A mean field game system of the form \[ -u_t^T-\Delta u^T+\frac{1}{2}|Du^T|^2=F(x,m^T(t)), \]
\[ m_t^T-\Delta m^T - \operatorname{div}(m^TDu^T)=0, \]
\[ m^T(0)=m_0,\quad u^T(T)=u_f \] is considered. Here, \(u^T\) and \(m^T\), defined on \([0,T]\times \mathbb R^d\), are the scalar unknowns, \(F\) is a coupling between two equations, \(m^T=m^T(t,x)\) is the probability density of the populations of players, \(u^T\) is the value function. Earlier, the convergence of the above system with a local coupling \(F=\tilde F(x,m^T(t,x))\), \(\tilde F: \mathbb R^d\times\mathbb R\to\mathbb R\), has been analyzed. In this article, the authors investigate the case of the nonlocal coupling \(F: \mathbb R^d\times L^1_{\sharp}(\mathbb R^d)\to\mathbb R\), where \( L^1_{\sharp}(\mathbb R^d)\) is the space of locally integrable periodic maps, under certain assumptions such as monotonicity, regularity, and continuity conditions.
Using the uniform Lipschitz continuity property of the map \(u^T(t,\cdot)\) and the regularizing of the equations, the authors prove that the solution \((u^T,m^T)\) of the mean field game system converges to the solution \((\overline{u},\overline{m})\) of the ergodic problem \[ \overline{\lambda}-\Delta\overline{u}+\dfrac{1}{2}|D\overline{u}|^2=F(x,\overline{m}), \]
\[ -\Delta \overline{m}-\operatorname{div}(\overline{m}D\overline{u})=0, \]
\[ \int_{Q}\overline udx=0,\quad \int_{Q}\overline mdx=1 \] as \(T\to\infty\), where \(\overline\lambda\) is the ergodic constant, \(\overline{u}\), \(\overline{m}\) are defined on \(\mathbb R^d\).
The exponential convergence rates for small initial conditions and for general data are also established.