Linear parabolic equation with Dirichlet white noise boundary conditions

2021/09/12 by Beniamin Gołdys, Goldys, Beniamin, Szymon Peszat +1 · 1 citation
Computer Science · Engineering · Mathematics · #60G15 #60H15 #60J99 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2109.05428

openalex publication_date 2021/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study inhomogeneous Dirichlet boundary value problems associated to a linear parabolic equation (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 C₀-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.

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