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Probability Theory and Related Fields

, Volume 70, Issue 1, pp 1–13

# Geometric bounds on the Ornstein-Uhlenbeck velocity process

• Christer Borell
Article

## Summary

Let X: Ω→C(ℝ+;ℝ n ) be the Ornstein-Uhlenbeck velocity process in equilibrium and denote by τ A =τ A (X) the first hitting time of $$A \subseteq \mathbb{R}^n$$. If A, B∈ℛn and ℙ(X(O)∈A=ℙ(X n (O)≦a), ℙ(X n (O)∈B=ℙ(X n (O)≧b)we prove that $$\mathbb{P}(\tau _A \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \leqslant } t)\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \geqslant } \mathbb{P}(\tau _{\{ \chi _n \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \leqslant } a\} } \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \leqslant } t)$$ and $$\mathbb{E}\left( {\int\limits_0^{t \wedge \tau A} {1_{\text{B}} (X({\text{s}})d{\text{s}}} } \right)\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \leqslant } \mathbb{E}\left( {\int\limits_0^{t \wedge \tau _{\left\{ {x_n \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \leqslant } a} \right\}} } {1_{\left\{ {x_n \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{ \geqslant } b} \right\}} (X({\text{s))}}d{\text{s}}} } \right)$$. Here X n denotes the n-th component of X.

## Keywords

Radon Weak Sense Isoperimetric Inequality Geometric Bound Unique Classical Solution
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## Copyright information

© Springer-Verlag 1985

## Authors and Affiliations

• Christer Borell
• 1
1. 1.Department of MathematicsChalmers University of Technology and The University of GöteborgGöteborgSweden