Isotropic diffeomorphisms: solutions to a differential system for a deformed random fields study

TL;DR

This paper solves a differential system related to isotropic deformations of random fields, showing that weak and strong notions of isotropy are equivalent through explicit mathematical resolution.

Contribution

It provides an explicit solution to a differential system arising from isotropic deformations of random fields, linking weak and strong isotropy notions.

Findings

Weak and strong isotropy notions coincide for deformed random fields.

Explicit differential system solution confirms geometrical properties of isotropic deformations.

The probability framework connects geometrical deformations with stochastic properties.

Abstract

This Note presents the resolution of a differential system on the plane that translates a geometrical problem about isotropic deformations of area and length. The system stems from a probability study on deformed random fields [J.Fournier '17], which are the composition of a random field with invariance properties defined on the plane with a deterministic diffeomorphism. The explicit resolution of the differential system allows to prove that a weak notion of isotropy of the deformed field, linked to its excursion sets, in fact coincides with the strong notion of isotropy. The present Note first introduces the probability framework that gave rise to the geometrical issue and then proposes its resolution.