Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Arturo Jaramillo and David Nualart

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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

\[I_{T}^{\varepsilon}=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})\,ds\,dt,\] where $p_{\varepsilon}(x)$ is the heat kernel. We prove that the process $\{I_{T}^{\varepsilon}-\mathbb{E}[I_{T}^{\varepsilon}]\}_{T\geq0}$, 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.


Soit $\{B_{t}\}_{t\geq0}$ un mouvement brownien fractionnaire $d$-dimensionel avec paramètre de Hurst $0<H<1$, où $d\geq2$. On considère l’approximation du temps local d’auto-intersection du processus $B$, défini comme

\[I_{T}^{\varepsilon}=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})\,ds\,dt,\] où $p_{\varepsilon}(x)$ est le noyau de la chaleur. Nous démontrons que le processus $\{I_{T}^{\varepsilon}-\mathbb{E}[I_{T}^{\varepsilon}]\}_{T\geq0}$, rééchelonné avec une normalisation convenable, converge en loi vers un mouvement brownien multiplié par une constante si $\frac{3}{2d}<H\leq\frac{3}{4}$ et vers une somme de processus de Hermite indépendants multipliée par une constante si $\frac{3}{4}<H<1$, dans l’espace $C[0,\infty)$, muni de la topologie de la convergence uniforme sur les compacts.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 480-527.

Received: 2 February 2017
Revised: 2 December 2017
Accepted: 5 February 2018
First available in Project Euclid: 18 January 2019

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Zentralblatt MATH identifier

Primary: 60G05: Foundations of stochastic processes 60H07: Stochastic calculus of variations and the Malliavin calculus 60G15: Gaussian processes 60F17: Functional limit theorems; invariance principles

Fractional Brownian motion self-intersection local time Wiener chaos expansion central limit theorem


Jaramillo, Arturo; Nualart, David. Functional limit theorem for the self-intersection local time of the fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 480--527. doi:10.1214/18-AIHP889.

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