We provide an Itô’s formula for \(C^1\) -functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the \(C^1\) -Itô’s formula in Gozzi and Russo (Stoch Process Appl 116(11):1563–1583, 2006) to this context. As the first application, we study a class of McKean–Vlasov optimal control problems, and establish a verification theorem which only requires \(C^1\) -regularity of its value function, which is equivalently the (viscosity) solution of the associated HJB master equation. It goes together with a novel duality result.
