Summary Article
Mathematics -- The Science of Patterns and Algorithms
Noise and Randomness
Noise and randomness are ubiquitous. The correspondence between random walks and diffusion differential equations has provided fertile territory for mathematical analysis, and for applications such as stochastic control, filtering, and predicting the likelihood of rare but catastrophic events. We have become adept at dealing with random perturbations of finite-dimensional systems, described by ordinary differential equations. In contrast, the analysis of similar issues for infinite-dimensional systems (those described by partial differential equations) is in its infancy. Learning how to deal with them is essential to our understanding of the consequences of uncertainty, imperfection, and thermal fluctuations in physical systems [KO], [GL]. By analogy with the existing theory, it will involve random walks and diffusion differential equations on infinite-dimensional spaces.
Related issues of infinite-dimensional analysis arise in the task of putting realistic quantum field theories on a mathematically sound foundation. Fresh insight in this area is emerging from links between string theoretic physics, topology, and geometry.
A different mandate for infinite-dimensional analysis comes from today's massive data sets, which must typically be interpreted using models with large numbers of parameters. Infinite-dimensional approximations provide one approach to get a handle on the behavior of statistical methods in the limit of increasingly large data sets and models [BI], [BO].
Most of these infinite-dimensional problems defeat us at present; gaining better insight would have extraordinary pay-off. What we can glimpse already has spectacular ramifications.