Asymmetric cooperative motion in one dimension

Addario-Berry, Louigi; Beckman, Erin; Lin, Jessica

doi:10.1090/tran/8581

Asymmetric cooperative motion in one dimension
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by Louigi Addario-Berry, Erin Beckman and Jessica Lin PDF

Trans. Amer. Math. Soc. 375 (2022), 2883-2913

Abstract:

We prove distributional convergence for a family of random processes on $\mathbb {Z}$ which we call asymmetric cooperative motions. The model generalizes the “totally asymmetric $q$-lazy hipster random walk” introduced by Addario-Berry et al. [Probab. Theory Related Fields 178 (2020), pp. 437–473]. We present a novel approach based on connecting a temporal recurrence relation satisfied by the cumulative distribution functions of the process to the theory of finite difference schemes for Hamilton-Jacobi equations, building off of the convergence results of Crandall and Lions [Math. Comp. 43 (1984), pp. 1–19]. We also point out some surprising lattice effects that can persist in the distributional limit and propose several generalizations and directions for future research.

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