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Brownian Motion and Potential Theory

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Book cover Partial Differential Equations II

Part of the book series: Applied Mathematical Sciences ((AMS,volume 116))

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Abstract

Diffusion can be understood on several levels. The study of diffusion on a macroscopic level, of a substance such as heat, involves the notion of the flux of the quantity. If u(t, x) measures the intensity of the quantity that is diffusing, the flux J across the boundary of a region O in x space satisfies the identity

$$ \frac{\partial } {\partial }\int\limits_O {u(t,x)dV(x) = - \int\limits_{\partial O} {v \cdot JdS(x)} ,} $$
((0.1))

as long as the substance is being neither created nor destroyed.

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References

  1. R. Azencott, Behavior of diffusion semi-groups at infinity, Bull Soc. Math. France 102(1974), 193–240.

    MathSciNet  MATH  Google Scholar 

  2. D. Bell, The Malliavin Calculus, Longman, Essex, 1987.

    MATH  Google Scholar 

  3. R. Blumenthal and R. Getoor, Markov Processes and Potential Theory, Academic Press, New York, 1968.

    MATH  Google Scholar 

  4. R. Cameron and W. Martin, Evaluation of various Wiener integrals by use of certain Sturm-Liouville differential equations, Bull. AMS 51(1945), 73–90.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Chernoff, Note on product formulas for operator semigroups, J. Func. Anal. 2(1968), 238–242.

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Chung and R. Williams, Introduction to Stochastic Integration, Birkhauser, Boston, 1990.

    Book  MATH  Google Scholar 

  7. J. Doob, The Brownian movements and stochastic equations, Ann. Math. 43(1942), 351–369.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Doob, Stochastic Processes, J.Wiley, New York, 1953.

    MATH  Google Scholar 

  9. J. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer- Verlag, New York, 1984.

    Book  MATH  Google Scholar 

  10. N. Dunford and J. Schwartz, Linear Operators, Wiley, New York, 1958.

    MATH  Google Scholar 

  11. R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, Calif., 1984.

    MATH  Google Scholar 

  12. A. Einstein, Investigations on the Theory of the Brownian Movement, Dover, New York, 1956.

    MATH  Google Scholar 

  13. K. Elworthy, Stochastic Differential Equations on Manifolds, LMS Lecture Notes #70, Cambridge Univ. Press, Cambridge, 1982.

    MATH  Google Scholar 

  14. M. Emery, Stochastic Calculus in Manifolds, Springer-Verlag, New York, 1989.

    Book  MATH  Google Scholar 

  15. M. Freidlin, Functional Integration and Partial Differential Equations, Princeton Univ. Press, Princeton, N. J., 1985.

    MATH  Google Scholar 

  16. A. Friedman, Stochastic Differential Equations and Applications, Vols. 1 & 2, Academic Press, New York, 1975.

    MATH  Google Scholar 

  17. E. Hille and R. Phillips, Functional Analysis and Semi-groups, Colloq. Publ. AMS, Providence, R. L., 1957.

    Google Scholar 

  18. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.

    Article  MathSciNet  MATH  Google Scholar 

  19. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam, 1981.

    MATH  Google Scholar 

  20. K. Ito, On Stochastic Differential Equations, Memoirs AMS #4, 1951.

    Google Scholar 

  21. K. Ito and H. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, New York, 1974.

    MATH  Google Scholar 

  22. M. Kac, Probability and Related Topics in Physical Sciences, Wiley, New York, 1959.

    MATH  Google Scholar 

  23. G. Kallianpur, Stochastic Filtering Theory, Springer-Verlag, New York, 1980.

    Book  MATH  Google Scholar 

  24. I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer- Verlag, New York, 1988.

    Book  MATH  Google Scholar 

  25. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

    MATH  Google Scholar 

  26. A. Kolmogorov, Uber die analytishen Methoden in Wahrscheinlichkeitsrechnung, Math. Ann. 104(1931), 415–458.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Larnperti, Stochastic Processes, Springer-Verlag, New York, 1977.

