A comparison of homogenization and large deviations, with applications to wavefront propagation

Quadratic and rate-independent limits for a large-deviations functional

“…The two limiting processes, n → ∞ and β → {0, ∞} can be interpreted as differently scaled combinations of two limiting processes: (a) the small-noise limit, (b) the limit of vanishing microstructure. In the case of SDEs [6,18,22], three regimes have been identified, corresponding to 'microstructure smaller than noise', 'noise smaller than microstructure', and the critical case. In the first of these, 'microstructure smaller than noise', a behaviour arises that resembles the quadratic limit of this paper, in which the microstructure is effectively swamped by the noise.…”

Limit of the Solution of a PDE in the Degenerate Case

“…Diédhiou and Manga in [2] studied the limit of ,  with a nondegenerate condition of the matrix. In Freidlin & Sowers [3], three cases are considered, with the assumption that the matrix is non, but we formulate here a hypoellipticity condition of. Since the parameter  (homogeneization parameter) decreases quickly than (large deviations principle parameter) to zero we must homogenize first and apply the large deviations principle.…”

Maximum likelihood estimation for small noise multiscale diffusions

“…When both parameters ǫ and δ go to zero together, then we need to consider three different regimes depending on how fast ǫ goes to zero relative to δ: We mention here that asymptotic problems for models like (1.1) have a long history in the mathematical literature. We refer the interested reader to classical manuscripts such as [4,13,24] for averaging and homogenization results and to the more recent articles [7,12] for large deviations results and [9,8] for importance sampling results on related rare event estimation problems.…”