Quartic variation of the solution to the semilinear stochastic heat equation: limit behavior and asymptotic independence with respect to the data

TL;DR

This work analyzes the temporal quartic variation of the mild solution to the semilinear stochastic heat equation driven by space–time white noise, proving a central limit theorem for the quartic variation and for the associated viscosity parameter estimator. Using a refined Stein–Malliavin framework, it derives quantitative rates in Wasserstein distance for the convergence to Gaussian limits and studies asymptotic independence between these time-based statistics and the data used to construct them, in both linear and semilinear settings. The results show that, under suitable growth conditions for the observed data subset, the quartic variation (and the estimator) become asymptotically independent of the data, with the nonlinear drift altering convergence rates in the semilinear case. The paper also introduces weaker notions of independence (K- and DK-type) to accommodate the estimator and provides explicit bounds in terms of the observation scheme, contributing to statistical inference for SPDEs via power variations.

Abstract

This work concerns the limit behavior of the quartic variation (i.e., the power variation of order four) with respect to the time variable of the solution to the semilinear stochastic heat equation with space-time white noise. In a first step, we prove that this sequence satisfies a Central Limit Theorem and we deduce a similar result for the viscosity parameter estimator associated with the quartic variation. Then, by using a recent variant of the Stein-Malliavin calculus, we analyze the asymptotic independence between the quartic variation (as well as the associated viscosity parameter estimator) and the data used to construct it.

Theorems & Definitions (31)

• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Proposition 3.4
• proof
• Definition 4.1
• Theorem 4.2