Weak Convergence to Equilibrium for Statistical Ensembles in Discrete Integrable Hamiltonian Systems with Markov Perturbations

We study the weak convergence of statistical ensembles for discrete integrable Hamiltonian systems under Markov perturbations. By introducing twisted Markov transition operators associated with angular Fourier modes, we establish the basic structural properties of the evolution, including preservation of positivity and mass, $$L^1$$-nonexpansiveness, and invariance of the action marginal. Under a natural decay assumption on all nonzero twisted modes, we prove convergence of bounded continuous observables to their angular averages and derive weak convergence of the induced probability measures toward the angularly averaged equilibrium state, without imposing any additional nonresonance condition on the frequency map. We further obtain joint weak convergence on the extended space $$\Omega \times \mathbb {T}^N\times E$$ when the driving Markov chain converges to stationarity. In the finite-state Markov switching case, the abstract decay condition reduces to a directly verifiable spectral-radius criterion, yielding exponential damping of nonzero angular Fourier modes. Numerical experiments for a two-state switching model illustrate the theory.