Second Order PDEs with Dirichlet White Noise Boundary Condition

Zdzisław Brzeźniak, Ben Goldys, Szymon Peszat, Francesco Russo

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 Lévy noise is considered.

Keywords: Partial differential equations, white noise, boundary conditions,Lévy process, fractional Brownian Motion.

AMS Subject Classification: Primary 60H15;; secondary 35J25; 35K10; 35K51; 60G20.