The Metastable Behavior of Infrequently Observed, Weakly Random, One-Dimensional Diffusion Processes

Consider the one-dimensional diffusion process $X_t $ satisfying $dX_t = \varepsilon ^{1/2} dW_t - G' ( X_t )dt$ where $W_t $ is the standard Wiener process. For suitable G’s with a double well (at m and M) shape and $G( m ) > G ( M )$ and for an appropriate choice of $\lambda_\varepsilon$, the scaled process $X_{{\lambda _\varepsilon }t } $, converges as $\varepsilon \to 0$ (in an appropriate sense) to the two-state (m and M) jump process with M an absorbing state and transitions from m to M at unit rate.