The developmentsin the recent yearsof the potential theoryemphasized a classof functions larger than that of excessive functions (i.e. the positive superharmonic functionsfromtheclassicalpotentialtheoryassociatedwiththeLaplaceoperator), namely the strongly supermedian functions. It turns out that a positive Borel function will be strongly supermedian if and only if it is the in?mum of all its excessive majorants. Apparently, these functions have been introduced by J.F. Mertens and then they have been studied mainly by P.A. Meyer, G. Mokobodzki, D. Feyel and recently by P.J. Fitzsimmons and R.K. Getoor. The aimofthis bookisamongothersto developa potential theoryappropriate to this new class of functions. Although our methods are analytical, we present also the probabilistic counterparts from the Markov processes theory. The natural frame in which this theory is settled is given by a sub-Markovian resolvent of kernels on a Radon measurable space. After a possible extension of the space, such a resolvent becomes that one associated with a right process on a Radon topological space, not necessary locally compact and without existing a reference measure. Intimately related to the excessive functions we present certain basic tools of the theory: the Ray topology and compacti?cation, the ?ne carrier and the reduction operation on measurable sets. We examine di?erent types of negligible sets with respect to a ?nite measure ?:the ?-polar, ?-semipolar and ?-mince sets. We take advantage of the cone of potentials structure for both excessive functions and measures.

1 Excessive Functions.- 1.1 Sub-Markovian resolvent of kernels.- 1.2 Basics on excessive functions.- 1.3 Fine topology.- 1.4 Excessive measures.- 1.5 Ray topology and compactification.- 1.6 The reduction operation and the associated capacities.- 1.7 Polar and semipolar sets. Nearly measurable functions.- 1.8 Probabilistic interpretations: Sub-Markovian resolvents and right processes.- 2 Cones of Potentials and H-Cones.- 2.1 Basics on cones of potentials and H-cones.- 2.2 ?-Balayages on cones of potentials.- 2.3 Balayages on H-cones.- 2.4 Quasi bounded, subtractive and regular elements of a cone of potentials.- 3 Fine Potential Theoretical Techniques.- 3.1 Cones of potentials associated with a sub-Markovian resolvent.- 3.2 Regular excessive functions, fine carrier and semipolarity.- 3.3 Representation of balayages on excessive measures.- 3.4 Quasi bounded, regular and subtractive excessive measures.- 3.5 Tightness for sub-Markovian resolvents.- 3.6 Localization in excessive functions and excessive measures.- 3.7 Probabilistic interpretations: Continuous additive functionals and standardness.- 4 Strongly Supermedian Functions and Kernels.- 4.1 Supermedian functionals.- 4.2 Supermedian ?-quasi kernels.- 4.3 Strongly supermedian functions.- 4.4 Fine densities.- 4.5 Probabilistic interpretations: Homogeneous random measures.- 5 Subordinate Resolvents.- 5.1 Weak subordination operators.- 5.2 Inverse subordination.- 5.3 Probabilistic interpretations: Multiplicative functionals.- 6 Revuz Correspondence.- 6.1 Revuz measures.- 6.2 Hypothesis (B) of Hunt.- 6.3 Smooth measures and sub-Markovian resolvents.- 6.4 Measure perturbation of sub-Markovian resolvents.- 6.5 Probabilistic interpretations: Positive left additive functionals.- 7 Resolvents under Weak Duality Hypothesis.- 7.1Weak duality hypothesis.- 7.2 Natural potential kernels and the Revuz correspondence.- 7.3 Smooth and cosmooth measures.- 7.4 Subordinate resolvents in weak duality.- 7.5 Semi-Dirichlet forms.- 7.6 Weak duality induced by a semi-Dirichlet form.- 7.7 Probabilistic interpretations: Multiplicative functionals in weak duality.- A Appendix.- A.1 Complements on measure theory, kernels, Choquet boundary and capacity.- A.2 Complements on right processes.- A.4 Basics on coercive closed bilinear forms.- Notes.