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We consider a semi-group of Markovian and symmetric operators to which we associate fractional Sobolev spaces $$\mathbb{D}$$ (0

In a first part, we present a potential theory constructed form a continuous kernel on a locally compact space. The notions of capacity, quasi-continuity, equilibrium measures and potentials are specially studied. In a second part, we particularize the framework, and, in the third part, we give probabilistic interpretations in this particular case. The process then involved is a sum of independent symmetric Levy processes in ℝ d , viewed as a multiparameter process. For instance, hitting probabilities for the process are estimated in terms of capacity.