Bregman-projected mirror methods for regularized stationary mean-field games
We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form $H(x,p,m)=H_0(x,p)-g(m)$, with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order $\barγ$-Laplacian regularization of the stationary MFG system as a variational inequality on $L^{\barβ}(\mathbb T^d)\times W^{1,\barγ}(\mathbb T^d)$. To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue-Sobolev exponents of the problem and analyze the exact constrained one-step mirror iteration and a frozen two-step variant. Under quantitative monotonicity of the coupling, both exact iterations converge strongly to the unique solution of the regularized variational inequality for each fixed $\varepsilon>0$, under natural summability conditions on the step sizes. Along weakly convergent vanishing-regularization sequences, we also prove strong convergence in the full product space. Numerical experiments in one and two dimensions show first-order accuracy on manufactured solutions and weak dependence of outer iteration counts on the tested grids. At matched effective step sizes, the frozen two-step and one-step implementations agree up to inner-solve accuracy, while the latter requires fewer mirror solves.