Open Access
2014 Quadratic variations for the fractional-colored stochastic heat equation
Soledad Torres, Ciprian Tudor, Frederi Viens
Author Affiliations +
Electron. J. Probab. 19: 1-51 (2014). DOI: 10.1214/EJP.v19-2698


Using multiple stochastic integrals and Malliavin calculus, we analyze the quadratic variations of a class of Gaussian processes that contains the linear stochastic heat equation on $\mathbf{R}^{d}$ driven by a non-white noise which is fractional Gaussian with respect to the time variable (Hurst parameter $H$) and has colored spatial covariance of $\alpha $-Riesz-kernel type. The processes in this class are self-similar in time with a parameter $K$ distinct from $H$, and have path regularity properties which are very close to those of fractional Brownian motion (fBm) with Hurst parameter $K$ (in the heat equation case, $K=H-(d-\alpha )/4$ ). However the processes exhibit marked inhomogeneities which cause naive heuristic renormalization arguments based on $K$ to fail, and require delicate computations to establish the asymptotic behavior of the quadratic variation. A phase transition between normal and non-normal asymptotics appears, which does not correspond to the familiar threshold $K=3/4$ known in the case of fBm. We apply our results to construct an estimator for $H$ and to study its asymptotic behavior.


Download Citation

Soledad Torres. Ciprian Tudor. Frederi Viens. "Quadratic variations for the fractional-colored stochastic heat equation." Electron. J. Probab. 19 1 - 51, 2014.


Accepted: 23 August 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1314.60132
MathSciNet: MR3256876
Digital Object Identifier: 10.1214/EJP.v19-2698

Primary: 60F05
Secondary: 60G18 , 60H05

Keywords: fractional Brownian motion , Hurst parameter , Malliavin calculus , Multiple stochastic integral , Non-central limit theorem , Quadratic Variation , selfsimilarity , statistical estimation , Stochastic heat equation

Vol.19 • 2014
Back to Top