On large deviations for the parabolic Anderson model

Abstract

The focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic Anderson model above the mean versus large deviations below the mean. The functionals we treat are the solution u(x, t) to the spatially discrete parabolic Anderson model and a functional A n which is used in analyzing the a.s. Lyapunov exponent for u(x, t). Both satisfy a "law of large numbers", with lim t→∞ 1 t log u(x, t) = λ(κ) and lim n→∞ A n n = α. We then think of αn and λ(κ)t as being the mean of the respective quantities A n and log u(t, x). Typically, the large deviations for such functionals exhibits a strong asymmetry; large deviations above the mean take on a different order of magnitude from large deviations below the mean. We develop robust techniques to quantify and explain the differences.

Properties of the Parabolic Anderson Model and the Anderson Polymer Model

“…In [73], a similar phenomenon was pointed out for the functional , defined in (93). The limit (95) holds as in the case of the first passage percolation.…”

“…The concentration results are thus closely related to large deviation results established in [39,[71][72][73]. For example, by means of a block argument, it was shown in [39] that for every > 0 there is a ( ) > 0 such that…”

Overlaps and pathwise localization in the Anderson polymer model

“…The model is also very appealing, with time-space iid environment (the Brownian increments) replaced by the configuration of an interacting particle system [16], modeling a chemical reaction with moving catalysers. It has non trivial large deviation properties [13], as a particular random growth model. The one-dimensional totally asymmetric case, where the walker only jumps to the right, has a lower complexity than the symmetric case, as shown by the computations of annealed Lyapunov exponents [4]; In this case an explicit solution was given in [23], with the strongly asymmetric case as a small perturbation [21].…”

An ergodic theorem of a parabolic Anderson model driven by Lévy noise

“…On the other hand, the rigorous analysis to some real world phenomenon, for example, intermittency effect has provided mathematical challenges, and often requires some new mathematical ideas and techniques. References [13,14,17] and the survey [12] provided the recent interesting progress on the mathematical aspects of parabolic Anderson model.…”