Summary
Two generalisations of Brownian motion to several-dimensional time are considered and the topology of their level sets is analysed. It is shown that for these maps non-trivial contours are quite rare — their union has Lebesgue measure zero. The boundedness of all contours is established for the generalisation due to Lévy. For the other, the Brownian sheet, a partial result concerning the behaviour of the zero contour near the boundary is established.
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Most of the results in this paper were obtained in the course of an S.R.C studentship at the University of Oxford, and appear in the ensuing D. Phil. thesis. I wish to acknowledge the encouragement of my supervisor John Kingman, and the stimulus of correspondence with J.B. Walsh and R. Pyke.
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Kendall, W.S. Contours of Brownian processes with several-dimensional times. Z. Wahrscheinlichkeitstheorie verw Gebiete 52, 267–276 (1980). https://doi.org/10.1007/BF00538891
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DOI: https://doi.org/10.1007/BF00538891