Poisson equation with measure data, reconstruction formula and Doob classes of processes

Abstract: We consider the Dirichlet problem for equation involving a general operator associated with a symmetric transient regular Dirichlet form and bounded Borel measure on the right-hand side of the equation. We introduce a new function space (depending on the form) which allows us to distinguish between solutions with diffuse measure and with general Borel measure. This new space can by characterized analytically in terms of the Poisson kernel associated with the underlying operator or probabilistically by using the notion of Doob class (D) of processes naturally associated with the operator. We also prove a reconstruction formula describing, in terms of the carr\'e du champ operator and jump measure associated with the underlying form, the behaviour of the solution on the set where it is very large.