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High-frequency analysis of parabolic stochastic PDEs with multiplicative noise

Carsten Chong

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 $α\in (0,1)$, 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.

High-frequency analysis of parabolic stochastic PDEs with multiplicative noise

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 , 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.

Paper Structure

This paper contains 9 sections, 27 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Set-up and main result
  3. Overview of the proof of Theorem \ref{['thm:CLT']}
  4. Martingale approximations and the core CLT
  5. Approximating the conditional expectation
  6. The bad terms
  7. Putting pieces together: the final step of the proof
  8. Details of the proof of Theorem \ref{['thm:CLT']}
  9. Auxiliary results

Key Result

Theorem 2.1

Suppose that $f$ is continuous with at most polynomial growth [i.e., there is $p\in(0,\infty)$ such that ${|f(z)|}\lesssim 1+|z|^p$ for all $z\in\mathbb{R}^{K\times L}$, where $|z|$ denotes the Euclidean norm of the matrix $z$ when viewed as a vector in $\mathbb{R}^{KL}$]. Further assume that the in Here, $\mu_f\colon [0,\infty)^K \to \mathbb{R}^M$, $(w_1,\dots,w_K)\mapsto \mathbb{E}[f(Z)]$, where

Theorems & Definitions (54)

  • Theorem 2.1: Law of large numbers
  • Theorem 2.2: Central limit theorem
  • Remark 2.3
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 44 more