Ergodic Control Bellman Equation with Neumann Boundary Conditions

Let O be an open bounded smooth domain of ℝ n , and let Γ = αO be its boundary. We denote by n the normal vector at the boundary Γ, oriented towards the outside of O. Let us consider the canonical process $$\begin{gathered} \Omega = C^0 ([0,\infty );\bar O), \hfill \\ y(t,w) \equiv w(t), if w \in \Omega , \hfill \\ \mathfrak{F}^t = \sigma (y(s),0 \leqslant s \leqslant t). \hfill \\ \end{gathered} $$