Summary
Let {W(t); 0≦t≦1} be a two-dimensional Wiener process starting from 0. We are interested in the almost sure asymptotic behaviour, asr tends to 0, of the processesX(r) andY(r), whereX(r) denotes the total time spent byW in the ball centered at 0 with radiusr andY(r) the distance between 0 and the curve {W(t);r≦t≦1}. While a characterization of the lower functions ofY was previously established by Spitzer [S], we characterize via integral tests its upper functions as well as the upper and lower functions ofX.
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Hu, Y., Shi, Z. Sojourns and future infima of planar Brownian motion. Probab. Th. Rel. Fields 103, 329–348 (1995). https://doi.org/10.1007/BF01195478
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DOI: https://doi.org/10.1007/BF01195478
Mathematics Subject Classification (1991)
- 60J65
- 60G17
- 60F15