Stochastic differential equations and some phenomena including noise

2015/07/03 Fri 15:30 - 17:00
Ta Viet Ton
Graduate School of Information Science and Technology, Osaka University

Nonequilibrium phenomena are very common in natural world such as self-organization of biological systems, phase transitions, pattern formation in crystal growth, to name a few. The pioneering works on the phenomena are those of Nicolis and Prigogine [1] and Haken [2]. Unlike the study of equilibrium systems where stationary states or periodic states are main concerns, attractors, chaos or fractal are of interest in investigating nonequilibrium systems. In other words, the process of dynamics including structural stability and robustness of the states, attracts much more attentions than the final states.
In this talk, I will introduce several of our results on two nonequilibrium problems. They are Animal Swarming and Forest Kinetics. Especially, we study the phenomena which include white noise. Our study is performed by means of mathematical models, called Animal Swarming Model and Forest Kinematic Model, which are formulated using stochastic differential equations (SDEs). For both models, I show the global existence and asymptotic behaviors of solutions. The obtained results suggest that the rules for constructing models, on one hand, are robust or structurally stable against the white noise for maintaining the systems in a favorable way, on the other hand, have limitation or may be fragile.
Our long-term objective is to establish a theory for abstract stochastic evolution equations. This is prompted by a need of a new theory to study our (diffusion) forest model. In the last part of the talk, I will present our results dealing with the existence, uniqueness and regularity of solutions to abstract stochastic linear and semilinear evolution equations, which is the first step on the way to the objective.

[1] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley, 1977.
[2] H. Haken, Synergetics: an Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry and Biology, Springer-Verlag, 1978.