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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Mathematical Reviews number (MathSciNet)

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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  • [1] S. Albeverio, Y. Hu and X.Y. Zhou. A remark on non-smoothness of the self-intersection local time of planar Brownian motion. Statist. Probab. Lett. 32 (1997) 57–65.
  • [2] P. Billingsley. Convergence of Probability Measures, 2nd edition. John Wiley & Sons, Inc., New York, 1999.
  • [3] S. Darses, I. Nourdin and D. Nualart. Limit theorems for nonlinear functionals of Volterra processes via white noise analysis. Bernoulli 16 (2010) 1262–1293.
  • [4] Y. Hu. On the self-intersection local time of Brownian motion – via chaos expansion. Publ. Mat. 40 (1996) 337–350.
  • [5] Y. Hu and D. Nualart. Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33 (2005) 948–983.
  • [6] P. Imkeller, V. Pérez-Abreu and J. Vives. Chaos expansions of double intersection local time of Brownian motion in ${\mathbb{R}}^{d}$ and renormalization. Stochastic Process. Appl. 56 (1995) 1–34.
  • [7] P. Jung and G. Markowsky. On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion. Stochastic Process. Appl. 124 (2014) 3846–3868.
  • [8] I. Nourdin, D. Nualart and C. Tudor. Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 1055–1079.
  • [9] I. Nourdin and G. Peccati. Normal Approximations with Malliavin Calculus. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2012.
  • [10] D. Nualart. The Malliavin calculus and related topics. In Probability and Its Applications. Springer-Verlag, Berlin, 2006.
  • [11] D. Nualart and G. Peccati. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005) 177–193.
  • [12] G. Peccati and C. Tudor. Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII 1857 247–262, 2005.
  • [13] J. Rosen. The intersection local time of fractional Brownian motion in the plane. J. Multivariate Anal. 23 (1987) 37–46.
  • [14] S. Varadhan. Appendix to Euclidean quantum field theory, by K. Symanzik. In Local Quantum Theory, R. Jost (Ed.). Academic Press, New York, 1969.
  • [15] M. Yor. Renormalisation et convergence en loi pour les temps locaux d’intersection du mouvement brownien dans ${\mathbb{R}}^{3}$. In Séminaire de probabilités, XIX 350–365. Lecture Notes in Math. 1123. Springer, Berlin, 1985.