    Book  Google Scholar 

  28. P. Lévy, Random functions, Univ. of Calif. Publ. in Statistics I(12)(1953), 331–388.

    Google Scholar 

  29. P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, Proc. Intern. Symp. SDE, Kyoto (1976), 195–263.

    Google Scholar 

  30. H. McKean, Stochastic Integrals, Academic Press, New York, 1969.

    MATH  Google Scholar 

  31. P. Meyer, Probability and Potentials, Blaisdell, Waltham, Mass., 1966.

    MATH  Google Scholar 

  32. E. Nelson, Operator Differential Equations, Graduate Lecture Notes, Princeton Univ., Princeton, N. J., 1965.

    Google Scholar 

  33. E. Nelson, Feynman integrals and the Schrödinger equation, J. Math. Phys. 5(1964), 332–343.

    Article  MATH  Google Scholar 

  34. E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, Princeton, N.J., 1967.

    MATH  Google Scholar 

  35. B. Øksendal, Stochastic Differential Equations, Springer-Verlag, New York, 1989.

    Book  Google Scholar 

  36. E. Pardoux, Stochastic partial differential equations, a review, Bull, des Sciences Math. 117(1993), 29–47.

    MathSciNet  MATH  Google Scholar 

  37. K. Petersen, Brownian Motion, Hardy Spaces, and Bounded Mean Oscillation, LMS Lecture Notes #28, Cambridge Univ. Press, Cambridge, 1977.

    MATH  Google Scholar 

  38. S. Port and C. Stone, Brownian Motion and Classical Potential Theory, Academic Press, New York, 1979.

    Google Scholar 

  39. J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Func. Anal. 18(1975), 27–59.

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Reed and B. Simon, Methods of Mathematical Physics, Academic Press, New York, Vols. 1,2,1975; Vols. 3,4, 1978.

    MATH  Google Scholar 

  41. Z. Schuss, Theory and Applications of Stochastic Differential Equations, Wiley, New York, 1980.

    MATH  Google Scholar 

  42. B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, 1979.

    MATH  Google Scholar 

  43. D. Stroock, The Kac approach to potential theory I, J. Math. Mech. 16(1967), 829–852.

    MathSciNet  MATH  Google Scholar 

  44. D. Stroock, The Malliavin calculus, a functional analytic approach, J. Func. Anal. 44(1981), 212–257.

    Article  MathSciNet  MATH  Google Scholar 

  45. D. Stroock and S. Varadhan, Multidimensional Diffusion Processes, Springer- Verlag, New York, 1979.

    MATH  Google Scholar 

  46. M. Taylor, Scattering length and perturbations of —A by positive potentials, J. Math. Anal. Appl. 53(1976), 291–312.

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Taylor, Estimate on the fundamental frequency of a drum, Duke Math. J. 46(1979), 447–453.

    Article  MathSciNet  MATH  Google Scholar 

  48. M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, N. J., 1981.

    MATH  Google Scholar 

  49. H. Trotter, On the product of semigroups of operators, Proc. AMS 10(1959), 545–551.

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Tsuji, Potential Theory and Modern Function Theory, Chelsea, New York, 1975.

    Google Scholar 

  51. G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev. 36(1930), 823–841.

    Article  MATH  Google Scholar 

  52. J. Walsch, An introduction to stochastic partial differential equations, pp.265–439 in Ecole d’été de Probabilité de Saint-Fleur XIV, LNM #1180, Springer-Verlag, New York, 1986.

    Google Scholar 

  53. N. Wiener, Differential space, J. Math. Phys. 2(1923), 131–174.

    Google Scholar 

  54. K. Yosida, Functional Analysis, Springer-Verlag, New York, 1965.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, 27599, Chapel Hill, NC, USA

    Michael E. Taylor

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  1. Michael E. Taylor
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© 1996 Springer Science+Business Media New York

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Taylor, M.E. (1996). Brownian Motion and Potential Theory. In: Partial Differential Equations II. Applied Mathematical Sciences, vol 116. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4187-2_5

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  • DOI: https://doi.org/10.1007/978-1-4757-4187-2_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4757-4189-6

  • Online ISBN: 978-1-4757-4187-2

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