Stochastic Analysis of Competing Opinion Dynamics with Trend-Following, Opposition, and Indifference

Abstract. We analyze a stochastic model for the diffusion of two competing opinions in a population composed of trend-followers, opposers, and indifferent individuals. The model introduces a reinforcement mechanism modulated by parameters representing these behavioral types, resulting in a rich asymptotic structure. We establish almost sure laws of large numbers, functional and almost sure central limit theorems, and laws of the iterated logarithm, characterizing stabilizing, critical, and superdiffusive regimes depending on the balance between trend-following and opposition. In particular, we derive explicit finite-time moments and limiting distributions via martingale techniques and identify the critical transition at [Formula: see text] where qualitative changes in fluctuation behavior occur. These results extend classical reinforced stochastic processes and provide analytical tools for modeling opinion dynamics, social influence, and information spread in complex systems.