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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

\[I_{T}^{\varepsilon}=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})\,ds\,dt,\] 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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802407

Digital Object Identifier
doi:10.1214/18-AIHP889

Mathematical Reviews number (MathSciNet)
MR3901653

Zentralblatt MATH identifier
07039777

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

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

Citation

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. https://projecteuclid.org/euclid.aihp/1547802407


Export citation

References

  • [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.