Abstract
In this chapter, equipped with our previously obtained knowledge of exit and transition times in the limit of small noise amplitude \(\varepsilon \rightarrow 0\), we shall investigate the global asymptotic behavior of our jump diffusion process in the time scale in which transitions occur, i.e. in the scale given by \({\lambda }^{0}(\varepsilon ) =\nu (\frac{1} {\varepsilon } B_{\delta }^{c}(0)),\varepsilon,\delta > 0\). It turns out that in this time scale, the switching of the diffusion between neighborhoods of the stable solutions ϕ ± can be well described by a Markov chain jumping back and forth between two states with a characteristic Q-matrix determined by the quantities \(\frac{\mu ({(D_{0}^{\pm })}^{c})} {\mu (B_{\delta }^{c}(0))}\) as jumping rates.
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Debussche, A., Högele, M., Imkeller, P. (2013). Localization and Metastability. In: The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise. Lecture Notes in Mathematics, vol 2085. Springer, Cham. https://doi.org/10.1007/978-3-319-00828-8_7
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