The density evolution of the killed McKean–Vlasov process

The density evolution of McKean–Vlasov stochastic differential equations in the presence of an absorbing boundary is analysed where the solution to such equations corresponds to the dynamics of partially killed large populations. By using a fixed point theorem, we show that the density evolution is characterized as the solution of an integro-differential Fokker–Planck equation with Cauchy–Dirichlet data. This problem arises naturally within mean field game theory.