# Stochastic perturbation of a cubic anharmonic oscillator

## TL;DR

This paper investigates the effects of Gaussian white noise perturbations on a cubic anharmonic oscillator, deriving explicit solutions and probabilistic bounds for the stochastic system’s behavior.

## Contribution

It introduces a formal power series expansion for the stochastic perturbation and analyzes the properties of its coefficients, linking them to Lamé's equation and providing convergence conditions.

## Key findings

- Explicit solutions in terms of Jacobi elliptic functions.
- Lower bounds for the probability of staying close to deterministic solutions.
- Conditions for convergence of the perturbation expansion.

## Abstract

We perturb with an additive Gaussian white noise the Hamiltonian system associated to a cubic anharmonic oscillator. The stochastic system is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We write a formal expansion in powers of the diffusion parameter for the candidate solution and analyze the probabilistic properties of the sequence of the coefficients. It turns out that such coefficients are the unique strong solutions of stochastic perturbations of the famous Lam\'e's equation. We obtain explicit solutions in terms of Jacobi elliptic functions and prove a lower bound for the probability that an approximated version of the solution of the stochastic system stay close to the solution of the deterministic problem. Conditions for the convergence of the expansion are also provided.