intersection properties of Wiener processes in the plane. For each positive integer k we show that k independent Wiener processes intersect almost surely in a set of Hausdorff dimension

http://www.sciencedirect.com/science/article/pii/0022123678900617

Wiener path intersections and local time

• Robert L Wolpert

• Department of Mathematics, Duke University, Durham, North Carolina 27706 U.S.A.

Received 1 May 1977, Available online 30 June 2004

Communicated by G. A. Hunt

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doi:10.1016/0022-1236(78)90061-7

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Abstract

We study intersection properties of Wiener processes in the plane. For each positive integer k we show that k independent Wiener processes intersect almost surely in a set of Hausdorff dimension two, and that the set of points a single process visits at least k distinct times also has dimension two. We construct a functional on configurations of k independent Wiener processes that measures the extent to which the trajectories of the k processes intersect. We prove certain L^p estimates for this functional and show that it is a local time for a certain vector-valued multiparameter stochastic process.

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