Let \(T^d=R^d/Z^d\) be a torus and \(\{P_x: x\in T^d\}\) probability measures on \(C([0,\infty): T^d)\) given by the infinitesmal generator \(L_0= {1 \over 2} \Delta+b\cdot\nabla\), for \(b\) a continuous function from \(T^d\) to \(R^d\). Previously E. Bolthausen, J.-D. Deuschel and Y. Tamura [Ann. Probab. 23, No. 1, 236-267 (1995; Zbl 0838.60023)] established, for a mapping \(\Phi\) from signed measures with finite total variation to \(R\), the large deviation asymptotics of \( E^{P_z}[ \exp(T\Phi\int_0^T\delta_{X_t} dt)\mid X_T=y] \) up to a factor of \((1+o(1))\). That result required a nuclearity assumption on the second Fréchet derivative of \(\Phi\), a condition that is removed in the paper under review.