n</i>+1</sub>(<i>x</i>) = <i>Q</i>[<i>u</i><sub><i>n</i></sub>](<i>x</i>), <i>n</i> ≥ 1, <i>x</i> ∈ ℝ, with a non-local operator <i>Q</i>[<i>u</i>] = <i>g</i>(<i>u * q</i>), where <i>g</i> is a polynomial, satisfying <i>g</i>(0) = 0, <i>g</i>(1) = 1, <i>g</i>((0, 1)) ⊆ (0, 1), and <i>q</i> 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 <i>g</i>, a probabilistic model based on branching random walk can be given to the solution of the recursion, while in case <i>g</i> 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 <i>n</i> centering.</p>

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Voting models and tightness for a family of recursion equations

  • 期刊论文

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  • 发表日期：

    2025年11月12日

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  • Xaver Kriechbaum,
  • Lenya Ryzhik,
  • Ofer Zeitouni

摘要

We consider recursion equations of the form un+1(x) = Q[un](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.