High-frequency analysis of parabolic stochastic PDEs with multiplicative noise

TL;DR

This paper studies the stochastic heat equation with multiplicative Gaussian noise, proving a central limit theorem for power variations and revealing unexpected bias cancellation effects due to noise regularity.

Contribution

It establishes a central limit theorem for power variations of the solution to a stochastic heat equation with spatially correlated noise, highlighting novel bias cancellation phenomena.

Findings

Central limit theorem for power variations

No asymptotic bias despite low regularity

Cancellation effects explain bias absence

Abstract

We consider the stochastic heat equation driven by a multiplicative Gaussian noise that is white in time and spatially homogeneous in space. Assuming that the spatial correlation function is given by a Riesz kernel of order $\alpha \in (0,1)$, we prove a central limit theorem for power variations and other related functionals of the solution. To our surprise, there is no asymptotic bias despite the low regularity of the noise coefficient in the multiplicative case. We trace this circumstance back to cancellation effects between error terms arising naturally in second-order limit theorems for power variations.