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August, 1973 Radial Processes
P. W. Millar
Ann. Probab. 1(4): 613-626 (August, 1973). DOI: 10.1214/aop/1176996890

Abstract

Let $X = \{X_t = (X_t^1,\cdots, X_t^d), t \geqq 0\}$ be an isotropic stochastic process with stationary independent increments having its values in $d$-dimensional Euclidean space, $d \geqq 2$. Let $R_t = |X_t|$ be the radial process. It is proved (except for a rather trivial exception) that the Markov process $\{R_t\}$ hits points if and only if the real process $\{X_t^1\}$ hits points; a simple analytic criterion for the latter possibility has been known now for some time. If $x > 0$, the sets $\{t: R_t = x\}$ and $\{t: X_t^1 = 0\}$ are then shown to have the same size in the sense that there is an exact Hausdorff measure function that works for both. Finally, if $X^1$ hits points, it is shown that then $X$ will hit any reasonable smooth surface.

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P. W. Millar. "Radial Processes." Ann. Probab. 1 (4) 613 - 626, August, 1973. https://doi.org/10.1214/aop/1176996890

Information

Published: August, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0265.60071
MathSciNet: MR353464
Digital Object Identifier: 10.1214/aop/1176996890

Subjects:
Primary: 60J30
Secondary: 60J25 , 60J40 , 60J45

Keywords: $\lambda$-capacity , exact Hausdorff measure function , hitting probability , isotropic process , Markov process , potential kernel , radial process , regular point , stationary independent increments

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 4 • August, 1973
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