Large deviations for processes on half-line

We consider a sequence of processes Xn(t) defined on the half-line 0 ≤ t < ∞, n = 1, 2, . . .. We give sufficient conditions for Large Deviation Principle (LDP) to hold in the space of continuous functions with metric ρκ(f, g) = sup t≥0 |f(t) − g(t)|1 + t 1+κ, κ ≥ 0. LDP is established for Random Walks and Diffusions defined on the half-line. LDP in this space is “more precise" than that with the usual metric of uniform convergence on compacts