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. Azéma, D. Revuz, J.B. Walsh, R.K. Getoor-M.J. Sharpe. Received: 24 July 2001 / Revised version: 29 December 2002 / Published online: 12 May 2003 Supported by the CNCSIS grant no. 33518/2002 (code 431), the CERES program of the Romanian Ministry of Ed. and Research, contract no. 152/2001 and the EURROMMAT program ICA1-CT-2000-70022 of the European Commission. Mathematics Subject Classification (2000): 60J40, 60J45, 60J55, 60J35, 31C15, 31C25, 31D05