Quadratic variations for the fractional-colored stochastic heat equation∗

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 R 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 α-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 − α)/4 ). However the processes exhibit marked inhomogeneities which cause naïve 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.