Absolute continuity of symmetric Markov processes (original) (raw)

Abstract

We study Girsanov's theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of "gradient type." We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the case of processes with jumps.

Figures (1)

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (27)

  1. ALBEVERIO, S. and SONG, S. (1993). Closability and resolvent of Dirichlet forms perturbed by jumps. Potential Anal. 2 115-130.
  2. CHEN, Z.-Q., MA, Z.-M. and RÖCKNER, M. (1994). Quasi-homeomorphisms of Dirichlet forms. Nagoya Math. J. 136 1-15.
  3. CHEN, Z.-Q. and ZHANG, T.-S. (2002). Girsanov and Feynman-Kac type transformations for symmetric Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 38 475-505.
  4. FITZSIMMONS, P. J. (1995). Even and odd continuous additive functionals. In Dirichlet Forms and Stochastic Processes (Z. M. Ma, M. Röckner and J. A. Yan, eds.) 139-154. de Gruyter, Berlin.
  5. FITZSIMMONS, P. J. (1997). Absolute continuity of symmetric diffusions. Ann. Probab. 25 230-258.
  6. FITZSIMMONS, P. J. (2001). On the quasi-regularity of semi-Dirichlet forms. Potential Anal. 15 151-185.
  7. FUKUSHIMA, M. (1982). On absolute continuity of multi-dimensional symmetrizable diffu- sions. Functional Analysis in Markov Processes. Lecture Notes in Math. 923 146-176. Springer, Berlin.
  8. FUKUSHIMA, M., OSHIMA, Y. and TAKEDA, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin.
  9. FUKUSHIMA, M. and TAKEDA, M. (1984). A transformation of symmetric Markov processes and the Donsker-Varadhan theory. Osaka J. Math. 21 311-326.
  10. GETOOR, R. K. and SHARPE, M. J. (1981). Two results on dual excursions. In Seminar on Stochastic Processes (E. Çinlar, K. L. Chung and R. K. Getoor, eds.) 31-52. Birkhäuser, Boston.
  11. GETOOR, R. K. and SHARPE, M. J. (1984). Naturality, standardness and weak duality for Markov processes. Z. Wahrsch. Verw. Gebiete 67 1-62.
  12. GONG, F., RÖCKNER, M. and WU, L. (2001). Poincaré inequality for weighted first order Sobolev spaces on loop spaces. J. Funct. Anal. 185 527-563.
  13. HE, S. W., WANG, J. G. and YAN, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Science Press, Beijing.
  14. JACOD, J. (1978). Projection prévisible et décomposition multiplicative d'une semi-martingale positive. Séminaire de Probabilités XII. Lecture Notes in Math. 649 22-34. Springer, Berlin.
  15. KUNITA, H. (1969). Absolute continuity of Markov processes and generators. Nagoya Math. J. 36 1-26.
  16. KUNITA, H. (1976). Absolute continuity of Markov processes. Séminaire de Probabilités X. Lecture Notes in Math. 511 44-77. Springer, Berlin.
  17. KUNITA, H. and WATANABE, S. (1967). On square integrable martingales. Nagoya Math. J. 30 209-245.
  18. KUWAE, K. (1998). Functional calculus for Dirichlet forms. Osaka J. Math. 35 683-715.
  19. MA, Z.-M. and RÖCKNER, M. (1992). Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin.
  20. OREY, S. (1974). Conditions for the absolute continuity of two diffusions. Trans. Amer. Math. Soc. 193 413-426.
  21. OSHIMA, Y. (1987). On absolute continuity of two symmetric diffusion processes. Stochastic Processes-Mathematics and Physics II. Lecture Notes in Math. 1250 184-194. Springer, Berlin.
  22. OSHIMA, Y. and TAKEDA, M. (1987). On a transformation of symmetric Markov processes and recurrence property. Stochastic Processes-Mathematics and Physics II. Lecture Notes in Math. 1250 171-183. Springer, Berlin.
  23. SHARPE, M. J. (1971). Exact multiplicative functionals in duality. Indiana Univ. Math. J. 21 27-60.
  24. SHARPE, M. J. (1988). General Theory of Markov Processes. Academic Press, San Diego.
  25. TAKEDA, M. (1999). Topics on Dirichlet forms and symmetric Markov processes. Sugaku Expositions 12 201-222.
  26. WALSH, J. B. (1972). Markov processes and their functionals in duality. Z. Wahrsch. Verw. Gebiete 24 229-246.
  27. YING, J. (1996). Bivariate Revuz measures and the Feynman-Kac formula. Ann. Inst. H. Poincaré Probab. Statist. 32 251-287.