On decomposition of additive functionals of reflecting Brownian motions

  • Masatoshi Fukushima
  • Matsuyo Tomisaki


Let X be a locally compact separable Hausdorff metric space and m be a positive Radon measure on X with full support. For an m-symmetric Hunt process M = (Xt, Px) on X with the associated Dirichlet form ,.F) being regular on L 2 (X; m), the following decomposition of additive functionals (AF’s in abbreviaton) is known ([11]):
P x — almost surely, which holds for quasi every (q.e. in abbreviation) xX Here u is a quasi- continuous function in the space F, Mat [u] is a martingale AF with quadratic variation being associated with the energy measure of u, Nt [u] is a continuous AF of zero energy and `for q.e. xX’ means `for every xX outside a set of zero capacity’. (1.1) is beyond a semimartingale decomposition in that N t [u] is of zero quadratic variation Pm-a.s. but not necessarily of bounded variation Px-a.s. on each finite time interval. Nevertheless both M t [u] and N t [u] are well computable from u through the Dirichlet form ε and accordingly the decomposition (1.1) has proved to be a useful substitute of Ito’s formula for symmetric Markov processes.


Brownian Motion Lipschitz Domain Dirichlet Form Additive Functional Bound Lipschitz Domain 
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Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • Masatoshi Fukushima
    • 1
  • Matsuyo Tomisaki
    • 2
  1. 1.Department of Mathematical Science, Faculty of Engineering ScienceOsaka UniversityToyonaka, Osaka 560Japan
  2. 2.Department of Mathematics, Faculty of EducationYamaguchi UniversityYamaguchi 753Japan

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