Asymptotic Transition Times

Abstract

The preceding chapter is concerned with the effect of small Lévy noise in H of intensity $\varepsilon$ of triggering exits from the reduced domains of attraction of the stable states ϕ ^± of a Chafee–Infante equation. Noise is seen quite generally to make stable states of deterministic systems given by ordinary or partial differential equations metastable. In this chapter, we shall investigate more closely the dynamics of the stochastic system, in particular the stochastic transition and wandering behavior between the metastable states. We shall ask questions about the reduced dynamics of the system, i.e. the reduction of the jump diffusion equation to a simple Markov chain in the small noise limit $\varepsilon \rightarrow 0+$ boiling down the dynamics to a simple switching between the metastable states. It will be seen that this reduction is related to a scaling limit of the jump diffusion in the polynomial scale ${\varepsilon }^{-\alpha }$ resulting from the asymptotic behavior of first exit times of domains of attraction encountered in the previous chapter.