Functional limit theorem for the self-intersection local time of the fractional Brownian motion

TL;DR

This paper establishes a functional limit theorem for the self-intersection local time of fractional Brownian motion, showing convergence to Brownian motion or Hermite processes depending on the Hurst parameter.

Contribution

It provides the first rigorous proof of the limit behavior of self-intersection local times for fractional Brownian motion across different Hurst regimes.

Findings

Convergence to Brownian motion for 3/2d < H ≤ 3/4.

Convergence to Hermite processes for 3/4 < H < 1.

Results hold in the space of continuous functions with uniform topology.

Abstract

Let $\{B_{t}\}_{t\geq0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $0<H<1$, where $d\geq2$. Consider the approximation of the self-intersection local time of $B$, defined as \begin{align*} I_{T}^{\varepsilon} &=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})dsdt, \end{align*} where $p_\varepsilon(x)$ is the heat kernel. We prove that the process $\{I_{T}^{\varepsilon}-\mathbb{E}\left[I_{T}^{\varepsilon}\right]\}_{T\geq0}$, rescaled by a suitable normalization, converges in law to a constant multiple of a standard Brownian motion for $\frac{3}{2d}<H\leq\frac{3}{4}$ and to a multiple of a sum of independent Hermite processes for $\frac{3}{4}<H<1$, in the space $C[0,\infty)$, endowed with the topology of uniform convergence on compacts.