| Abstract: |
| A mean-field game system is a system consisting of a backward Hamilton--Jacobi equation and a forward Fokker--Planck equation. It was proposed 20 years ago by Huang, Malham\`{e}, Caines, Lasry, and Lions, as a means to study large population games. We will discuss the well-posedness results for mean-field game systems and the classical solutions to associated master equations in the whole space $\mathbb{R}^d$, driven by individual L\`{e}vy noise of order $\alpha \in (1,2]$, without \textit{a priori} assuming the existence of any moments. We will also present a rather optimal way to express the order $\alpha$ of the noise in terms of the heat kernel of its generator. In this regime, we prove Schauder estimates for the Hamilton--Jacobi equations in the subcritical case $\alpha\in (1,2]$, where the noise dominates the gradient nonlinearity, and we give partial results for the critical case $\alpha=1$. Our results are purely deterministic, but extensions to the framework of SPDEs are possible and interesting. In this connection, we will briefly discuss mean-field games with common noise. |
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