Second Order PDEs with Dirichlet White Noise Boundary Conditions

Zdzislaw Brzezniak; Ben Goldys; Szymon Peszat; Francesco  Russo

doi:10.1007/s00028-014-0246-2

Second Order PDEs with Dirichlet White Noise Boundary Conditions

Zdzislaw Brzezniak, Ben Goldys, Szymon Peszat, Francesco Russo

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we study inhomogeneous Dirichlet boundary problems associated to the Poisson and heat equations on bounded and unbounded domains with smooth boundary and random boundary data. The main novelty of this work is a convenient framework for the analysis of 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. We also 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 are considered.

Original language	English
Pages (from-to)	1-26
Number of pages	26
Journal	Journal of Evolution Equations
Volume	15
Issue number	1
Early online date	2 Sept 2014
DOIs	https://doi.org/10.1007/s00028-014-0246-2
Publication status	Published - 2015

Keywords

Heat equation
Poisson equation
Dirichlet problem
White noise
Boundary conditions
Fractional Brownian motion.

Access to Document

10.1007/s00028-014-0246-2

Cite this

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title = "Second Order PDEs with Dirichlet White Noise Boundary Conditions",

abstract = "In this paper, we study inhomogeneous Dirichlet boundary problems associated to the Poisson and heat equations on bounded and unbounded domains with smooth boundary and random boundary data. The main novelty of this work is a convenient framework for the analysis of 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. We also 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{\'e}vy noise are considered.",

keywords = " Heat equation, Poisson equation, Dirichlet problem , White noise, Boundary conditions, Fractional Brownian motion.",

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T1 - Second Order PDEs with Dirichlet White Noise Boundary Conditions

AU - Brzezniak, Zdzislaw

AU - Goldys, Ben

AU - Peszat, Szymon

AU - Russo, Francesco

PY - 2015

Y1 - 2015

N2 - In this paper, we study inhomogeneous Dirichlet boundary problems associated to the Poisson and heat equations on bounded and unbounded domains with smooth boundary and random boundary data. The main novelty of this work is a convenient framework for the analysis of 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. We also 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 are considered.

AB - In this paper, we study inhomogeneous Dirichlet boundary problems associated to the Poisson and heat equations on bounded and unbounded domains with smooth boundary and random boundary data. The main novelty of this work is a convenient framework for the analysis of 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. We also 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 are considered.

KW - Heat equation

KW - Poisson equation

KW - Dirichlet problem

KW - White noise

KW - Boundary conditions

KW - Fractional Brownian motion.

U2 - 10.1007/s00028-014-0246-2

DO - 10.1007/s00028-014-0246-2

M3 - Article

VL - 15

SP - 1

EP - 26

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

IS - 1

ER -