On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion

2025/10/01 by Das, Kaustav, Markowsky, Gregory, Wu, Binghao +1
#60G22 #60J55 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.00456

Abstract

We give the correct condition for existence of the k-th derivative of the intersection local time for fractional Brownian motion, which was originally discussed 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]. We also show that the k-th derivative of the intersection and self-intersection local times of fractional Brownian motion are exponentially integrable for certain parameter values. In addition, we show convergence in distribution when the existence condition is violated for the k-th derivative of self-intersection local time of fractional Brownian motion under scaling.

Citations

• Exact convergence rates to derivatives of local time for some self-similar Gaussian processes
• On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes
• Central limit theorems for the derivatives of self-intersection local time for d-dimensional Brownian motion
• Derivatives of local times for some Gaussian fields II
• Higher order derivative of self-intersection local time for fractional Brownian motion
• Derivatives of local times for some Gaussian fields
• Higher-order derivative of intersection local time for two independent fractional Brownian motions
• Functional limit theorem for the self-intersection local time of the\n fractional Brownian motion
• Asymptotic properties of the derivative of self-intersection local time\n of fractional Brownian motion
• Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency
• Holder continuity and occupation-time formulas for fBm self-intersection local time and its derivative
• Integral representation of renormalized self-intersection local times
• Intersection local time for two independent fractional Brownian motions
• Renormalization and convergence in law for the derivative of intersection local time in R²
• Renormalized self-intersection local time for fractional Brownian motion
• Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm
• Brownian intersection local times: Exponential moments and law of large masses

Related

• On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes
• Higher order derivative of self-intersection local time for fractional Brownian motion
• Derivative of self-intersection local time for multidimensional fractional Brownian motion
• Higher-order derivative of intersection local time for two independent fractional Brownian motions
• Asymptotic Properties of the Derivative of Self-Intersection Local Time of Multidimensional Fractional Brownian Motion