An averaging principle for stochastic dynamical systems with Lévy noise

Stochastic stability of quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

“…It is seen from these equations that I 1 ; …; I n ; ψ 1 ; …; ψ α are slowly varying processes while θ 1 ; …; θ n À α are rapidly varying processes. Based on the theorems in [32,33], I r and ψ u converge in probability to an ðn þ αÞÀdimensional Markov process as ε-0, in time interval 0 r t r T, where T $ Oðε À 1 Þ. The same symbols I r (r ¼ 1; …; n) and ψ u (u ¼ 1; …; α) will be used to denote the new ðn þ αÞÀdimensional Markov process for simplicity.…”

L p $L^{p}$ ( p ≥ 2 ) $(p\geq2)$ -strong convergence in averaging principle for multivalued stochastic differential equation with non-Lipschitz coefficients

“…A series of useful theories and methods have been proposed to explore stochastic differential equations, such as invariant manifolds [1][2][3], averaging principle [3][4][5][6][7][8][9][10][11][12], homogenization principle, and so on. All these theories and methods develop to extract an effective dynamics from these stochastic differential equations, which is more effective for analysis and simulation.…”

“…Stoyanov and Bainov [11] investigated the averaging method for a class of stochastic differential equations with Poisson noise, proving that under some conditions the solutions of averaged systems converge to the solutions of the original systems in mean square and in probability. Xu, Duan, and Xu [4] established an averaging principle for stochastic differential equations with general non-Gaussian Lévy noise. Quite recently, L 2 (mean square) strong averaging principle for multivalued stochastic differential equations with Brownian motion was established by Xu and Liu [14].…”

Some results in stochastic functional integro-differential equations with infinite delays

“…Recently, few authors studied the averaging principle for stochastic differential equation under some restrictive conditions with non-Lipschitz conditions Mao et al (2015); Tan and Lei (2013). In Xu, Duan and Xu (2011), the author established averaging principle for dynamical systems with Lẽvy noise. Likewise stochastic differential equation, stochastic integro-differential equation also has wide applications in the areas of mechanics, electrical engineering and so on [see Liu and Ezzinbi (2003); Ding, Liang and Xiao (2012) and the references therein].…”