Wednesday 30th May, 2012 at 5pm – Lecture Theatre T.0.003 (Wolfson Building)
Stochastic (Partial) Differential Equations and Stochastic Dynamical Systems
Professor Huaizhong Zhao, Department of Mathematical Sciences
Uncertainty and randomness occurs naturally in many real world problems. Probability theory provides mathematical tools to study and to model them. Random processes were noted by Brown, Bachelier and Einstein et al to describe random movements in various contexts such as, particle movement in a liquid, and economics.
On the other hand, many physical systems are nonlinear in nature. Classical probability theory and canonical stochastic processes are basically linear. To understand random and nonlinear phenomena, one needs to study stochastic differential equations and stochastic partial differential equations.
Mathematicians developed many new mathematics to understand random phenomena. Stochastic analysis began with its ground-breaking work in the study of stochastic integrals, together with some pioneering work of Kolmogorov, Markov and Wiener on probability theory, Markov process and rigorous mathematical construction of Brownian motion.
It has been profoundly developed as a vital fundamental research area in mathematics in recent decades. It has been discovered to have intrinsic connections with many areas of mathematics, and vast applications in physics, engineering, finance, biology etc.
The theory of stochastic differential equations (SDEs) and stochastic partial differential equations (SPDE) is a fascinating area of research. The theory of existence and uniqueness is pretty much developed after work of many mathematicians. With these results, it is natural to regard SDEs and SPDEs as random dynamical systems.
In fact, it is not trivial to prove that SDEs or SPDEs can generate a random dynamical system. To understand random phenomena, we study the long-term behaviour and invariant set of stochastic dynamical systems.
In this inaugural lecture, Professor Huaizhong Zhao will introduce the stochastic dynamical system generated by SDEs and SPDEs and discuss recent developments especially invariant manifolds, stationary solutions, random periodic solutions. Professor Zhao will also talk about probabilistic/analysis tools which have developed in recent years.