Quadratic variations for the fractional-colored stochastic heat equation. (English) Zbl 1314.60132

Summary: 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 \(\mathbb{R}^{d}\) 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.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
60G18 Self-similar stochastic processes
60G22 Fractional processes, including fractional Brownian motion
60F05 Central limit and other weak theorems
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