SpecialityUniversità dell'Aquila, IT; University of Reading, UK

Linear Response for dissipative SPDEsAn easy to apply framework to establish response theory is presented which is tailored for non-linear dissipative SPDEs of Navier-Stokes type. With response theory we mean in this context the following: one considers a dynamical system whose dynamical law depends on a parameter (here given by an SPDE where the parameter is in the forcing) and we say that one has a response theory if one can show a regularity in the dependence of the invariant measure on the parameter (here differentiability in weak topology or H\”older continuity in a Wasserstein metric). The method presented allows to consider observables which are not necessarily differentiable. This is useful, because the main hypothesis to be checked in order to obtain a response theory is a spectral gap result for the Markov semigroup associated with the dynamics. Hairer, Mattingly, Scheutzow and Butkovsky, Kulik, Scheutzow developed a framework to establish spectral gap results for Markov semigroup, which can be comparably easily verified in the case of SPDEs, but works only for a class of observables which are not necessarily differentiable. Finally, for expository reasons, in the case of 2D Navier-Stokes we will demonstrate how the two frameworks can be combined to establish response theory. For 2D Navier-Stokes this has been done already by Hairer and Majda, but using a different type of spectral gap result based on Malliavin calculus and differentiable observables. As physically, the response theory justifies the study of climate predictions via different forcing scenarios, our main application for this framework was to prove response theory for the 2 layer quasi-geostrophic model which is a key-model in geosciences to study atmosphere and ocean dynamics. Because of time constrains we cannot go into this in this talk. This work is jointly with Jochen Br\”ocker (University of Reading) and Giulia Carigi (Universit\’a dell’Aquila)