The paper is concerned with ergodic-type Bellman equations arising typically in linear (exponential) quadratic Gaussian control. We are interested in giving recurrence-transience criteria for associated optimal feedback diffusions in terms of qualitative properties of solutions to the Bellman equation. To establish such criteria, we propose a new approach which is based on the Lyapunov method. It turns out that a certain convexity of the equation plays a key role in our arguments.