# The density evolution of the killed Mckean-Vlasov process

## TL;DR

This paper investigates how the density of large populations evolves in McKean-Vlasov stochastic differential equations with absorbing boundaries, using a fixed point approach to characterize the solution as a unique integro-differential Fokker-Planck equation.

## Contribution

It introduces a novel analysis of density evolution for killed McKean-Vlasov processes, linking it to a unique solution of an integro-differential equation.

## Key findings

- Density evolution characterized by a unique solution
- Application of fixed point theorem to establish existence and uniqueness
- Connection to integro-differential Fokker-Planck equations

## Abstract

The study of the density evolution naturally arises in Mean Field Game theory for the estimation of the density of the large population dynamics. In this paper, we study the density evolution of McKean-Vlasov stochastic differential equations in the presence of an absorbing boundary, 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 unique solution of an integro-differential Fokker-Planck equation with Cauchy-Dirichlet data.