We consider a multidimensional time-homogeneous dynamical system and add a randomly perturbed time-dependent deterministic signal to some of its components, giving rise to a high-dimensional system of stochastic differential equations, which is driven by possibly very low-dimensional noise. Equations of this type commonly occur in biology when modelling neurons or in statistical mechanics for certain Hamiltonian systems. We provide verifiable conditions on the original deterministic dynamical system under which the solution to the respective stochastic system features a point in the interior of its state space, which can be proved to be attainable by deterministic control arguments, and at which a local H\"ormander condition holds. Together with a Lyapunov condition, it follows that the corresponding process is positive Harris recurrent.

中文翻译：

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具有内部变量和随机扰动的时间周期输入的退化扩散的正哈里斯递归

我们考虑一个多维时间齐次动力系统，并将随机扰动的与时间相关的确定性信号添加到其某些组件中，从而产生随机微分方程的高维系统，该系统可能由非常低维的噪声驱动。在对神经元建模时或在某些哈密顿系统的统计力学中，这种类型的方程通常出现在生物学中。我们为原始确定性动力系统提供了可验证的条件，在该条件下，相应随机系统的解决方案在其状态空间内部具有一个点，这可以通过确定性控制参数证明是可实现的，并且在该条件下局部 H\"命令条件成立，与李雅普诺夫条件一起，相应的过程是正哈里斯循环的。