Convergence analysis of probability density function for second-order stochastic systems with Markovian switching and Lévy noise under the local Lipschitz conditions

Based on the multidimensional Fokker–Planck equation with regime-switching and nonlocal diffusion and the new established Feynman–Kac formula, we mainly investigate the dynamic evolution of the probability density function corresponding to a class of second-order stochastic systems with Lévy noise and Markovian switching by varying the system parameter μ and the noise intensity ɛ in (1.1). First, after deriving the multidimensional Fokker–Planck equation associated with the second-order stochastic differential system (1.1), we prove the existence and uniqueness of the solutions for the regime-switching and nonlocal Fokker–Planck equation by utilizing the techniques from stochastic analysis. Then, from the Feynman–Kac formula and the convergence analysis of the second-order stochastic differential system (4.1) under the local Lipschitz conditions, the continuous dependence (w.r.t. ɛ ∈ (0,1]) and the limit behaviour (i.e., ɛ → 0) or the continuous dependence [w.r.t. μ ∈ (0, ∞)] of solutions for the Fokker–Planck equation are analysed for the fixed case μ = 1 or ɛ = 1. Finally, as a significant class of second-order nonlinear differential equations which can be used to model physical systems with nonlinear damping, the stochastic Liénard equation driven by Lévy noise and Markovian switching is adopted to reveal the impacts of ɛ and μ on the probability density function by Monte Carlo simulations. In particular, some surprising research findings are presented in Examples 7.1–7.6, especially that for the fixed case ɛ = 1 while varying μ from 10 to 0.003, Fig. 2 shows a non-monotonic relationship between μ and kurtosis of the probability density function for the stochastic Liénard system, which is completely different with the fixed case μ = 1, the probability density function exhibits progressively sharper peaks and narrower distributions with the decrease of ɛ, and then reaches a steady state for sufficiently small ɛ as shown in Fig. 1.