Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-04T21:37:26.190Z Has data issue: false hasContentIssue false

A decomposition of additive functionals of finite energy

Published online by Cambridge University Press:  22 January 2016

Masatoshi Fukushima*
Affiliation:
College of General Education Osaka University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The celebrated Ito formula for the n-dimensional Brownian motion Xt and for u ∈ C2(Rn) runs as follows:

(0.1)

In § 6 of this paper we extend this to the case where u is any element of the Sobolev space H1R(n) and accordingly Δu is a tempered distribution which is not even a signed measure in general. As a consequence the second term of the right hand side of (0.1) may not be of bounded variation in t.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1979

References

[1] Deny, J., Méthodes hilbertiennes en theory du potential, Potential Theory, C.I.M.E., Edizioni Cremonese, Roma, 1970.Google Scholar
[2] Fukushima, M., On the generation of Markov processes by symmetric forms, Proc. 2nd Japan-USSR Symp. on Prob. Th., Lecture Notes in Math. 330, Springer-Verlag, Berlin-Heidelberg-New York, 1973.Google Scholar
[3] Fukushima, M., Potential theory of symmetric Markov processes and its applications, Proc. 3rd Japan-USSR Symp, on Proc. Th., Lecture Notes in Math. 550, Springer-Verlag, Berlin-Heidelberg-New York, 1976.Google Scholar
[4] Fukushima, M., On additive functionals admitting exceptional sets, to appear in J. Math. Kyoto Univ.Google Scholar
[5] Getoor, R.K. and Sharpe, M. J., An extension of Ito’s formula, Preprint.Google Scholar
[6] Kunita, H. and Watanabe, S., On square integrable martingales, Nagoya Math. J., 30 (1967), 209245.CrossRefGoogle Scholar
[7] Krylov, N. V., On Ito’s stochastic integral equations, Theory Prob. Applications, 14 (1969), 330336.CrossRefGoogle Scholar
[8] LeJan, Yves, Mesures associées a une forme de Dirichlet—Applications, Bull. Soc. Math. France 106 (1978), 61112.CrossRefGoogle Scholar
[9] McKean, H. P., Stochastic integrals, Academic press, New York, 1969.Google Scholar
[10] Meyer, P. A., Intégrales stochastiques I, II, III, Seminaire de Probabilités, 1, Lecture Notes in Math. 39, Springer-Verlag, Berlin-Heidelbert-New York, 1967.Google Scholar
[11] Motoo, M. and Watanabe, S., On a class of additive functionals of Markov processes, J. Math. Kyoto Univ. 4 (1965), 429469.Google Scholar
[12] Silverstein, M. L., Symmetric Markov processes, Lecture Notes in Math. 426, Springer-Verlag, Berlin-Heidelberg-New York, 1974.Google Scholar
[13] Tsuchiya, M., On an extension of Ito’s formula, Ann. Sci. Kanazawa Univ. 14 (1977), 713.Google Scholar