Linear Response Theory for Nonlinear Stochastic Differential Equations with $\alpha$-stable L\'{e}vy Noises

TL;DR

This paper extends linear response theory to nonlinear stochastic differential equations driven by non-Gaussian $$-stable Le9vy noises, establishing regularity of invariant measures and response functions.

Contribution

It introduces a novel linear response framework for systems influenced by non-Gaussian stable noises, generalizing classical theories to new stochastic dynamics.

Findings

Established regularity results for invariant measure densities.

Derived response functions for non-Gaussian stochastic systems.

Generalized linear response theory to $$-stable Le9vy driven equations.

Abstract

We consider a nonlinear stochastic differential equation driven by an $\alpha$-stable L\'{e}vy process ($1<\alpha<2$). We first obtain some regularity results for the probability density of its invariant measure via establishing the a priori estimate of the corresponding stationary Fokker-Planck equation. Then by the a priori estimate of Kolmogorov backward equations and the perturbation property of Markov semigroup, we derive the response function and generalize the famous linear response theory in nonequilibrium statistical mechanics to non-Gaussian stochastic dynamic systems.