The large deviations of the generalised Parabolic Anderson Model (gPAM) in the small noise limit have been quantified by Hairer and Weber. A classical result due to Varadhan therefore characterises the asymptotics of so-called Laplace functionals on a logarithmic scale. In this talk, I will explain how to generalise the log-asymptotics to a precise asymptotic expansion of arbitrary order.
The talk consists of three parts:
(1) I will motivate the problem, review the above-mentioned results, and explain how our work fits into the existing literature.
(2) In a very simple toy example, I will present an (almost) complete proof to highlight some of the core features of our argument.
(3) I will explain how to generalise the methodology to the regularity structures setting.
Parts (1) and (2) do not require any familiarity with Hairer’s regularity structures but only basic knowledge of large deviations.
Part (3) uses Hairer’s setting but I will present a “dictionary” to translate the encountered objects back to the toy example whenever possible. This part also contains a Taylor expansion of gPAM in the noise intensity which might be of independent interest.
Based on joint work with Peter Friz, available at https://doi.org/10.1016/j.jfa.2022.109446,
and the recent preprint article arXiv:2202.03358.
