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Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds

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Abstract

We find sufficient conditions for the coefficients of a diffusion equation on a noncompact manifold that guarantee the nonexplosion of solutions in finite time. This property leads to the existence and uniqueness of solutions for the corresponding stochastic differential equation with globally non-Lipschitz coefficients.

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References

  1. E. P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Society, Providence, RI (2002).

    MATH  Google Scholar 

  2. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Dordrecht (1981).

    MATH  Google Scholar 

  3. K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin (1965).

    MATH  Google Scholar 

  4. H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge (1990).

    MATH  Google Scholar 

  5. D. W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold, American Mathematical Society, Providence, RI (2000).

    MATH  Google Scholar 

  6. M. Emery, Stochastic Calculus in Manifolds, Springer, Berlin (1989).

    MATH  Google Scholar 

  7. A. Grigor’yan, “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds,” Bull. Amer. Math. Soc., 36, No. 2, 135–249 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  8. N. V. Krylov and B. L. Rozovskii, “On evolutionary stochastic equations,” in: VINITI Series in Contemporary Problems in Mathematics [in Russian], Vol. 14, VINITI, Moscow (1979), pp. 71–146

    Google Scholar 

  9. E. Pardoux, “Stochastic partial differential equations and filtering of diffusion processes,” Stochastics, 3, 127–167 (1979).

    MATH  MathSciNet  Google Scholar 

  10. A. L. Besse, Manifolds All of Whose Geodesics are Closed, Springer, Berlin (1978).

    MATH  Google Scholar 

  11. J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland, Dordrecht (1975).

    MATH  Google Scholar 

  12. A. Val. Antoniouk, “Upper bounds on second order operators acting on metric function,” Ukr. Math. Bull., 4, No. 2, 161–171 (2007).

    Google Scholar 

  13. P. A. Meyer, Probability and Potentials, Blaisdell, New York (1966).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

    A. Val. Antonyuk & A. Vik. Antonyuk

Authors
  1. A. Val. Antonyuk
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  2. A. Vik. Antonyuk
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Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1454–1472, November, 2007.

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Antonyuk, A.V., Antonyuk, A.V. Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds. Ukr Math J 59, 1632–1652 (2007). https://doi.org/10.1007/s11253-008-0016-9

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  • DOI: https://doi.org/10.1007/s11253-008-0016-9

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