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.
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