| Abstract: |
| We investigate a Mean-Field Game (MFG) posed in an infinite-dimensional Hilbert space and driven by degenerate noise. The associated MFG system consists of a Hamilton--Jacobi--Bellman (HJB) equation for the value function coupled with a nonlinear Fokker--Planck (FP) equation for the distribution of the particles, both governed by a degenerate Kolmogorov operator.
The degeneracy of the noise introduces significant analytical challenges. In particular, the HJB equation is treated in the viscosity sense, while the FP equation is interpreted in a suitable weak formulation. A major difficulty stems from the degeneracy of the Kolmogorov operator, which makes the uniqueness of solutions to the FP equation particularly delicate.
Under appropriate structural assumptions, we establish well-posedness of the MFG system. As an application, we consider Mean-Field Games arising from stochastic delay differential equations, highlighting how delay effects naturally lead to infinite-dimensional and degenerate dynamics.
This talk is based on joint work with Andrzej \`{S}wi\k{e}ch. |
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