| Abstract: |
| To rigorously take the limit of N-player games as N goes to infinity, one important step is to establish existence of solutions for the mean field games PDE system allowing measure-valued data. To do so, we consider two classes of initial data, each of which has probability measures as a subset. The two classes of data are the space of pseudomeasures, and Sobolev spaces with sufficiently negative regularity index. To allow such rough data, we consider a class of non-separable Hamiltonians which are regularizing with respect to the measure, but which arise naturally in a number of applications. In these Hamiltonians, the measure variable appears only in an integral over the whole spatial domain. We present existence theorems for the mean field games PDE system with these Hamiltonians and these classes of initial data. In the negative Sobolev case, this includes the first-order case, i.e. without diffusion. |
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