Linear-Quadratic Mean Field Games in Hilbert Spaces

The paper proves existence of an LQM mild solution by a Yosida-approximation/decoupling scheme under Assumption 3.2 (existence and uniform boundedness of strict solutions to a regularized Riccati equation for ηn), plus uniqueness either for small horizon (a contraction with the explicit constant CT) or, for all T, under dissipativity/nondegeneracy of −QS and −QTST. These are stated and proved in Proposition 3.1, Theorem 3.4, Proposition 4.1, and Theorem 4.2, with key definitions of mild solutions and the formula for s given in (2.11)–(2.14) . By contrast, the model’s Phase 2 claims unconditional (all-horizon) existence for the coupled forward–backward system via a “Volterra resolvent” fixed-point on z. This overlooks that composing the backward (t→T) and forward (0→t) Volterra resolvents produces a two-sided (non-Volterra) operator on z, so the unconditional Neumann-series argument does not apply. The paper’s careful use of Assumption 3.2 (and, alternatively, small T or dissipativity) precisely addresses this gap, and is consistent with their emphasis that the coupled system is “completely new” and must be handled delicately .