A framework to establish response theory for a class of nonlinear stochastic
partial differential equations (SPDEs) is provided. More specifically, it is
shown that for a certain class of observables, the averages of those
observables against the stationary measure of the SPDE are differentiable
(linear response) or, under weaker conditions, locally H\”older continuous
(fractional response) as functions of a deterministic additive forcing. The
method allows to consider observables that are not necessarily differentiable.
For such observables, spectral gap results for the Markov semigroup associated
with the SPDE have recently been established that are fairly accessible. This
is important here as spectral gaps are a major ingredient for establishing
linear response. The results are applied to the 2D stochastic Navier-Stokes
equation and the stochastic two-layer quasi-geostrophic model, an intermediate
complexity model popular in the geosciences to study atmosphere and ocean
dynamics. The physical motivation for studying the response to perturbations in
the forcings for models in geophysical fluid dynamics comes from climate change
and relate to the question as to whether statistical properties of the dynamics
derived under current conditions will be valid under different forcing
scenarios.
