The Annals of Probability

Radial Processes

P. W. Millar

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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.

Article information

Ann. Probab., Volume 1, Number 4 (1973), 613-626.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J30
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J40: Right processes 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

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


Millar, P. W. Radial Processes. Ann. Probab. 1 (1973), no. 4, 613--626. doi:10.1214/aop/1176996890.

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