Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T02:20:57.587Z Has data issue: false hasContentIssue false
  • English
  • Français

On conditional diffusion processes

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.
Get access Rights & Permissions [Opens in a new window]

Synopsis

If Xt is the diffusion process associated with a second-order uniformly elliptic operator L in divergence form, then without assuming smoothness in L we prove that for each x and y in ℝd,

where p is the fundamental solution to the heat equation associated with L. This allows one to control p when bounded drift terms are added to L; and also allows one to do Stratonovich integration with respect to the process conditioned to start at any point; previous work only dealt with quasi-every starting point.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 3-4 , 1990 , pp. 243 - 255
Copyright
Copyright © Royal Society of Edinburgh 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Fukushima, M.. Dirichlet forms and Markov processes (Amsterdam: North-Holland, 1980).Google Scholar
2Lyons, T. J. and Zheng, W.. A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Asterisque 157–158 (1988), 249–271.Google Scholar
3Lyons, T. J. and Zheng, W.. Diffusion processes with non-smooth diffusion coefficients and their density function. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 231–242.CrossRefGoogle Scholar
4Nakao, S.. Z. Wahrscheinlichkeitstheorie Verw., 68 (1985), 557–578.Google Scholar
5Stroock, D.. Diffusion semigroups corresponding to uniformly elliptic divergence form operators. Seminaire de Probabilites XXII. (Springer Lecture Notes in Mathematics, 1321, 316–347.)Google Scholar