We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian motion that drives the system. We establish a version of the geometric dynamic programming principle for the associated reachability sets and prove that the corresponding value function is a viscosity solution of a geometric partial differential equation. This provides a characterization of the initial masses that can be almost-surely transported towards a given target, along the paths of a stochastic differential equation. Our results extend [16] to our setting.

中文翻译：

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粒子向目标的淬火传质

我们考虑随机目标问题，即寻找平均场随机微分方程的初始定律的集合，以便我们可以控制它的演化，以确保它在固定的时间范围内达到一组规定的终端概率分布。在这里，定律被有条件地考虑到驱动系统的布朗运动的路径。我们为相关的可达集建立了几何动态规划原理的一个版本，并证明相应的值函数是几何偏微分方程的粘性解。这提供了几乎可以肯定地沿着随机微分方程的路径向给定目标传输的初始质量的表征。我们的结果将 [16] 扩展到我们的设置。