Asymptotic Behaviour of Solutions to the Fokker-Planck Equation: Naval Dynamics Under Stochastic Influence

This study investigates the asymptotic dynamics of solutions to the Fokker-Planck-Kolmogorov (FPK) equation, with a specific focus on ship roll stability in dynamic sea conditions. Utilizing a fourth-order filter, we conduct a thorough analysis of the time evolution of the probability distributions for roll angles, roll speeds, and roll excitations. Our theoretical framework provides new insights into the long-term behavior of these systems, emphasizing the role of stochastic perturbations. Key findings reveal that the probability of capsizing remains constant over time, offering significant contributions to the stability assessment of maritime vessels under uncertain environmental conditions. This work paves the way for more robust models in maritime engineering and dynamic stability analysis.