The dynamics of nonlinear reaction-diffusion equations with small Lévy noise
These lecture notes investigate the stochastically perturbed Chafee-Infante equation \[ du=\bigl(\frac{\partial^2}{\partial x^2}u+\lambda(u-u^3)\bigr)\,dt +\varepsilon dL(t),\qquad x\in(0,1),\;t>0, \eqno(*) \] with \(u(t,0)=u(t,1)=0\), where~\(L\) is a pure jump space-time Lévy process and~\(\lambda\) is a parameter with \(0<\lambda\neq k^2\pi^2\) for \(k\in\mathbb N\), for small \(\varepsilon\). The unperturbed system \(\varepsilon=0\) has a unique stable fixed point for \(0<\lambda<\pi^2\) and two stable fixed points together with a number of unstable fixed points for \(\pi^2<\lambda\), increasing as \(\lambda\) passes~\(k^2\pi^2\), \(k\geq2\). These fixed points together with their unstable manifolds form an attracting set. The major results state that the exit and transition times between domains of attraction depend polynomially on the noise intensity~\(\varepsilon\), provided that some non-degeneracy conditions on the Lévy process \(L\) are satisfied. Furthermore, the solutions behave in a metastable way insofar as they can be approximated by a finite-state Markov chain on a suitable polynomial time scale. The polynomial dependence on~\(\varepsilon\) in the case of a Lévy perturbation is substantially different from the well-investigated case of Gaussian perturbations, where the dependence on~\(\varepsilon\) is exponential. The presentation is organized as follows: after an introduction with a detailed outline of the results, Chapter 2 provides a discussion of the deterministic Chafee-Infante equation \(\varepsilon=0\). Chapter 3 describes the Lévy process, giving a decomposition into ``large'' and ``small jump'' components, and a brief introduction into the basic existence and uniqueness results for solutions of \((*)\). Chapter 4 provides small deviation results of the solution of the stochastic equation with only ``small jump'' components from the deterministic solution. In Chapter 5, probabilities of exit events from neighbourhoods of the stable fixed points of the deterministic equation are discussed, while Chapter~6 uses exit times to investigate transition times between different domains of attraction of the deterministic equation. Chapter~7 introduces an additional assumption on the Lévy process in order to establish metastable behaviour of the solutions of~\((*)\). An appendix describes some aspects of the climate physics background which lead to the study of stochastic differential equations perturbed by Lévy noise.
stochastically perturbed Chafee-Infante equation
scientific article; zbMATH DE number 6175547