By Feng J., Kurtz T.

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Proof. Let Km = {x : I(x) ≤ m}. Then F is continuous at each point in Km and hence F (Km ) is compact in S . Exponential tightness for {Yn } follows since for each η > 0, there exists > 0 such that x ∈ Km implies F (x) ∈ F (Km )η and hence {Yn ∈ / F (Km )η } ⊂ {Xn ∈ / Km } and lim sup 1 log P {Yn ∈ / F (Km )η } n 1 log P {Xn ∈ / Km } n inf c I(x) ≤ −m. ≤ lim sup ≤ − x∈(Km ) 48 3. LDP AND EXPONENTIAL TIGHTNESS Let I denote the rate function for {Yn } (or a subsequence). Suppose F (x) = y. If I(x) < ∞, then for each > 0, there exists δ > 0 such that F (Bδ (x)) ⊂ B (y).

2) Verify exponential tightness. 10). 6). Alternatively, one can avoid verifying the compact containment condition by compactifying the state space and verifying the large deviation principle in the compactified space. 11). (3) Verify the comparison principle for the limiting operator H (or the pair (H† , H‡ )). The comparison principle asserts a strong form of uniqueness for the equation (I − αH)f = h. If the comparison principle holds for a sufficiently large class of functions h, then one can conclude that the nonlinear semigroups {Vn } converge.

A short version is summarized in Chapter 2. Further generalizations of these results are also given. For instance, we discuss situations where certain functionals of the Markov processes satisfy a large deviation principle, while the full processes may not. We also discuss the use of a general notion of convergence of test functions and operators, to handle processes with multiple scales. 3). Verification is an analytic issue and often gives the impression of being rather involved and disconnected from the probabilistic large deviation problems.