A mesh-free generalized method for stochastic stability analysis: Mean exit time and escape probability of dynamical systems driven by Gaussian white noise and Lévy noise

Randomness and nonlinearity are integral properties of the real world, and their interaction gives rise to highly complex phenomena. In dynamical systems driven by random processes, mean first exit time and exit probability distribution are of great significance for comprehending evolutionary trends and evaluating structure lifespans. However, the traditional finite difference method encounters difficulties when dealing with non-rectangular irregular regions and high-dimensional problems, especially under the influence of Lévy noise. To overcome this challenge, we enhance the conventional mesh-free method, enabling it to adeptly handle arbitrary boundary shapes and non-local complexity for the exit problem. Additionally, error estimates are derived for this improved mesh-free method, a task that is challenging to achieve for the traditional method with random node distribution. Furthermore, we explore the noise coupling effect in systems driven by both Gaussian white noise and Lévy noise. Notably, our method provides valuable insights for solving boundary value problems with non-local complexity and arbitrary boundary shapes.