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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

Kaustav Das, Gregory Markowsky, Binghao Wu, Qian Yu

TL;DR

This work analyzes the derivative of intersection local time for fractional Brownian motion, establishing the correct $L^p$ existence condition for the $k$-th derivative under $2|k|H+Hd<2$ and proving exponential integrability for $0<\beta<\frac{1}{|k|+|k|H+dH}$. It extends these results to the self-intersection local time via the DSLT and provides exponential integrability under comparable regimes, along with scaling limit theorems when the existence criteria fail, including explicit planar and spatial CLTs with variances expressed through Beta-functions. The authors develop sharp covariance-structure estimates using local nondeterminism, Malliavin calculus, and chaos expansions to obtain these results for fractional Brownian motion with same $H$. This advances understanding of the stochastic regularity and limit behavior of derivatives of intersection local times in non-Markovian Gaussian settings, with implications for related polymer and self-interacting models.

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.

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

TL;DR

This work analyzes the derivative of intersection local time for fractional Brownian motion, establishing the correct existence condition for the -th derivative under and proving exponential integrability for . It extends these results to the self-intersection local time via the DSLT and provides exponential integrability under comparable regimes, along with scaling limit theorems when the existence criteria fail, including explicit planar and spatial CLTs with variances expressed through Beta-functions. The authors develop sharp covariance-structure estimates using local nondeterminism, Malliavin calculus, and chaos expansions to obtain these results for fractional Brownian motion with same . This advances understanding of the stochastic regularity and limit behavior of derivatives of intersection local times in non-Markovian Gaussian settings, with implications for related polymer and self-interacting models.

Abstract

We give the correct condition for existence of the -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 -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 -th derivative of self-intersection local time of fractional Brownian motion under scaling.

Paper Structure

This paper contains 5 sections, 28 theorems, 189 equations.

Table of Contents

  1. Introduction and main results
  2. Proofs of existence and exponential integrability, Theorems \ref{['theorem of existence of DILT of fbm']}, \ref{['theorem of EI of DILT of fbm']}, and \ref{['theorem of EI of DSLT of fbm']}.
  3. Proof of \ref{['thm1', 'thm2']}
  4. Malliavin calculus preliminaries
  5. Technical lemmas

Key Result

Theorem 1.1

Let $k=(k_{1},\dots,k_{d}) \in \mathbb{N}^{d}$ and $|k|=k_{1}+\cdots+k_{d}$. Suppose that $2|k|H+Hd<2$. Then $\alpha_{\epsilon}^{(k)}$ defined in molifiled version converges in $L^{n}(\Omega)$ as $\epsilon\to 0$ for all $n \geq 1$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • proof : Proof of \ref{['theorem of existence of DILT of fbm', 'theorem of EI of DILT of fbm']}
  • proof : Proof of Theorem \ref{['theorem of EI of DSLT of fbm']} (sketch)
  • ...and 43 more