Abstract
We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L 2-infinitesimal generator \(\mathcal {L}\) of a general symmetric Markov process. An illuminating concrete example for \(\mathcal {L}\) is \(\Delta_D-(-\Delta)^s_D\), where D is a bounded Euclidean domain in \(\mathbb {R}^d, s\in [0, 1], \Delta_D\) is the Laplace operator in D with zero Dirichlet boundary condition and \(-(-\Delta)^s_D\) is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to \(\mathcal {L}\) is a Lévy process that is the sum of Brownian motion in \(\mathbb {R}^d\) and an independent symmetric (2s)-stable process in \(\mathbb {R}^d\) killed upon exiting the domain D. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is the use of an extension of Nakao’s stochastic integral for zero-energy additive functionals and the associated Itô formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)].
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In Memory of Professor Martin L. Silverstein.
The research of Z.-Q. Chen is supported in part by NSF Grant DMS-0600206.
The research of P. J. Fitzsimmons is supported by a foundation based on the academic cooperation between Yokohama City University and UCSD.
The research of K. Kuwae is supported by a foundation based on the academic cooperation between Yokohama City University and UCSD, and partially supported by a Grant-in-Aid for Scientific Research (C) No. 16540201 from Japan Society for the Promotion of Science.
The research of T.-S. Zhang is supported by the British EPSRC.
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Chen, Z.Q., Fitzsimmons, P.J., Kuwae, K. et al. Perturbation of symmetric Markov processes. Probab. Theory Relat. Fields 140, 239–275 (2008). https://doi.org/10.1007/s00440-007-0065-2
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DOI: https://doi.org/10.1007/s00440-007-0065-2
Keywords
- Perturbation
- Symmetric Markov process
- Time reversal
- Girsanov transform
- Feynman-Kac transform
- Stochastic integral for Dirichlet processes
- Martingale
- Revuz measure
- Dual predictable projection
Mathematics Subject Classification (2000)
- Primary: 31C25
- Secondary: 60J57
- Secondary: 60J55
- Secondary: 60H05