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The aim of this paper is to study first order Mean field games subject to alinear controlled dynamics on $\mathbb R^{d}$. For this kind of problems, wedefine Nash equilibria (called Mean Field Games equilibria), as Borelprobability measures on the space of admissible trajectories, and mildsolutions as solutions associated with such equilibria. Moreover, we prove theexistence and uniqueness of mild solutions and we study their regularity: weprove H\"older regularity of Mean Field Games equilibria and fractionalsemiconcavity for the value function of the underlying optimal control problem.Finally, we address the PDEs system associated with the Mean Field Gamesproblem and we prove that the class of mild solutions coincides with a suitableclass of weak solutions.

Mild and weak solutions of Mean Field Games problem for linear control systems