In this article, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of radicals (stubborn individuals). The distribution of radical opinions serves as an infinite-dimensional exogenous input to the FP equation, visibly influencing the steady opinion profile. We establish mathematical properties of the FP equation. In particular, we, first, show the well-posedness of the dynamic equation, second, provide existence result accompanied by a quantitative global estimate for the corresponding stationary solution, and, third, establish an explicit lower bound on the noise level that guarantees exponential convergence of the dynamics to stationary state. Combining the results in second and third readily yields the input-output stability of the system for sufficiently large noises. Next, using Fourier analysis, the structure of opinion clusters under the uniform initial distribution is examined. The results of analysis are validated through several numerical simulations of the continuum-agent model (partial differential equation) and the corresponding discrete-agent model (interacting stochastic differential equations) for a particular distribution of radicals.

中文翻译：

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具有分布式激进观点的宏观噪声有界置信模型


在本文中，我们研究了非线性福克-普朗克（FP）方程，该方程作为有界置信意见动态的平均场（宏观）近似而出现，其中意见受到环境噪音和激进分子（顽固个体）意见的影响。激进意见的分布作为 FP 方程的无限维外生输入，明显影响稳定的意见分布。我们建立了 FP 方程的数学性质。特别是，我们首先证明了动态方程的适定性，其次提供了存在性结果，并附有相应稳态解的定量全局估计，第三，建立了噪声水平的显式下界，以保证指数动力学收敛到稳态。结合第二个和第三个的结果很容易得出系统在足够大的噪声下的输入输出稳定性。接下来，使用傅里叶分析，检查均匀初始分布下的意见聚类结构。通过对特定自由基分布的连续主体模型（偏微分方程）和相应的离散主体模型（相互作用随机微分方程）进行多次数值模拟来验证分析结果。