多尺度马尔可夫过程的随机平均及其在波动选择Wright-Fisher模型中的应用
Stochastic averaging for multiscale Markov processes with an application to a Wright-Fisher model with fluctuating selection
摘要：
设$Z = (Z_t)_{t\in[0,\infty)}$是遍历马尔可夫过程，对于每个$n\in\mathbb{N}$，让$Z^n = (Z_{n^2 t})_{t\in[0,\infty)}$驱动一个过程$X^n$。经典结果证明了在适当的条件下，非马尔可夫过程序列$(X^n)_{n\in\mathbb{N}}$收敛于一个马尔可夫过程，并给出了它的无穷小特征.在这里，我们考虑一个一般序列$(Z^n)_{n\in\mathbb{N}}$。使用随机平均的一般结果[Kur 92]，我们推导出的条件，确保序列$(X^n)_{n\in\mathbb{N}}$收敛的经典情况。作为应用，我们考虑了具有波动选择的Wright-Fisher模型的扩散极限。
英文摘要：
Let $Z = (Z_t)_{t\in[0,\infty)}$ be an ergodic Markov process and, for every $n\in\mathbb{N}$, let $Z^n = (Z_{n^2 t})_{t\in[0,\infty)}$ drive a process $X^n$. Classical results show under suitable conditions that the sequence of non-Markovian processes $(X^n)_{n\in\mathbb{N}}$ converges to a Markov process and give its infinitesimal characteristics. Here, we consider a general sequence $(Z^n)_{n\in\mathbb{N}}$. Using a general result on stochastic averaging from [Kur92], we derive conditions which ensure that the sequence $(X^n)_{n\in\mathbb{N}}$ converges as in the classical case. As an application, we consider the diffusion limit of a Wright-Fisher model with fluctuating selection.
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