On the behavior of solutions to certain parabolic SPDEʹs driven by wiener processes

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear stochastic partial differential equations driven by finite-dimensional Wiener processes. This class encompasses important equations that occur in the mathematical analysis of certain migration phenomena in population dynamics and population genetics. The solutions to such equations are generalized random fields whose long-time behavior we investigate in detail. In particular, we unveil the mechanism whereby these random fields approach the global attractor by proving that their asymptotic behavior is entirely controlled by that of their spatial average. We also show how to determine explicitly the corresponding Lyapunov exponents when the nonlinearities of the noise-term of the equations are subordinated to the nonlinearity of the drift-term in some sense. The ultimate picture that emerges from our analysis is one that displays a phenomenon of exchange of stability between the components of the global attractor. We provide a very simple interpretation of this phenomenon in the case of Fisherʹs equation of population genetics. Our method of investigation rests upon the use of martingale arguments, of a comparison principle and of some simple ergodic properties for certain Lebesgue- and Itô integrals.