Nonhomogeneous Noise and Q-Wiener Processes on Bounded Domains

Abstract

In this article, we comment on the well known connection between noise and Q-Wiener processes. In physical applications, noise is usually given as a generalized Gaussian process and all assumptions are formulated in terms of the correlation functional. In contrast, the mathematical theory of stochastic partial differential equations on bounded domains is in many cases formulated with Q-Wiener processes. Our aim is to relate frequently made assumptions on the covariance operator Q to assumptions on the correlation function of the noise process, which is the kernel of Q. One of our main results gives necessary and sufficient conditions when the Wiener process is given as a series expansion in terms of eigenfunctions of the Laplacian.

Key Words:

• Bounded domains
• Generalized stochastic processes
• Non-homogeneous noise
• Q-Wiener process
• White noise

Mathematics Subject Classification:

• 60G20
• 60G60

aCKNOWLEDGMENTS

The author was supported by the DFG Forschungsstipendium BL535/5-1. He would like to thank Christoph Gugg for many discussions, and the referee for carefully reading the manuscript.