Long-time behavior of deterministic mean field games with nonmonotone interactions (Q6598452)

The authors consider deterministic mean field game problems and the corresponding systems of first order PDEs in two cases: problems with finite horizon \(T\), written as: \(\partial _{t}u^{T}+\frac{1}{2}\left\vert Du^{T}(x,t)\right\vert ^{2}=F(x,m^{T}(t))\), \(\partial _{t}m^{T}-\operatorname{div}(m^{T}\nabla u^{T})=0\), in \(\mathbb{R}^{n}\times (0,T)\), with the conditions \(u^{T}(x,T)=0\), \(m^{T}(0)=m_{0}\), in the unknowns \( (u^{T},m^{T})\), and problems with ergodic cost functional written as: \(c+ \frac{1}{2}\left\vert \nabla v(x)\right\vert ^{2}=F(x,m)\), \(\operatorname{div}(m\nabla v)=0\), in \(\mathbb{R}^{n}\), with the condition \(\int_{\mathbb{R}^{n}}dm(x)=1\), in the unknowns \(c\in \mathbb{R}\), \(v\in C(\mathbb{R}^{n})\), and \(m\in \mathcal{ P}^{1}(\mathbb{R}^{n})\), the set of probability measures with finite first moment endowed with the Kantorovich-Rubinstein distance \(d_{1}\). Here the cost function \(F:\mathbb{R}^{n}\times \mathcal{P}^{1}(\mathbb{R} ^{n})\rightarrow \mathbb{R}\) is supposed to be continuous, and such that \(F\) attains a minimum in \(x\) for each fixed measure \(m\) because of the gap condition: \(\liminf_{\left\vert x\right\vert \rightarrow \infty }F(x,m)>\min_{x\in K_{0}}F(x,m)\), for some compact set \(K_{0}\). \N\NThe authors consider the static problem which consists to find \(\overline{m}\in \mathcal{ P}^{1}(\mathbb{R}^{n})\) such that \(\operatorname{supp}(\overline{m})\subseteq \operatorname{argmin}_{x\in \mathbb{R}^{n}}F(x,\overline{m})\), where \(\operatorname{supp}(\overline{m})\) denotes the support of the measure \(\overline{m}\). They prove that if \(F\) is continuous in \(x\) for every \(m\in \mathcal{P}^{1}(\mathbb{R}^{n})\), \(F\) is continuous in \(m\) with respect to the Kantorovich-Rubinstein distance \(d_{1}\) in \( \mathcal{P}^{1}(\mathbb{R}^{n})\), locally uniformly in \(x\), and satisfies \( \exists K_{o}\subset \mathbb{R}^{n}\) compact, \(\delta _{o}>0\), such that \( \forall m\in \mathcal{P}^{1}(\mathbb{R}^{n})\), \(\inf_{x\notin K_{o}}F(x,m)-\min_{x\in K_{o}}F(x,m)\geq \delta _{o}\), the static problem has a solution. Concerning the ergodic mean field game problem, the authors define a solution as a triple \((c,v,m)\in \mathbb{R}\times C(\mathbb{R} ^{n})\times \mathcal{P}^{1}(\mathbb{R}^{n})\) such that \(v\) is a Lipschitz viscosity solution to the first equation, \(\nabla v(x)\) exists for \(m\)-a.e. \( x\in \mathbb{R}^{n}\), and the second equation is satisfied in the sense of distributions. \N\NThe first main result proves that any measure \(m\) solving the static mean field game problem can be used to build a solution to the ergodic mean field game problem with critical value \(c=\min F(\cdot,m)\), and the function \(v\) can be uniquely characterized. The authors then analyze the finite horizon mean field game problem and the limit of its solutions as \(T\) tends to \(+\infty \). Assuming stronger assumptions, among which \(F\) is uniformly \(C^{2}\) in \(x\) and uniformly Lipschitz in \(m\), the support of the initial measure \(m_{o}\) is compact and \(\bigcap_{m\in \mathcal{P}^{1}(\mathbb{R }^{n})}\operatorname{argmin}F(\cdot,m)\) is non empty, they prove that for any \(s\in (0,1]\) the family \( \{m^{T}(sT)\), \(T>1\}\) is precompact and any weak-* limit \(\overline{m}(s)\) of a subsequence with \(T_{n}\rightarrow \infty \) has its support in \(\bigcup_{m\in \mathcal{P}^{1}(\mathbb{R}^{n})}\operatorname{argmin}F(\cdot,m)\). Then, under the additional assumption that for some \(\mathcal{A}\subseteq \mathbb{R}^{n}\) \(\operatorname{argmin}_{x}F(x,m)=\mathcal{A}\) \(\forall m\in \mathcal{P}^{1}(\mathbb{R}^{n})\), the authors prove that \(\operatorname{supp}(\overline{m}(s))\subseteq \mathcal{A}\), and therefore the limit \(\overline{m}(s)\) solves the static game, as well as the ergodic mean field game problem. They finally describe the limit behavior of \(u^{T}\) and of \(v^{T}\).