2023/01/17 by Henderson, Iain, Noble, Pascal, Roustant, Olivier · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2301.06895
Let L be a linear differential operator acting on functions defined over an open set D⊂ ℝd. In this article, we characterize the measurable second order random fields U = (U(x))x\inD whose sample paths all verify the partial differential equation (PDE) L(u) = 0, solely in terms of their first two moments. When compared to previous similar results, the novelty lies in that the equality L(u) = 0 is understood in the sense of distributions, which is a powerful functional analysis framework mostly designed to study linear PDEs. This framework enables to reduce to the minimum the required differentiability assumptions over the first two moments of (U(x))x\inD as well as over its sample paths in order to make sense of the PDE L(Uω)=0. In view of Gaussian process regression (GPR) applications, we show that when (U(x))x\inD is a Gaussian process (GP), the sample paths of (U(x))x\inD conditioned on pointwise observations still verify the constraint L(u)=0 in the distributional sense. We finish by deriving a simple but instructive example, a GP model for the 3D linear wave equation, for which our theorem is applicable and where the previous results from the literature do not apply in general.