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Some geometric aspects of potential theory

  • John Hawkes
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1095)

Keywords

Brownian Motion Markov Process Potential Theory Equilibrium Measure Resolvent Operator 
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References

  1. [1]
    BERG, C. and G. FORST. Potential Theory on Locally Compact Abelian Groups. Springer-Verlag, Berlin, 1975.CrossRefzbMATHGoogle Scholar
  2. [2]
    BLUMENTHAL, R.M. Some relationships involving subordination. Proc. Amer. Math. Soc. 10 (1959) 502–510.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    BLUMENTHAL, R.M. and R.K. GETOOR. Markov Processes and Potential Theory. Academic Press, New York, 1968.zbMATHGoogle Scholar
  4. [4]
    BLUMENTHAL, R.M. and R.K. GETOOR. Dual processes and potential theory. Proc. 12th Biennial Seminar, Canad. Math. Congress (1970) 137–156.Google Scholar
  5. [5]
    BRELOT, M. Les étapes et les aspects multiples de la théorie du potentiel. L'Enseignement mathém. XVIII (1972) 1–36.MathSciNetzbMATHGoogle Scholar
  6. [6]
    BRETAGNOLLE, J. Résultats de Kesten sur les processus à accroissements indépendants. Séminaire de Probabilités V, Lecture Notes in Mathematics 191 21–36, Springer-Verlag, Berlin, 1971.Google Scholar
  7. [7]
    CHUNG, K.L. Probabilistic approach to the equilibrium problem in potential theory. Ann. Inst. Fourier 23 (1973) 313–322.CrossRefzbMATHGoogle Scholar
  8. [8]
    CHUNG, K.L. Remarks on equilibrium potential and energy. Ann. Inst. Fourier 25 (1975) 131–138.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    CHUNG, K.L. Lectures from Markov Processes to Brownian Motion. Springer-Verlag, New York, 1982.CrossRefzbMATHGoogle Scholar
  10. [10]
    CHUNG, K.L. and M. RAO. Equilibrium and energy. Probab. Math. Statist. 1 (1980) 99–108.MathSciNetzbMATHGoogle Scholar
  11. [11]
    COURANT, R., K. FRIEDRICHS and H. LEWY. Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Annalen 100 (1928) 32–74.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    FUKUSHIMA, M. Potential theory of symmetric Markov processes and its applications. Lecture Notes in Mathematics 550 119–133, Springer-Verlag, Berlin, 1976.zbMATHGoogle Scholar
  13. [13]
    FUKUSHIMA, M. Dirichlet Forms and Markov Processes. North-Holland, Amsterdam, 1980.zbMATHGoogle Scholar
  14. [14]
    GETOOR, R.K. Some Asymptotic Formulas Involving Capacity. Z. Wahrscheinlichkeitstheorie 4 (1965) 248–252.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    GLOVER, J. Energy and the maximum principle for non symmetric Hunt processes. Theory Probab. and its Applications XXVI (1981) 745–757.MathSciNetzbMATHGoogle Scholar
  16. [16]
    HAWKES, J. Polar sets, regular points and recurrent sets for the symmetric and increasing stable processes. Bull. London Math. Soc. 2 (1970) 53–59.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    HAWKES, J. On the Hausdorff Dimension of the Intersection of the Range of a Stable Process with a Borel Set. Z. Wahrscheinlichkeitstheorie 19 (1971) 90–102.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    HAWKES, J. Potential theory of Lévy processes. Proc. London Math. Soc. (3) 38 (1979) 335–352.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    HAWKES, J. Transition and resolvent densities for Lévy processes. J. London Math. Soc. To appear.Google Scholar
  20. [20]
    HAWKES, J. Fourier methods in the geometry of small sets. In preparation.Google Scholar
  21. [21]
    HAWKES, J. Energy, capacity and polar sets for some Markov processes. In preparation.Google Scholar
  22. [22]
    HAWKES, J. Harmonic analysis of Lévy sets. In preparation.Google Scholar
  23. [23]
    HELMS, L.L. Introduction to Potnetial Theory. Wiley, New York, 1969.zbMATHGoogle Scholar
  24. [24]
    HUNT, G.A. Markov processes and potentials I and II. Illinois J. Math. 1 (1957) 44–93 and 316–369.MathSciNetzbMATHGoogle Scholar
  25. [25]
    HUNT, G.A. Markov processes and potentials III. Illinois J. Math. 2 (1958) 151–213.MathSciNetGoogle Scholar
  26. [26]
    ITÔ, K. and H.P. McKEAN. Diffusion processes and their sample paths. Springer-Verlag, Berlin, 1965.CrossRefzbMATHGoogle Scholar
  27. [27]
    KAC, M. Aspects Probabilistes de la théorie du potentiel. Publications du séminaire de Mathématiques Supérieures, Montreal, 1970.zbMATHGoogle Scholar
  28. [28]
    KAHANE, J.-P. Ensembles parfaits et processus de Lévy. Periodica Math. Hungar. 2 (1972) 49–59.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    KAKUTANI, S. Two-dimensional Brownian motion and harmonic functions. Proc. Acad. Tokyo 20 (1944) 706–714.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    KAKUTANI, S. Markoff processes and the Dirichlet problem. Proc. Acad. Tokyo 21 (1945) 227–233.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    KANDA, M. Two Theorems on Capacity for Markov Processes with Stationary Independent Increments. Z. Wahrscheinlichkeitstheorie 35 (1976) 159–165.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    KANDA, M. Characterization of Semipolar Sets for Processes with Stationary Independent Increments. Z. Wahrscheinlichkeitstheorie 42 (1978) 141–154.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    KANDA, M. On the class of polar sets for a certain class of Lévy processes on the line. J. Math. Soc. Japan 35 (1983) 221–242.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    KELLOGG, O.D. Foundations of Potential Theory. Springer-Verlag, Berlin, 1929; reprinted by Dover, New York, 1955.CrossRefzbMATHGoogle Scholar
