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Derivatives of self-intersection local times. (English) Zbl 1063.60110

Émery, Michel (ed.) et al., 38th seminar on probability. Dedicated Jacques Azéma on the occasion on his 65th birthday. Berlin: Springer (ISBN 3-540-23973-1/pbk). Lecture Notes in Mathematics 1857, 263-281 (2005).
Summary: We show that the renormalized self-intersection local time \(\gamma_t(x)\) for both the Brownian motion and symmetric stable process in \(R^1\) 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 Dririchlet process is the potential of a random Schwartz distribution. Analogous results for fractional derivatives of self-intersection local times in \(R^1\) and \(R^2\) are also discussed.
For the entire collection see [Zbl 1055.60001].

MSC:

60J55 Local time and additive functionals
60G52 Stable stochastic processes
60J65 Brownian motion
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