A non-linear Riesz respresentation in probabilistic potential theory

For a nice Markov process such as Brownian motion on a bounded domain, we introduce a non-linear potential operator defined in terms of running suprema, and we prove a non-linear Riesz representation of a given function as the sum of a harmonic function and a non-linear potential. The proof involves a family of optimal stopping problems in analogy to the general construction of Bank and El Karoui [Ann. Probab. 32 (1B) (2004) 1030-1067], but here the analysis is carried out in terms of probabilistic potential theory. (c) 2005 Elsevier SAS. All rights reserved.