Second Order PDEs with Dirichlet White Noise Boundary Condition

TL;DR

This paper develops a framework for analyzing second order PDEs with random Dirichlet boundary conditions driven by white noise, establishing existence, uniqueness, regularity, and boundary blow-up rates of solutions.

Contribution

It introduces a new approach for studying PDEs with boundary white noise, covering various noise types and providing detailed solution properties.

Findings

Proved existence and uniqueness of weak solutions.

Established solutions are smooth inside the domain.

Estimated boundary blow-up rates for solutions.

Abstract

In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises including Wiener and fractional Wiener space time white noise, homogeneous noise and L\'evy noise is considered.