Harmonic Functions And Linear Elliptic Dirichlet Problems With Random Boundary Values--Stochastic Extensions Of Some Classical Theorems And Estimates

2020/05/06 by Miller, Steven D
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2005.02556

Abstract

Let ψ:D→R be a harmonic function such that Δψ(x)=0 for all x\inD⊂Rⁿ. There are then many well-established classical results:the Dirichlet problem and Poisson formula, Harnack inequality, the Maximum Principle, the Mean Value Property etc. Here, a 'noisy' or random domain is one for which there also exists a classical scalar Gaussian random field (GRF) \mathscrJ(x) defined for all x∈D or x∈∂ D with respect to a probability space [Ω,F,I P]. The GRF has vanishing mean value E[ [\mathscrJ(x)] ] = 0 and a regulated covariance E[ [\mathscrJ(x) ⊗ \mathscrJ(y)] ] = αJ(x,y;ξ) for all (x,y)∈D and/or (x,y)∈\partialD, with correlation length ξ and E[ [\mathscrJ(x) ⊗ \mathscrJ(x)] ] = α

Related

• Systems of stochastic Poisson equations: hitting probabilities
• A higher-order large-scale regularity theory for random elliptic\n operators
• Boundary estimates and Green function's expansion for elliptic systems with random coefficients
• Random weighted Sobolev inequalities on ℝ^d and application to Hermite functions
• Stochastic Partial Differential Equations, Space-time White Noise and Random Fields