Weak convergence of two-time-scale stochastic delay differential equations driven by L\'evy processes

We focus on the asymptotic behavior of two-time-scale delay systems driven by Levy processes. There are several difficulties in these problems, such as ${\left( {{x^\varepsilon }\left( t \right),{\xi ^\varepsilon }\left( t \right)} \right)^\prime }$ being not Markov, the state-dependence of the noises and the dispose of the infinitesimal operator. To overcome these difficulties, we go the following steps. Firstly, we investigate stochastic differential delay systems whose "slow component" has a memory, and prove the existence and uniqueness of the solution to the systems. Then under dissipative conditions, we exhibit exponential ergodicity of the "fast component" with the aid of the B-D-G inequality and some other inequalities. Based on the exponential ergodicity of the "fast component" and tightness which we can obtain by virtue of the Ascoli-Arzela theorem, weak convergence is studied by using martingale methods, truncation technique and the property of the Levy jump measure. Finally we extend some acquired results for a "fast component" with memory.

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