Stabilisation of stochastic single-file dynamics using port-Hamiltonian systems

The paper establishes (i) weak convergence of Z̃(t) to a Gaussian with mean [L/N·1; 0] for γ>0 (Proposition 9) and (ii) the stationary covariance by solving the Lyapunov equation BΣ+ΣB^⊤=−GG^⊤ together with block relations, yielding V2=0, V3 given by the circulant formula (21), and V1=α^{−2}(V3−σ^2/(2γN)·11^⊤) . It also proves the large-N limit v_j→σ^2 a^j/(2F) using a contour-integral argument, matching the model’s closed form . The candidate solution reaches the same conclusions via DFT-based OU mode decomposition and a standard trigonometric integral identity; no substantive conflicts were found.