Current Research Interests

Stochastic Systems

By stochastic systems we mean stochastic differential equations, stochastic differential delay equations, stochastic partial differential equations and stochastic evolution equations in Hilbert spaces which arise from engineering, economy, biology etc. Prof. Mao's research interests in this area include the existence and uniqueness of solutions, comparison theorems, random fixed point theorem, stochastic integral inequalities, asymptotic properties. (return to index)

Stochastic Stability and Attraction

Stochastic stability is one of the most active areas on stochastic analysis and is important on stochastic control. Prof. Mao has made a number of significant contributions to this field and has written three research texts in this area. He is currently investigating the robustness of stability of non-linear stochastic systems, stability of large-scale stochastic systems, stability of neutral-type stochastic functional differential equations. Recently his interest in this direction has been extended to the study of stochastic attraction. (return to index)

Approximate Solutions

In Ito's classical theory of stochastic differential equations where the coefficients are assumed to be Lipschitz continuous, the solutions are constructed through the Picard successive approximation procedure. However, it is still open whether this procedure works on stochastic differential equations where the coefficients are not Lipschitz continuous, for example, Holder continuous. This is a very hard and important problem. Apart from the Picard approximation procedure, the Euler-Maruyama and Caratheodory approximation procedures are both interesting research topics. Recently the emphasis of research in this direction is to find out asymptotic behaviours of approximate solutions. (return to index)

Stochastic Modelling in Biochemical Science

This is a new research area. Biochemical systems are highly nonlinear and are often subject to environmental noise. Their study involves many disciplines e.g. biological science, space-time modelling, stochastic analysis, dynamical systems and computer simulation. The research in this direction is concerned with linking experimental and theoretical analysis of biochemical systems subject to environmental noise, and will specifically address:

Stochastic Neural Networks

Much of the current interest in artificial networks stems not only from their richness as a theoretical model of collective dynamics but also from the promise they have shown as a practical tool for performing parallel computation. Theoretical understanding of neural-network dynamics has advanced greatly in the last ten years. Although the stability of neural networks had been studied by many authors, the problem of stochastic effects to the stability was not investigated until 1996 by Professors Liao and Mao, where the exponentially stability and instability of stochastic neural networks were discussed. Recent research is to cope with time delay, parameter uncertainty, numerical analysis, global stochastic attractor. (return to index)
(return to Xuerong Mao's Home Page)