We present an investigation of stochastic evolution in which a family of evolution equations in $L^1$ are driven by continuous-time Markov processes. These are examples of so-called piecewise deterministic Markov processes (PDMP's) on the space of integrable functions. We derive equations for the first moment and correlations (of any order) of such processes. We also introduce the mean of the process at large time and describe its behaviour. The results are illustrated by some simple, yet generic, biological examples characterized by different one-parameter types of bifurcations.

中文翻译：

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随机切换进化方程

我们对随机演化进行了研究，其中 $L^1$ 中的一系列演化方程由连续时间马尔可夫过程驱动。这些是可积函数空间上所谓的分段确定性马尔可夫过程 (PDMP) 的示例。我们推导出此类过程的一阶矩和相关性（任何阶数）的方程。我们还大量介绍了过程的平均值并描述了它的行为。结果通过一些简单但通用的生物学示例来说明，这些示例以不同的单参数类型的分叉为特征。