Linear parabolic equation with Dirichlet white noise boundary conditions

TL;DR

This paper investigates linear parabolic equations with white noise boundary conditions, establishing well-posedness, smoothing, and stability properties, and reformulating the problem as an evolution equation in weighted spaces.

Contribution

It introduces a framework for analyzing parabolic equations with white noise boundary data using weighted $L^p$-spaces and proves existence of Markovian solutions.

Findings

Heat kernel satisfies Gaussian estimates related to boundary distance.

The operator generates a $C_0$-semigroup with smoothing and stability properties.

Existence of Markovian solutions for the stochastic boundary value problem.

Abstract

We study inhomogeneous Dirichlet boundary value problems associated to a linear parabolic equation $\frac{du}{dt}=Au$ with strongly elliptic operator $A$ on bounded and unbounded domains with white noise boundary data. Our main assumption is that the heat kernel of the corresponding homogeneous problem enjoys the Gaussian type estimates taking into account the distance to the boundary. Under mild assumptions about the domain, we show that $A$ generates a $C_0$-semigroup in weighted $L^p$-spaces where the weight is a proper power of the distance from the boundary. We also prove some smoothing properties and exponential stability of the semigroup. Finally, we reformulate the Cauchy-Dirichlet problem with white noise boundary data as an evolution equation in the weighted space and prove the existence of Markovian solutions.