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.
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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.
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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.
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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:
- the experimental study of the dynamics of
biochemical systems in the presence of noise;
- the representation of biochemical systems through the
development
of stochastic models, including stochastic space-time models and
stochastic
functional differential equations;
- mathematical and experimental analysis of the dynamical properties of
stochastic biochemical models, e.g. stability, attractors and
oscillation, especially those relating to the role of enzymes in the stability
of
biochemical systems under environmental noise;
- numerical analysis of the consequences of
noise in biochemical systems with experimental validation.
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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.
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