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

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 $\mathbb{R}^{m}$ with unbounded and merely measurable data and where $H$ is a Bellman Hamiltonian. The method we use is different from classical approaches. It relies on duality theory and optimization in abstract Banach spaces together with maximal dissipativity of the diffusion operator.