Convergence toward equilibrium of the first-order consensus model with random batch interactions

Abstract We study convergence analysis of the first-order stochastic consensus model with random batch interactions. The proposed model can be obtained via random batch method (RBM) from the first-order nonlinear consensus model. This model has two competing mechanisms, namely intrinsic free flow and nonlinear consensus interaction terms. From the competition between the two mechanisms, the original (full batch) model can admit relative equilibria and relaxation of the dynamics to the relative equilibrium in a large coupling regime. In authors' earlier work, we have studied the RBM approximation and its uniform error analysis. In this paper, we present two convergence analysis of RBM solutions toward the relative equilibrium. More precisely, we show that the variances of displacement processes between the full batch and random batch solutions tend to zero exponentially fast, as time goes to infinity. Second, we also show that, almost surely, the diameter process of displacement tends to zero exponentially.