Radial Processes

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.