Perturbation theory for stochastic nonlinear oscillators and Feynman diagram

Abstract

We show that a systematic perturbation theory for the class of stochastically driven nonlinear oscillator ̈ 𝑥 + 2𝛤 ̇ 𝑥 + 𝜔20 𝑥 + 𝜇𝑥2 + 𝜆𝑥3 = 𝑓(𝑡) where f(t) is a Gaussian white noise can be effectively developed by writing down the relevant Feynman diagrams using response function and correlation functions. For 𝜇 = 0, we find that the response function acquires a non- Lorentzian shape at 𝑂(𝜆2). This is a qualitative shift in the response. For 𝜆 << 1 and 𝜇 ≠ 0, we show that a particle located at the minimum of a potential well can cross the barrier within a perturbative approach.