Relaxation Kernel and Global Convergence of the Blahut-Arimoto Dynamics

Motivated by a continuous-time formulation of the Blahut-Arimoto scheme, we study a nonlinear dissipative flow on the probability simplex generated by a Gibbs-type self-consistent evolution. We establish an exact dissipation identity showing that the free energy decreases according to a weighted $\chi^2$-type fluctuation, yielding an explicit entropy-production formula for the nonlinear dynamics. Linearization around a nondegenerate stationary state reveals that the same fluctuation is governed by a symmetric positive semidefinite relaxation kernel built from equilibrium conditional covariances. This kernel determines both the local linearized flow and the quadratic expansion of the free energy. We further show that it coincides with the Fisher-Rao Hessian of the free energy at equilibrium,so that its spectral gap characterizes the local relaxation rate. Combining the exact dissipation identity with local spectral contraction, we obtain convergence of the flow toward equilibrium within the connected component of a nondegenerate stationary state. In the Gaussian quadratic case, the dynamics admits an explicit finite-dimensional reduction for which the relaxation kernel, spectral gap, and asymptotic relaxation law can be computed in closed form. These results identify a common structure linking entropy dissipation, local equilibrium geometry, and spectral relaxation in a class of nonlinear Gibbs-type probability flows.