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We address the problem of existence and uniqueness of solutions$(c,u(\cdot))$ to ergodic Hamilton-Jacobi-Bellman (HJB) equations of the form$H(x,\nabla u(x), D^{2}u(x)) = c$ in the whole space $\mathbb{R}^{m}$ withunbounded and merely measurable data and where $H$ is a Bellman Hamiltonian.The method we use is different from classical approaches. It relies on dualitytheory and optimization in abstract Banach spaces together with maximaldissipativity of the diffusion operator.

A viscous ergodic problem with unbounded and measurable ingredients. Part 1: HJB Equation