We construct an importance sampling method for computing statistics related to rare events for weakly interacting diffusions. Standard Monte Carlo methods behave exponentially poorly with the number of particles in the system for such problems. Our scheme is based on subsolutions of a Hamilton–Jacobi–Bellman (HJB) equation on Wasserstein space which arises in the theory of mean-field (McKean–Vlasov) control. We identify conditions under which such a scheme is asymptotically optimal. In the process, we make connections between the large deviations principle for the empirical measure of weakly interacting diffusions, mean-field control, and the HJB equation on Wasserstein space. We also provide evidence, both analytical and numerical, that with sufficient regularity of the HJB equation, our scheme can have vanishingly small relative error in the many particle limit.

我们构建了一种重要性采样方法，用于计算与弱相互作用扩散的罕见事件相关的统计数据。对于此类问题，标准蒙特卡罗方法的表现随着系统中粒子数量的增加呈指数级下降。我们的方案基于 Wasserstein 空间上的 Hamilton-Jacobi-Bellman (HJB) 方程的子解，该方程出现在平均场 (McKean-Vlasov) 控制理论中。我们确定了这种方案渐近最优的条件。在此过程中，我们将弱相互作用扩散的经验测量的大偏差原理、平均场控制和 Wasserstein 空间上的 HJB 方程联系起来。我们还提供了分析和数值证据，表明只要 HJB 方程具有足够的规律性，我们的方案在多粒子极限下可以具有非常小的相对误差。