Description
Let $\Omega \subset \mathbb{R}^d$ be a bounded smooth domain and $b : \Omega \rightarrow \mathbb{R}^d$ be a smooth vector field. We focus on the associated overdamped Langevin equation :
$$\dot{X_t}=b(X_t) + \sqrt{h} \dot{B_t} $$
in the low regime temperature where $h \rightarrow 0$ and in the case where $b$ admits the decomposition $b = - \nabla f - l$ with :
$$\bullet ~ l\text{ a smooth vector field ; }$$
$$\bullet~ f\text{ a Morse function on }\overline{\Omega}\text{ admitting several local minima ; }$$
$$\bullet~ \nabla f \cdot l =0\text{ on }\overline{\Omega}.$$
In this framework, minima of the function $f$ correspond to metastable states for this Langevin dynamics.
In this context, we analyse the spectrum of the infinitesimal generator of the dynamics :
\begin{align*}
L_h = - \frac{h}{2} \Delta + \nabla f \cdot \nabla + l \cdot \nabla
\end{align*}
with Neumann boundary conditions. In this case, moving particles will remain trapped inside the domain. More specifically, we will consider additional hypotheses ensuring that the measure $e^{-\frac{2f}{h}}dx$ is invariant.
