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Geometric bounds on the Ornstein-Uhlenbeck velocity process

Research paper by Christer Borell

Indexed on: 01 Mar '85Published on: 01 Mar '85Published in: Probability Theory and Related Fields



Abstract

Let X: Ω→C(ℝ+;ℝn) be the Ornstein-Uhlenbeck velocity process in equilibrium and denote by τA=τA(X) the first hitting time of \(A \subseteq \mathbb{R}^n \). If A, B∈ℛn and ℙ(X(O)∈A=ℙ(Xn(O)≦a), ℙ(Xn(O)∈B=ℙ(Xn(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 Xndenotes the n-th component of X.