Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, we find that the rate function (or action functional) does not involve with the solution of the skeleton equation, which describes unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution-dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.
翻译:取决于分布的随机随机变化的动态系统在工程和科学中广泛出现。 我们考虑了一组此类系统,它们模拟在矢量字段中移动的相互作用粒子的极限行为,我们的目标是检查矢量字段平衡稳定状态之间最可能的过渡路径。在小噪音系统中,我们发现率函数(或行动功能)与骨架方程式的解决方案无关,该方程式描述的是因零距离的相互作用而转移的矢量字段的无间断的确定性流动。因此,我们引导我们研究不依赖分布的随机变化的随机偏差方程的最可能过渡路径。这能够通过适应性最低行动方法计算出这些依赖分布的随机动态系统最可能的过渡路径,我们用两个例子来说明我们的方法。