标题： 具有随机负反馈的时滞方程不变测度的存在性 显示英文标题

标题： Existence of Invariant Measures for Delay Equations with Stochastic Negative Feedback

摘要： 我们给出了具有限时滞的一般随机微分方程存在不变概率测度的充分条件。应用实例包括带有一（小）乘性噪声项扰动的麦凯-格拉斯方程和尼科尔森苍蝇方程。这些随机负反馈系统的解全局持久且所有解以概率方式均被上界限制。结果表明，有限时间爆破的发生与解及其片段以概率方式有界密切相关。若存在至少一个初始条件使得解以概率方式远离零，则证明了非平凡不变测度的存在。驱动动力系统的噪声允许为可积的Lévy过程。 显示英文摘要

摘要： We provide sufficient conditions for the existence of invariant probability measures for generic stochastic differential equations with finite time delay. Applications include the Mackey--Glass equations and Nicholson's blowflies equation, each perturbed by a (small) multiplicative noise term. Solutions to these stochastic negative feedback systems persist globally and all solutions are bounded above in probability. It turns out that the occurrence of finite time blowups and boundedness in probability of solutions and solution segments are closely related. A non-trivial invariant measure is shown to exist if and only if there is at least one initial condition for which the solution remains bounded away from zero in probability. The noise driving the dynamical system is allowed to be an integrable L\'evy process.