On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes

We show that the derivative of the intersection and self-intersection local times of alpha-stable processes are exponentially integrable for certain parameter values. This includes the Brownian motion case. We also discuss related results present in the literature for fractional Brownian motion, and in particular give a counter-example to a result in Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection local time for two independent fractional Brownian motions, Journal of Theoretical Probability 32, (2019), pp. 1190-1201 related to this question.