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Contents of PMS, Vol. 26, Fasc. 2,
pages 379 - 394
 

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 Rd, 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.

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