Asymptotic evolution of a stochastic control problem

Here we are studying the minimization problem of an integral discounted functional, on a set of non-explosive and non constrained diffusions. The integrand is "weakly coercive", which leads us, using the dynamic programming method, to characterize the optimal cost among the solutions of the solving equation, with radiative conditions expessing the centripetal aspect of the optimal control. The evolution of the problem when the discount vanishes is then considered: the integrand being "strongly coercive" (i.e. its gradient being outward), a limit problem is defined and similarly solved; an inward optimal control exists, which is the limit of the ones of the initial problems. The existence properties are obtained by means of a priori estimates concerning suitable solutions of the solving equations in the whole space.