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 [3], but here the analysis is carried out in terms of
probabilistic potential theory.