Full $\mathrm{\Gamma }-$expansion for the level-two large deviation rate functionals of non-reversible one-dimensional diffusions with periodic boundary conditions

Consider the diffusion process \begin{equation*} dX_ε(t) = \mss b(X_ε(t)) , dt + \sqrt{2, ε, \mss a(X_ε(t))} , dW_{t}, \end{equation*} on the one-dimensional torus $\text{bb}T=[0,1)$. Here $\epsilon$ is the temperature, $W_t$ a Brownian motion on $\text{bb}T$ and $\text{mss}a$, $\text{mss}b$ functions of class $C^2(\text{bb}T)$ satisfying further conditions. Denote by $\text{mss}P(\text{bb}T)$ the set of probability measures on $\text{bb}T$ equipped with the weak topology, and by $\text{ms}{I}_{\epsilon }\text{\hspace{0.17em}⁣}:\text{mss}P(\text{bb}T)\to [0,+\mathrm{\infty })$ the level two large deviation rate functional of the diffusion $X_{\epsilon }(\cdot )$. We derive a full $\mathrm{\Gamma }-$expansion of $\text{ms}{I}_{\epsilon }$, as $\epsilon \to 0$, expressing it as \begin{equation*} \ms I_ε = \frac{1}ε ;\ms J^{(-1)} ; +; \ms J^{(0)} ;+; \sum_{p=1}^{\widehat{\mf q}}\frac{1}{θ^{(p)}_ε};\ms J^{(p)},, \end{equation*} where $\text{ms}{J}^{(-1)}$, $\text{ms}{J}^{(0)}$, $\text{ms}{J}^{(p)}\text{\hspace{0.17em}⁣}:\text{mss}P(\text{bb}T)\to [0,+\mathrm{\infty }]$ represent rate functionals, independent of $\epsilon$, and ${\theta }_{\epsilon }^{(p)}$ are the time-scales at which the Markov process $X_{\epsilon }(\cdot )$ exhibits a metastable behaviour.