  35. [35]
    KESTEN, H. Hitting probabilities of single points for processes with stationary independent increments. Mem. Amer. Math. Soc. 93 (1969).Google Scholar
  36. [36]
    KINGMAN, J.F.C. Recurrence properties of processes with stationary independent increments. J. Austral. Math. Soc. 4 (1964) 223–228.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    KINGMAN, J.F.C. Subadditive ergodic theory. Ann. Probab. 1 (1973) 883–909.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    LAMPERTI, J. Wiener's test and Markov chains. J. Math. Anal. Appl. 6 (1963) 58–66.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    LANDKOF, N.S. Foundations of Modern Potential Theory. Springer-Verlag, Berlin, 1972.CrossRefzbMATHGoogle Scholar
  40. [40]
    McKEAN, H.P. A probabilistic interpretation of equilibrium charge distribution. J. Math. Kyoto Univ. 4 (1965) 617–625.MathSciNetzbMATHGoogle Scholar
  41. [41]
    OREY, S. Polar sets for processes with independent increments. In Markov processes and potential theory, ed. J. Chover, Wiley, New York, 1967.Google Scholar
  42. [42]
    PORT, S.C. and C.J. STONE. The asymmetric Cauchy process on the line. Ann. Math. Statist. 40 (1969) 137–143.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    PORT, S.C. and C.J. STONE. Infinitely divisible processes and their potential theory I. Ann. Inst. Fourier (Grenoble) 21 (2) (1971) 157–275.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    PORT S.C. and C.J. STONE. Infinitely divisible processes and their potential theory II. Ann. Inst. Fourier (Grenoble) 21 (4) (1971) 179–265.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    PORT, S.C. and C.J. STONE. Brownian Motion and Classical Potential Theory. Academic Press, New York, 1978.zbMATHGoogle Scholar
  46. [46]
    PHILIPS, H.B. and N. WIENER. Nets and the Dirichlet problem. J. Math. and Phys. 2 (1923) 105–124.CrossRefzbMATHGoogle Scholar
  47. [47]
    PRUITT, W.E. The Hausdorff dimension of the range of a process with stationary independent increments. J. Math. Mech. 19 (1969) 371–378.MathSciNetzbMATHGoogle Scholar
  48. [48]
    PRUITT, W.E. Some Dimension Results for Processes with Independent Increments. In Stochastic Processes and Related Topics, ed. M. Puri. Academic Press, New York, 1975.Google Scholar
  49. [49]
    RAO, M. On a result of M. Kanda. Z. Wahrscheinlichkeitstheorie 41 (1977) 35–37.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    SILVERSTEIN, M.L. Symmetric Markov Processes. Lecture Notes in Mathematics 426, Springer-Verlag, Berlin, 1974.zbMATHGoogle Scholar
  51. [51]
    SILVERSTEIN, M.L. Boundary Theory for Symmetric Markov Processes. Lecture Notes in Mathematics 516, Springer-Verlag, Berlin, 1976.zbMATHGoogle Scholar
  52. [52]
    SILVERSTEIN, M.L. The sector condition implies that semipolar sets are quasi-polar. Z. Wahrscheinlichkeitstheorie 41 (1977) 13–33.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    SPITZER, F. Electrostatic capacity, heat flow and Brownian motion. Z. Wahrscheinlichkeitstheorie 3 (1964) 110–121.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    STEIN, E.M. and G. WEISS. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, 1971.zbMATHGoogle Scholar
  55. [55]
    STRATTON, H.H. On dimension of support for stochastic processes with independent increments. Trans. Amer. Math. Soc. 132 (1968) 1–29.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    TAYLOR, S.J. Sample path properties of a transient stable process. J. Math. Mech. 16 (1967) 1229–1246.MathSciNetzbMATHGoogle Scholar
  57. [57]
    WERMER, J. Potential Theory. Lecture Notes in Mathematics 408, Springer-Verlag, Berlin, 1974.CrossRefzbMATHGoogle Scholar
  58. [58]
    WHITMAN, W. Some strong laws for random walks and Brownian motion. Ph. D. Thesis, Cornell, 1964.Google Scholar
  59. [59]
    WIENER, N. The Dirichlet problem. J. Math. Phys. 3 (1924) 127–146.CrossRefzbMATHGoogle Scholar
  60. [60]
    ZABCZYCK, J. Sur la théorie semi-classique du potentiel pour les processus à accorissements indépendants. Studia Math. 35 (1970) 227–247.MathSciNetGoogle Scholar
  61. [61]
    ZABCZYCK, J. A note on semipolar sets for processes with independent increments. Lecture Notes in Mathematics 472 277–283, Springer-Verlag, Berlin, 1975.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • John Hawkes
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity College of SwanseaSwanseaGreat Britain

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