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This paper focuses on a mean-field optimal stopping problem with non-Markovdynamics and common noise, inspired by Talbi, Touzi, and Zhang\cite{TalbiTouziZhang1,TalbiTouziZhang3}. The goal is to establish the limittheory and demonstrate the equivalence of the value functions between weak andstrong formulations. The difference between the strong and weak formulationslies in the source of randomness determining the stopping time on a canonicalspace. In the strong formulation, the randomness of the stopping timeoriginates from Brownian motions. In contrast, this may not necessarily be thecase in the weak formulation. Additionally, a $(H)$-Hypothesis-type conditionis introduced to guarantee the equivalence of the value functions. The limittheory encompasses the convergence of the value functions and solutions of thelarge population optimal stopping problem towards those of the mean-fieldlimit, and it shows that every solution of the mean field optimal stoppingproblem can be approximated by solutions of the large population optimalstopping problem.

On the limit theory of mean field optimal stopping with non-Markov dynamics and common noise