Asymptotic Behavior of Solutions to the ε-Perturbed Fokker–Planck–Kolmogorov Equation for Long-Time Ship Stability Analysis

This paper investigates the long-time behaviour of solutions to the $$\varepsilon $$ -perturbed Fokker–Planck–Kolmogorov (FPK) equation in the context of ship roll dynamics under stochastic sea excitation. Starting from a stochastic model of a ship navigating in a transverse sea, we derive the associated FPK equation governing the time evolution of the joint probability density function (PDF) of the roll angle, angular velocity, and wave-induced excitation. The theoretical analysis focuses on the asymptotic behaviour of the PDF as time tends to infinity, demonstrating that the probability of capsize stabilises and eventually becomes time-invariant. This result offers new insights into the probabilistic stability of ships subjected to random sea conditions. To support these theoretical findings, we implement a finite-difference numerical scheme that accurately captures the transient dynamics and confirms convergence towards the steady-state distribution. The simulations validate the analytical predictions and underline the robustness of the proposed approach for long-term stability assessments in marine engineering applications.