The exterior Dirichlet problem for the class of bounded functions

We consider different settings of the exterior Dirichlet problem for the class of bounded functions for an elliptic operator of the second order. It is known that if the Markov process corresponding to a given operator is a non return one the solution of the exterior problem may not be unique when no additional conditions are imposed at infinity. We study the conditions at infinity which secure the uniqueness of the solution in the class of bounded functions. It comes out that there is a nontrivial boundary at infinity. This boundary is constructed as the set of equilibrium points of some vector field on the unit sphere. The aforementioned vector field is constructed from the coefficients of the operator. All the results are obtained by investigating the behaviour of the trajectories of the corresponding Markov process at $t\to\infty$.