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Probability Theory and Related Fields

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

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Baernstein II, A.: Integral means, univalent functions and circular symmetrization. Acta Math. 133, 139–169 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bandle, C.: Isoperimetric Inequalities and Applications. Boston, London, Melbourne: Pitman Advanced Publishing Program 1980zbMATHGoogle Scholar
  3. 3.
    Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. New York, London: Academic Press 1968zbMATHGoogle Scholar
  4. 4.
    Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ehrhard, A.: Inégalités isopérimétriques et intégrales de Dirichlet Gaussiennes. Ann. Sci. Éc. Norm. Sup. 17, 317–332 (1984)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ehrhard, A.: Symétrisation dans l'espace de Gauss. Math. Scand. 53, 281–301 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Essén, M.: The cos πλ Theorem. Lecture Notes in Math. 467. Berlin-Heidelberg-New York: Springer 1975Google Scholar
  8. 8.
    Friedman, A.: Stochastic Differential Equations and Applications. Vol 1. New York-San Francisco-London: Academic Press 1975zbMATHGoogle Scholar
  9. 9.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Englewood Cliffs: Prentice Hall, N.J. 1964zbMATHGoogle Scholar
  10. 10.
    Friedman, A.: Classes of solutions of linear systems of partial differential equations of parabolic type. Duke Math. J. 24, 433–442 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nelson, E.: Dynamical Theories of Brownian Motion. Math. Notes, Princeton: Princeton University Press 1967zbMATHGoogle Scholar
  12. 12.
    Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. New York, San Francisco, London: Academic Press 1978zbMATHGoogle Scholar
  13. 13.
    Simon, B.: Functional Integration and Quantum Physics. New York, San Francisco, London: Academic Press 1979zbMATHGoogle Scholar

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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