2023/12/28 by Alexander Dunlap, Dunlap, Alexander, Lenya Ryzhik +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35K57 #60J70 #60J85 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2312.17139
openalex publication_date 2023/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a voting model on a branching Brownian motion process on ℝ in which the diffusivity of each child particle is increased from that of the parent by a factor of γ>1. The probability distribution of the overall vote is given in terms of the solution to a nonlocal nonlinear PDE. We exhibit conditions on the nonlinearity such that the long-time behavior of the distribution undergoes a phase transition in γ. If γ is sufficiently large, then the long-time distribution converges to uniform. If γ is close enough to 1, then the long-time distribution depends in a nontrivial way on the location of the initial particle. The limiting dependence is given by a steady-state solution to the nonlocal PDE. Our study gives a probabilistic interpretation of a class of semilinear nonlocal PDEs. Interestingly, while the PDE are nonlocal, the underlying random process does not require any non-local interactions.