Wick Ordering and Kinetic Energy Renormalization for L\'evy White Noise Fields

Let $S={\mathbb{T}}^d$ be a torus and $\mu$ the probability distribution of a L'evy white noise field $x:S\to \mathbb{R}$. Using projective limit measures we address the problem of making sense of ${\mathrm{e}}^{-T(x)}$, where $T(x)={\int }_S\mid \mathrm{\nabla }x(s){\mid }^2\mathrm{d}s$ is the kinetic energy, as a function in $L^1(\mu )$. We start by making sense of $T(x)$ itself as a sort of distribution, which is achieved by a generalization of Wick ordering. Then we specify to the case of a $\mathrm{\Gamma }$ field, finding that Wick ordering does not eliminate all divergences. Higher order renormalization would be required, but the model seems to be non-renormalizable.