arXiv:2610.37911v1 Announce Type: new Abstract: Let $(U_t)_{t\in\mathbb R}$ be a stationary Ornstein--Uhlenbeck process in $\mathbb R^d$ whose coordinate processes $(U_t^{(i)})_{t\in\mathbb R}$, $1\le i\le d$, are independent, centered, and have covariance function $\mathbb E [U_s^{(i)}U_t^{(i)}]=e^{-|t-s|/2}$. For every $T>0$, we obtain an explicit formula for the probability that the convex hull $\mathrm{conv}\{U_t:0\le t\le T\}$ contains the origin. We obtain a similar formula for a circular Ornstein--Uhlenbeck loop whose covariance is the normalized $T$-periodization of this kernel. For each fixed dimension $d$, we determine the precise long-time asymptotics of the non-absorption probabilities, which decay as explicit constants times $T^{d-1} e^{-T/2}$ as $T\to\infty$. We also show that, as $d\to\infty$, the absorption probabilities in both models undergo a transition from $0$ to $1$ around $T=2d$, with a common Gaussian limiting profile on the scale $\sqrt d$. We also obtain explicit formulas for the probability that $\sigma Z$ belongs to the convex hull of Brownian motion or Brownian bridge on $[0,1]$ in $\mathbb R^d$, where $Z$ is an independent standard Gaussian vector and $\sigma\ge0$. The large-$\sigma$ asymptotics recover the known formulas for the expected volumes of these convex hulls. Finally, we compute the conic intrinsic volumes of the closed positive hulls of deterministic Ornstein--Uhlenbeck spirals and their circular analogues in Hilbert spaces. Their Grassmann angles coincide with the corresponding Ornstein--Uhlenbeck absorption probabilities. After suitable rescaling, the small-time limits of these conic intrinsic volumes recover the formulas of Gao--Vitale and Gao for the intrinsic volumes of the Brownian body and its Brownian-bridge analogue.
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