We present a theory of optimal control for McKean-Vlasov stochastic differential equations with infinite time horizon and discounted gain functional. We first establish the well-posedness of the state equation and of the associated control problem under suitable hypotheses for the coefficients. We then especially focus on the time invariance property of the value function \(V(t,\xi )\) , stating that \(V\) is in fact independent of the initial time of the dynamics. This property can easily be derived if the class of controls can be restricted, forgetting the past of the Brownian noise, without modifying the value. This result is not trivial in a general McKean-Vlasov case; in fact, we provide a counterexample where changing the class of controls worsens the value. We thus require appropriate continuity assumptions in order to prove the time invariance property for the value function in our infinite time horizon setting and for the value of an analogous finite time horizon optimal control problem. Furthermore, we show that the value function only depends on the initial random condition through its probability distribution. The function \(V\) can thus be rewritten as a map \(v\) on the Wasserstein space of order \(2\) . After establishing a dynamic programming principle for \(v\) , we derive an elliptic Hamilton-Jacobi-Bellman equation, solved by \(v\) in the viscosity sense. Finally, using a finite horizon approximation of our optimal control problem, we prove that the aforementioned equation admits a unique viscosity solution under stronger assumptions.
