Asymptotic behavior of the generalized principal eigenvalue for Hamilton–Jacobi–Bellman equations with asymptotically radial data

We study the ergodic problem for Hamilton–Jacobi–Bellman equations with convex and superlinear Hamiltonians and inward-pointing drifts. Compared to the existing literature, our results allow for (i) non-polynomial growth of the data; (ii) a more general diffusion operator than the usual Laplacian; and (iii) coefficients which are only asymptotically radially symmetric. We develop a new method for estimating the asymptotic behavior of the generalized principal eigenvalue under large perturbations of the potential function. This method reduces the asymptotic evaluation to a simple one-dimensional optimization problem in terms of the given data. We also discuss the stochastic control interpretation of our results.