Long-term Effects of Small Perturbations and Other Multiscale Asymptotic Problems

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Long term effects of small perturbations, both deterministic and stochastic are of great interest in real life and for various intellectual enterprises from history, sociology, and economics to biology, physics, and engineering. When one wants to understand the time evolution of a complex system, a simplified model of the system, that can be analyzed, is usually a way forward. In this case one chooses a relatively small number of main 'factors' or variables which are guiding the evolution of the system while neglecting other factors that are relatively insignificant. On a short time interval, these small factors are not essential. However, on long time scales, the factors, which were considered as negligible, can become important and even critical for determining the system's behavior. The investigator and his colleagues are developing new models and methods for studying such problems. 'Problem' examples include climate change, biological evolution, and phase transitions in physical or economic systems as well as the appearance of 'stable' oscillations, equilibriums, and other patterns caused by small perturbations.The investigator and his colleagues study deterministic and stochastic perturbations of various dynamical systems as well as semi-flows corresponding to the evolution of PDEs. A general approach to, at first glance, different problems is suggested. Metastability and stochastic resonance, which are manifestation of the large deviation theory, as well as the averaging principle and its modifications can be considered from this point of view. This general approach is based on the study of limiting slow evolution as a motion on the cone of invariant measures of the non-perturbed system. The approach underlines the importance of joint consideration of deterministic and stochastic perturbations. Perturbations of the generalized Landau-Lifshitz equation for multiple spins, incompressible flows in 3D having a conservation law, reaction-advection-diffusion equations in narrow channels arising in models of molecular motors can be studied using this approach. Various boundary problems for second order elliptic equations with a small parameter can be considered as well. Finally, the PI will also study the small mass asymptotics for the Langevin equation.