Consider a Feller semigroup of operators \(p_ t\) on C(X), the space of bounded continuous functions on a metric space X. The authors investigate the possibility of stating the Donsker and Varadhan limit relation \[ (A)\quad \lim_{t\uparrow \infty}t^{-1}\log p_ t1(x)=-\lambda_ 0 \] without assuming compactness of X. It is shown that (A) holds under a certain condition on the generator L of the semigroup \(p_ t\). Generally, ''\(\geq ''\) is proved for \(''\lim_{t\uparrow \infty}''\) replaced by \(''\underline{\lim}_{t\to \infty}''\) in (A). In the case of a symmetric Markov process \(p_ t\), two sufficient conditions for (A) are stated. It is interesting to see that (A) can fail even in the latter case. A complete answer is given for the one- dimensional diffusion, in particular for a time changed Brownian motion on (0,1).