By Hermann Thorisson

It is a booklet on coupling, together with self-contained remedies of stationarity and regeneration. Coupling is the valuable subject within the first 1/2 the e-book, after which enters as a device within the latter part. the 10 chapters are grouped into 4 components.

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For the case where the average system is globally exponentially stable and all the other assumptions are valid globally, a global result is obtained for the original system. The chapter is organized as follows. 1 describes the investigated problem. 2 presents results for two cases: uniform strong ergodic perturbation process, and exponentially φ-mixing and exponentially ergodic perturbation process, respectively. In Sect. 3, we give the detailed proofs for the results in Sect. 2. In Sect. 4, we give three examples.

Then for 0 < ε ≤ ε2 1 and any t ≥ 0, Aˆ εδ V ε Xτε ε (t) , t ≤ 0. 29)). Suppose ε ∈ (0, ε2 ], r ∈ (0, δ), and X0ε = x is such that |x| ≤ r. For t ≥ 0, define two stopping times τrε and τrε (t) by τrε = inf s ≥ 0 : Xsε > r and τrε (t) = τrε ∧ t. 107) and τδε τrε (t) = τδε ∧ τrε (t) = τδε ∧ τrε ∧ t = τδε ∧ t ∧ τrε ∧ t = τδε (t) ∧ τrε (t) = τrε (t). 105), E V ε Xτε ε (t) , τrε (t) − V ε (x, 0) r = E V ε Xτε ε (τ ε (t)) , τrε (t) − V ε (x, 0) r δ = E E V ε Xτε ε (τ ε (t)) , τrε (t) − V ε (x, 0)|F0ε r δ =E E0ε V ε τrε (t) = E E0ε 0 τrε (t) =E Xτε ε (τ ε (t)) , τrε (t) δ r − V ε (x, 0) Aˆ εδ V ε Xτε ε (u) , u du δ Aˆ εδ V ε Xτε ε (u) , u du ≤ 0.

8 The vector field a(x, y) satisfies 1. a(x, y) and its first-order partial derivatives with respect to x are continuous and supy∈SY |a(0, y)| < ∞; 2. There is a constant k > 0 such that, for all x ∈ Rn and y ∈ SY , | ∂a(x,y) ∂x | ≤ k. 7. 31) ∈ Rn , P lim Xtε = 0 = 1. 1) has no equilibrium, we obtain the following result. 8. , lim sup P Xtε > r = 0. 6 are aimed at globally Lipschitz systems and can be viewed as an extension of the deterministic averaging principle [121] to the stochastic case. We present the results for the global case not only for the sake of completeness but also because of the novelty relative to [21]: (i) ergodic Markov process on some compact space is replaced by an exponential φ-mixing and exponentially ergodic process; (ii) for the case without equilibrium condition the weak convergence is considered in [21], while here we obtain the result on boundedness in probability.