Linear and fractional response for nonlinear dissipative SPDEs

TL;DR

This paper develops a response theory framework for nonlinear stochastic PDEs, showing how observables' averages respond to forcing changes, with applications to climate-relevant geophysical fluid models.

Contribution

It introduces a novel approach to establish linear and fractional response for nonlinear SPDEs, including non-differentiable observables, with applications to complex geophysical models.

Findings

Established differentiability of observable averages for certain SPDEs.

Proved local Hölder continuity of responses under weaker conditions.

Applied results to stochastic Navier-Stokes and quasi-geostrophic models.

Abstract

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.