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Diffusion processes with non-smooth diffusion coefficients and their density functions

Published online by Cambridge University Press:  14 November 2011

T. J. Lyons  and
W. A. Zheng
T. J. Lyons
Affiliation:
Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland, U.K.
W. A. Zheng
Affiliation:
Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland, U.K.
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Synopsis

Denote by Xt an n-dimensional symmetric Markov process associated with an elliptic operator

where (aij) is a bounded measurable uniformly positive definite matrix-valued function of x. Let f(x, t) be a measurable function defined on Rn × [0, 1]. In this paper, we prove that f(Xt, t) is a regular Dirichlet process if and only if the following two conditions are satisfied:

(i) For almost every and

(ii) Let be a sequence of subdivisions of [0,1] so that

Then

As an application of the above result, we prove the following fact: Let p(y, t) be the probability density of the diffusion process Yt, associated with the elliptic operator

where (bi) are bounded measurable functions of x and we suppose that . Then, p(Yt, t) is a regular Dirichlet process and therefore p(.,.) satisfies (i) and (ii).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 3-4 , 1990 , pp. 231 - 242
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Lyons, T. J. & Zheng, W. A.. A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Asterisque 157–158 (1988), 249–271.Google Scholar
2Fukushima., M.. Dirichlet Forms and Markov Processes (Amsterdam: North-Holland Publishing Company, 1980).Google Scholar