Temporal quadratic and higher order variation for the nonlinear stochastic heat equation and applications to parameter estimation

TL;DR

The paper analyzes a nonlinear stochastic heat equation with fractional Laplacian $- heta(-\Delta)^{α/2}$, $α∈(1,2]$, driven by space-time white noise and Lipschitz diffusion $σ$, focusing on temporal two- and higher-order power variations. It proves that the renormalized quadratic variation $V_{N,x}(u)$ and the power variation of order $\frac{2α}{α-1}$ converge to nontrivial limits, enabling consistent estimation of the anomaly parameter $α$ and the drift parameter $θ$ from observations on $[0,1]$ at a fixed $x$. The approach leverages a comparison with the linear equation, viewing the linear solution as a perturbed fractional Brownian motion, to control increments and derive convergence rates. The results advance statistical inference for nonlinear SPDEs with fractional Laplacians, providing estimators that identify $α$ from temporal data and $θ$ via $V_{N,x}$ and $U_{N,x}$, with extensions to parametric variations via rescaling and asymptotic normality in special cases.

Abstract

We consider the stochastic heat equation which includes a fractional power of the Laplacian of order $α\in (1, 2]$ and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the solution to this equation: the renormalized quadratic variation and the power variation of order $\frac{2α}{α-1}$, both over an equidistant partition of the unit interval. We prove that these two sequences admit nontrivial limits when the mesh of the partition goes to zero. We apply these results to identify certain parameters of the stochastic heat equation.

Theorems & Definitions (15)

• Lemma 1
• Proposition 1
• Lemma 2
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 1
• Lemma 3
• Theorem 2
• Remark 1