On a Stability Property of Harmonic Measures

  • Peter March
Part of the Progress in Probability book series (PRPR, volume 17)


The purpose of this note is to prove that the exit distributions of a diffusion from a somewhat smooth domain are stable under a large class of perturbations. That this need not be so in general had been observed already by M.V. Keldyš [5], who constructed a Jordan domain D containing the origin, with the property that if D n ,n ≥ 1, is any sequence of smooth domains such that
$$ \overline D \subset {D_n} \subset {D_{{n - 1}}}\;{\text{and}}\;\overline D = \mathop{ \cap }\limits_n {D_n} $$
then the classical harmonic measures h Dn (0,dy),n ≥ 1, do not converge weakly to the harmonic measure h D (0,dy). Of course, these measures are the exit distributions of Brownian motion starting from the origin.


Brownian Motion Harmonic Measure Exit Time Smooth Domain Jordan Domain 
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  1. [1.]
    P. Bauman, A wiener test for nondivergence structure, second order elliptic equations, Indiana U. Math. J., 34, (1985), 825–844.MathSciNetMATHCrossRefGoogle Scholar
  2. [2.]
    C. Caratheodory, Conformal Representation, Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, London, second edition, reprinted, 1958.Google Scholar
  3. [3.]
    R.M. Hervé, Recherches axiomatiques sur la theorie des fonctions surharmoniques et du potential, Ann. Inst. Fourier (Grenoble), 12, (1962), 415–571.MathSciNetMATHCrossRefGoogle Scholar
  4. [4.]
    H. Hueber and M. Sieveking, Uniform bounds for quotients of green functions on C1,1 — domains, Ann. Inst. Fourier, 32, (1982), 105–117.MathSciNetMATHCrossRefGoogle Scholar
  5. [5.]
    M.V. Keldyš, On the solvability and stability of the dirichlet problem, A.M.S. Translations, Series 2, Vol. 51, 1966.Google Scholar
  6. [6.]
    E.M. Landis, s—capacity and the behavior of a solution of a second order elliptic equation with discontinuous coefficients in the neighbourhood of a boundary point, Soviet Math. Dokl., 9 (1968), 582–586.MATHGoogle Scholar
  7. [7.]
    K. Miller, Barriers on cones for uniformly elliptic operators, Ann. Math. Pura. Applicata IV (1967), 93–105.CrossRefGoogle Scholar
  8. [8.]
    K. Miller, Non equivalence of regular boundary points for the Laplace and non divergence equations, even with continuous coefficients, Ann. Scuola. Norm. Sup. Pisa, (3), 24, 1970, 159– 163.Google Scholar
  9. [9.]
    D.W. Stroock, Penetration times and passage times, in Markov Processes and Potential Theory, J. Chover, ed., Publ. 19, M.R.C., U. of Wisconsin, J. Wiley and Son, 1967.Google Scholar
  10. [10].
    D.W. Stroock and S.R.S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math., 24, 1972, 651–714.MathSciNetCrossRefGoogle Scholar
  11. [11.]
    D.W. Stroock, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, Heidelberg, New York, 1979.MATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • Peter March
    • 1
  1. 1.Department of Mathematics & StatisticsCarleton UniversityOttawaCanada

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