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On the possibility of defining the Chapman–Kolmogorov semi-group on L∞

Published online by Cambridge University Press:  14 November 2011

Allen Devinatz  and
Paul Malliavin
Allen Devinatz
Affiliation:
Northwestern University, Evanston, Illinois, U.S.A.
Paul Malliavin
Affiliation:
Université de Paris VI, France
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Synopsis

If the diffusion matrix coefficient of an Itô stochastic differential equation is everywhere non-singular, then the corresponding Chapman-Kolmogorov semi-group may be defined on L∼(Rn), the space of Lebesgue equivalence classes of essentially bounded Borei measurable functions. However, if the diffusion matrix is singular at some points of Rn, it is not clear that this can always be done. We show that in certain situations it is possible to do so.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 3-4 , 1979 , pp. 327 - 331
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Devinatz, A.. On an inequality of Tosio Kato for degenerate-elliptic operators. J. Functional Analysis (to appear in July, 1979, issue).CrossRefGoogle Scholar
2Malliavin, P.. Stochastic calculus of variations and hypoelliptic operators. Proc. Internat. Symp. of Stochastic Differential Equations, Kyoto 1976, pp. 195–263.Google Scholar