Random attractors for stochastic partly dissipative systems

Abstract

We prove the existence of a global random attractor for a certain class of stochastic partly dissipative systems. These systems consist of a partial and an ordinary differential equation, where both equations are coupled and perturbed by additive white noise. The deterministic counterpart of such systems and their long-time behaviour have already been considered but there is no theory that deals with the stochastic version of partly dissipative systems in their full generality. We also provide several examples for the application of the theory.

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

“…In order to show existence of solutions to (2.4), one could use the concept of mild solutions as employed for the FitzHugh-Nagumo SPDE with additive noise in [61,90]. The approach presented in [55], which we build upon, uses variational solutions [57,66,75] for equations with locally monotone coefficients [65,66].…”

“…Also the SODE variant for ν = 0 is quite well studied, mainly due to a flurry of activity since the mid 1990s; see, for example, [3,7,9,63,64,72]. Yet, the full SPDE variant (1.1) has only attracted major attention quite recently, including a large number of numerical studies [67,80,81,83,[86][87][88] as well as analytical studies regarding existence, regularity, invariant measures and attractors [4,8,12,61,62,90]. The question regarding stochastic stability of pulses for additive noise is far less studied.…”