The Sharp Markov Property of the Brownian Sheet and Related Processes

Many papers concerned with the Brownian sheet contain a statement to the effect that this process satisfies the sharp Markov property with respect to almost no curve in the plane, and thus is not Markovian in a natural way. The objective of this paper is to show quite the opposite: the Brownian sheet satisfies the sharp Markov property with respect to almost every Jordan curve in the plane. The "almost every " can be interpreted both in the sense of Baire category and with respect to appropriate reference measures. These results follow from simple geometric conditions on the curve which are necessary and sufficient for the sharp Markov property to hold. These conditions turn out to be sufficient not only for the Brownian sheet but also for a large class of processes with independent increments. For processes in this class, we give the miniimal splitting field for an arbitrary open set, and sufficient conditions on the boundary of an open set for it to have the sharp Markov property. In particular, many sets with a fractal boundary have this property. *Department of Statistics, University of California, Berkeley, CA 94720. The research of this author is supported by a National Science Foundation Postdoctoral Fellowship. **Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T1Y4, Canada.