Milton da Costa Lopes Filho
Universidade Federal do Rio de Janeiro
Title
Nonseparable mean field games in spaces of pseudomeasures
Abstract
Mean field games can be modeled by a system of PDE of the form $u_t + H(t,x,Du,m) = -\Delta u$ and $m_t + div(m H_p(t,x,Du,m) = \Delta m$,
with initial data $m(0,x) = m_0(x)$ and a terminal condition of the form $u(T,x) = G(m(T,x),x)$. The nonlinearity $H$ is the Hamiltonian and $H_p$ is the gradient of $H$ with respect to the $Du$ variable. The $G$ is called the payoff, and it is a nonlocal operator. The variable $m$ represents a distribution of "agents" and $u$ is an action function. Separable Hamiltonians are those of the form $H_1(.,m) + H_2(.,Du)$ and most of the results known in this context apply to separable, convex nonlinearities. However, there are interesting applications for which these structural hypothesis do not apply. The results we will present apply to nonlocal, nonseparable Hamiltonians of the form $H = g(Du) \int f(du) m dx$, which are relevant for certain models of resource extraction and a model for management of household wealth.
Under appropriate conditions on the Hamiltonian and the payoff, we prove well-posedness of this problem on the d-dimensional torus for $m_0$ in the space of pseudomeasures, i.e., measures whose Fourier coefficients are globally bounded. In addition, we prove that the solution depends continuously on the initial measure $m_0$ in the weak-star topology of the space of bounded measures. This is relevant since mean field games model the limiting behavior of games with a finite number of agents, which the weak-star topology on initial data captures in a more natural way than the stronger topology of pseudomeasures