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 

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References

  1. 1.
    Baernstein II, A.: Integral means, univalent functions and circular symmetrization. Acta Math. 133, 139–169 (1974)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bandle, C.: Isoperimetric Inequalities and Applications. Boston, London, Melbourne: Pitman Advanced Publishing Program 1980MATHGoogle Scholar
  3. 3.
    Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. New York, London: Academic Press 1968MATHGoogle Scholar
  4. 4.
    Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ehrhard, A.: Inégalités isopérimétriques et intégrales de Dirichlet Gaussiennes. Ann. Sci. Éc. Norm. Sup. 17, 317–332 (1984)MathSciNetMATHGoogle Scholar
  6. 6.
    Ehrhard, A.: Symétrisation dans l'espace de Gauss. Math. Scand. 53, 281–301 (1983)MathSciNetCrossRefMATHGoogle 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 1975MATHGoogle Scholar
  9. 9.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Englewood Cliffs: Prentice Hall, N.J. 1964MATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Nelson, E.: Dynamical Theories of Brownian Motion. Math. Notes, Princeton: Princeton University Press 1967MATHGoogle Scholar
  12. 12.
    Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. New York, San Francisco, London: Academic Press 1978MATHGoogle Scholar
  13. 13.
    Simon, B.: Functional Integration and Quantum Physics. New York, San Francisco, London: Academic Press 1979MATHGoogle 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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