Metastability in a continuous mean-field model at low temperature and strong interaction

We consider a system of $N \in \mathbb{N}$ mean-field interacting stochastic differential equations that are driven by a single-site potential of double-well form and by Brownian noise. The strength of the noise is measured by a small parameter $\varepsilon >0$ (which we interpret as the \emph{temperature}), and we suppose that the strength of the interaction is given by $J>0$. Choosing the \emph{empirical mean} ($P:\mathbb{R}^N \rightarrow \mathbb{R}$, $Px =1/N \sum_i x_i$) as the macroscopic order parameter for the system, we show that the resulting macroscopic Hamiltonian has two global minima, one at $-m^\star_\varepsilon <0$ and one at $m^\star_\varepsilon>0$. Following this observation, we are interested in the average transition time of the system to $P^{-1}(m^\star_\varepsilon)$, when the initial configuration is drawn according to a probability measure (the so-called \emph{last-exit distribution}), which is supported around the hyperplane $P^{-1}(-m^\star_\varepsilon)$. Under the assumption of strong interaction, $J>1$, the main result is a formula for this transition time, which is reminiscent of the celebrated Eyring-Kramers formula up to a multiplicative error term that tends to $1$ as $N \rightarrow \infty$ and $\varepsilon \rightarrow 0$. The proof is based on the \emph{potential-theoretic approach to metastability.} In the last chapter we add some estimates on the metastable transition time in the high temperature regime, where $\varepsilon =1$, and for a large class of single-site potentials.