Quadratic variations for the fractional-colored stochastic heat equation

Soledad Torres (Universidad de Valparaíso)
Ciprian A. Tudor (Université de Lille 1)
Frederi G. Viens (Purdue University)


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.

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Pages: 1-51

Publication Date: August 23, 2014

DOI: 10.1214/EJP.v19-2698


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