Large deviations for interacting multiscale particle systems

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

“…Stated more concretely, [73] shows 23) in the classical sense must be given by (24). Note that while [73] restricts to the case of Markovian feedback controls for the infimization problem (9), we know thanks to the convexity of the running cost in the definition of the rate function that the restricted infimization problem is equivalent to the full one (see Proposition 5.8 in [9]).…”

“…Proof. The result follows from Theorem 3.1 in [16] (see also Theorem 5.1 in [45]) by applying the contraction principle to pose the rate function on C([s, T ]; P(R d )) rather than P(C([s, T ]; R d )) as in Proposition 5.6 of [9]. □…”

“…Indeed, via the equivalence with the small-noise regime as discussed in Remark 3.2, such a choice of control can be seen to solve the minimization problem (9) with…”

“…Assumption 1.1 is taken directly from [16] with the exception of (A4), which is a mildly less restrictive form of weak uniqueness under which the arguments from [16] still hold (see Lemma 3.4 in [15] or Definition 3.6 in [9]). For verifiable conditions on the data of the problem under which (A4) holds, from Appendix C in [45] we have that: are enough to imply (A2)-(A5).…”