Sub-Markovian resolvents under weak duality hypothesis

We study the weak duality between two sub-Markovian resolvents of kernels on a Lusin topological space with respect to a given measure m. Our frame covers the probabilistic context of two Borel Markov processes in weak duality. The main results are related to: the coincidence of the m-semipolar and m-cosemipolar sets, the Revuz correspondence between the measures charging no cofinely open m-polar set and the potential kernels associated with the homogeneous random measures of the process, the equivalence between smoothness and cosmoothness (a smooth measure is the Revuz measure of a continuous additive functional). We extend and improve results of R.M. Blumenthal-R.K. Getoor, J. Azema, D. Revuz, J.B. Walsh, R.K. GetoorM.J. Sharpe.