SummaryLet 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.
[1]
Classes of solutions of linear systems of partial differential equations of parabolic type
,
1957
.
[2]
Edward Nelson.
Dynamical Theories of Brownian Motion
,
1967
.
[3]
M. Bartlett,et al.
Markov Processes and Potential Theory
,
1972,
The Mathematical Gazette.
[4]
A. Baernstein.
Integral means, univalent functions and circular symmetrization
,
1974
.
[5]
C. Borell.
The Brunn-Minkowski inequality in Gauss space
,
1975
.
[6]
Sidney C. Port,et al.
Brownian Motion and Classical Potential Theory
,
1978
.
[7]
B. Simon.
Functional integration and quantum physics
,
1979
.
[8]
C. Bandle.
Isoperimetric inequalities and applications
,
1980
.
[9]
A. Ehrhard.
Symétrisation dans l'espace de Gauss.
,
1983
.
[10]
A. Friedman.
Partial Differential Equations of Parabolic Type
,
1983
.
[11]
A. Ehrhard.
Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes
,
1984
.