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

TL;DR

This paper investigates the limit behavior of the quartic variation of solutions to the semilinear stochastic heat equation driven by space-time white noise, establishing a CLT and asymptotic independence results.

Contribution

It proves a CLT for the quartic variation and viscosity parameter estimator, and analyzes their asymptotic independence using Stein-Malliavin calculus.

Findings

Quartic variation satisfies a Central Limit Theorem.

Viscosity parameter estimator converges in distribution.

Quartic variation and data are asymptotically independent.

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.