# Optimal control of path-dependent McKean-Vlasov SDEs in infinite dimension

arXiv: https://arxiv.org/abs/2012.14772 · HTML: https://mlsanity.com/paper/2012.14772/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2020.

## Built on by (most cited first, 2 of 2)

- [Continuous-time q-learning for mean-field control problems](https://mlsanity.com/paper/2306.16208/) (2023)
- [Actor-Critic Learning for Extended Mean Field Control with Deterministic Policies](https://mlsanity.com/paper/2607.11005/) (2026)

## Abstract (arXiv)

We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [41], and prove a related functional Ito formula in the spirit of Dupire [24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.