forcing from alternating to synchronised saddles an additional resonance at double the forcing frequency starts to appear. The conditional KS test reliably detects the stochastic resonance. This paper is mainly based on the thesis [@tommy_thesis].'
author:
- 'Tommy Liu[^1]'
bibliography:
- 'paper\_reportreferences.bib'
title: 'A comparative study of stochastic resonance for a model with two pathways by escape times, linear response, invariant measures and the conditional Kolmogorov-Smirnov Test'
---

Introduction
============

Outline of Problem
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Consider the following problem. Let $X_t^\epsilon$ be the random variable describing the trajectory of a diffusion process in $\mathbb{R}^r$ where $t$ is the time and $\epsilon^2$ is the variance level. More precisely we consider processes described by the following type of stochastic differential equation $$\begin{aligned}
dX^\epsilon_t=b\left(X^\epsilon_t,t\right)dt+\epsilon\,dW_t\end{aligned}$$ where $b:\mathbb{R}^r\times \mathbb{R}\longrightarrow \mathbb{R}^r$ and $W_t$ is a Wiener process in $\mathbb{R}^r$. We suppose that the drift term $b$ has the form $$\begin{aligned}
b(x,t)=-\nabla V_0 (x) + F\cos \Omega t \end{aligned}$$ where $F,x\in\mathbb{R}^r$ and $V_0:\mathbb{R}^r\longrightarrow \mathbb{R}$ is called the unperturbed potential. We consider unperturbed potentials with two or more minimas (wells). Most importantly, we consider potentials where there are multiple pathways between the wells. To our knowledge systems with two pathways have not been studied in the context of stochastic 