Characterize the limiting consensus point in finite-N anisotropic CBO

Characterize the asymptotic limit within the consensus manifold \(\mathcal{G}^{\star} = \{ z \in \mathbb{R}^{N\times D} : z^{1,d} = \cdots = z^{N,d},\ d=1,\ldots,D\} \) for the finite-agent anisotropic Consensus-Based Optimization dynamics governed by the stochastic differential equation \(dZ_t = -\lambda L Z_t\, dt + \sigma L Z_t \circ dW_t\), i.e., determine which specific element of \(\mathcal{G}^{\star}\) the trajectories converge to as \(t \to \infty\) under finite \(N\) and finite \(\alpha\).

Background

The paper proves almost sure and mean square global exponential convergence of the finite-N anisotropic CBO dynamics toward the consensus manifold $\mathcal{G}^\star$. However, the analysis is restricted to the dynamics orthogonal to $\mathcal{G}^\star$ and does not identify the particular consensus point reached.

In contrast, prior mean-field analyses have established convergence to the global minimizer under additional assumptions and asymptotic regimes. Bridging this gap at finite $N$ and finite $\alpha$ requires a precise characterization of the limiting point within $\mathcal{G}^\star$ for the finite-agent dynamics studied here.