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Abstract and Applied Analysis
Volume 2006 (2006), Article ID 39786, 17 pages
http://dx.doi.org/10.1155/AAA/2006/39786

Necessary and sufficient conditions for global-in-time existence of solutions of ordinary, stochastic, and parabolic differential equations

Faculty of Mathematics, Voronezh State University, Universitetskaya pl. 1, Voronezh 394006, Russia

Received 26 June 2005; Accepted 1 July 2005

Copyright © 2006 Yuri E. Gliklikh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Azencott, “Behavior of diffusion semi-groups at infinity,” Bulletin de la Société Mathématique de France, vol. 102, pp. 193–240, 1974. View at Zentralblatt MATH · View at MathSciNet
  2. R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, vol. 15 of Pure and Applied Mathematics, Academic Press, New York, 1964. View at Zentralblatt MATH · View at MathSciNet
  3. K. D. Elworthy, Stochastic Differential Equations on Manifolds, vol. 70 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1982. View at Zentralblatt MATH · View at MathSciNet
  4. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, vol. 260 of Fundamental Principles of Mathematical Sciences, Springer, New York, 1984. View at Zentralblatt MATH · View at MathSciNet
  5. I. I. Gikhman and A. V. Skorohod, Introduction to the Theory of Random Processes, Nauka, Moscow, 1977. View at Zentralblatt MATH · View at MathSciNet
  6. Yu. E. Gliklikh, “On conditions of non-local prolongability of integral curves of vector fields,” Differential equations, vol. 12, no. 4, pp. 743–744, 1977 (Russian).
  7. Yu. E. Gliklikh, “A necessary and sufficient condition of completeness for a certain class of stochastic flows on manifolds,” in Trudy seminara po vektornomu i tenzornomu analizu, vol. 26, pp. 130–138, Moscow State University, Moscow, 2005.
  8. Yu. E. Gliklikh, Global and Stochastic Analysis in Problems of Mathematical Physics, URSS (KomKniga), Moscow, 2005.
  9. Yu. E. Gliklikh, “A necessary and sufficient condition for completeness of stochastic flows continuous at infinity,” Warwick preprint, August 2004, no. 8, pp. 1–7.
  10. Yu. E. Gliklikh and L. A. Morozova, “Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 17–20, pp. 901–912, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Yu. E. Gliklikh and A. V. Sinelnikov, “On a certain necessary and sufficient condition for completeness of a vector field on a Banach manifold,” in Trudy matematicheskogo fakul'teta Voronezhskogo gosudarstvennogo universiteta, no. 9, pp. 46–50, 2005.
  12. C. Godbillon, Géométrie différentielle et mécanique analytique, Hermann, Paris, 1969. View at Zentralblatt MATH · View at MathSciNet
  13. S. Lang, Differential Manifolds, Springer, New York, 1985. View at Zentralblatt MATH · View at MathSciNet
  14. X.-M. Li, “Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers,” Potential Analysis, vol. 3, no. 4, pp. 339–357, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J.-P. Penot, “Weak topology on functional manifolds,” in Global Analysis and Its Applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), vol. 3, pp. 75–84, International Atomic Energy Agency, Vienna, 1974. View at Zentralblatt MATH · View at MathSciNet
  16. L. Schwartz, “Processus de Markov et désintégrations régulières,” Université de Grenoble. Annales de l'Institut Fourier, vol. 27, no. 3, pp. xi, 211–277, 1977. View at Zentralblatt MATH · View at MathSciNet
  17. L. Schwartz, “Le semi-groupe d'une diffusion en liaison avec les trajectoires,” in Séminaire de Probabilités, XXIII, vol. 1372 of Lecture Notes in Math., pp. 326–342, Springer, Berlin, 1989. View at Zentralblatt MATH · View at MathSciNet