On the semigroup generated by the renormalized Nelson Hamiltonian

We consider the renormalized Nelson model at a fixed total momentum P: $H_{\mathrm{ren}}(P)$; the Hamiltonian $H_{\mathrm{ren}}(P)$ is defined through an infinite energy renormalization. We prove that $e^{?\beta H_{\mathrm{ren}}(P)}$ is positivity improving for all $P\in {\mathbb{R}}^3$ and $\beta >0$ in the Fock representation.