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Article Dans Une Revue Bernoulli Année : 2023

Characterization of the second order random fields subject to linear distributional PDE constraints

Résumé

Let $L$ be a linear differential operator acting on functions defined over an open set $\mathcal{D}\subset \mathbb{R}^d$. In this article, we characterize the measurable second order random fields $U = (U(x))_{x\in\mathcal{D}}$ 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 $(U(x))_{x\in\mathcal{D}}$ as well as over its sample paths in order to make sense of the PDE $L(U_{\omega})=0$. In view of Gaussian process regression (GPR) applications, we show that when $(U(x))_{x\in\mathcal{D}}$ is a Gaussian process (GP), the sample paths of $(U(x))_{x\in\mathcal{D}}$ 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.
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Dates et versions

hal-03770715 , version 1 (06-09-2022)
hal-03770715 , version 2 (16-01-2023)

Identifiants

Citer

Iain Henderson, Pascal Noble, Olivier Roustant. Characterization of the second order random fields subject to linear distributional PDE constraints. Bernoulli, 2023, 29 (4), pp.3396-3422. ⟨10.3150/23-BEJ1588⟩. ⟨hal-03770715v2⟩
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