A REPRESENTATION OF EXCESSIVE FUNCTIONS AS EXPECTED
SUPREMA
Hans Föllmer
Thomas Knispel
Abstract: For a nice Markov process such as Brownian motion on a domain in we prove
a representation of excessive functions in terms of expected suprema. This is motivated
by recent work of El Karoui [5] and El Karoui and Meziou [8] on the max-plus
decomposition for supermartingales. Our results provide a singular analogue to the
non-linear Riesz representation in El Karoui and Föllmer [6], and they extend the
representation of potentials in Föllmer and Knispel [10] by clarifying the role of the
boundary behavior and of the harmonic points of the given excessive function.
2000 AMS Mathematics Subject Classification: 60J45, 60J25, 31C05.
Key words and phrases: Markov processes, excessive functions, expected suprema.