Probability Theory and Related Fields

, Volume 70, Issue 1, pp 1–13 | Cite as

Geometric bounds on the Ornstein-Uhlenbeck velocity process

  • Christer Borell


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.


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

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