Quadratic variations for the fractional-colored stochastic heat equation

Abstract

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.