Behavior in strong fields of Euclidean gauge theories

We represent the Green's functions and functional determinants in Euclidean gauge theories by means of the Feynman-Kac formula. Such a representation is useful in the investigation of the determinant as a functional of the gauge field. As a first application of these formulas we show (for the Abelian gauge field A) that lndet(-∇A²+m²)^-1=c∫F²lnF²+r(∂F²) in four dimensions, where $c>0$ and the remainder r(∂F²) is bounded by a quadratic form of ∂_ρF_μν. This behavior for large F is crucial for the stability of quantum models of gauge theories.