SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1320-1356, January 2021.
We prove that viscosity solutions of Hamilton--Jacobi--Bellman (HJB) equations, corresponding either to deterministic optimal control problems for systems of $n$ particles or to stochastic optimal control problems for systems of $n$ particles with a common noise, converge locally uniformly to the viscosity solution of a limiting HJB equation in the space of probability measures. We prove uniform continuity estimates for viscosity solutions of the approximating problems which may be of independent interest. We pay special attention to the case when the Hamiltonian is convex in the gradient variable and equations are of first order and provide a representation formula for the solution of the limiting first order HJB equation. We also propose an intrinsic definition of viscosity solution on the Wasserstein space.

中文翻译：

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概率测度空间中Hamilton-Jacobi-Bellman方程的有限维逼近

SIAM数学分析杂志，第53卷，第2期，第1320-1356页，2021年1月。
我们证明了Hamilton-Jacobi-Bellman（HJB）方程的粘度解，既对应于$ n $粒子系统的确定性最优控制问题，又对应于具有常见噪声的$ n $粒子系统的随机最优控制问题，在概率测度空间内，局部均匀收敛于一个极限HJB方程的粘度解。我们证明了关于近似问题的粘度解的一致连续性估计，这可能是独立引起关注的。我们特别注意梯度变量中的哈密顿量为凸且方程为一阶的情况，并为极限一阶HJB方程的求解提供了表示公式。我们还提出了Wasserstein空间上粘度解的内在定义。