Abstract
We show that the renormalized self-intersection local time \(\gamma_t(x)\) for both the Brownian motion and symmetric stable process in R1 is differentiable in the spatial variable and that \(\gamma'_t(0)\) can be characterized as the continuous process of zero quadratic variation in the decomposition of a natural Dirichlet process. This Dirichlet process is the potential of a random Schwartz distribution. Analogous results for fractional derivatives of self-intersection local times in R1 and R2 are also discussed.
Keywords
- Brownian Motion
- Local Time
- Fractional Derivative
- Quadratic Variation
- Dirichlet Process
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Jay Rosen: This research was supported, in part, by grants from the National Science Foundation and PSC-CUNY.
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© 2005 Springer-Verlag Berlin/Heidelberg
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Rosen, J. (2005). Derivatives of Self-intersection Local Times. In: Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics, vol 1857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31449-3_18
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DOI: https://doi.org/10.1007/978-3-540-31449-3_18
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23973-4
Online ISBN: 978-3-540-31449-3
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