The importance of stochastic modeling is increasing in all engineering and scientific disciplines. The inclusion of random effects can serve a number of distinct purposes. On one hand, randomness can be used to compensate for elements of a system not fully modeled or for insufficiently resolved scales in the system. This research proposal concentrates on the analysis of how randomness spreads through large dimensional, extended dynamical systems. One of the central questions is how passing through many nonlinear interactions or scales shapes the randomness. Stochastically forced partial differential equations (SPDEs) such as stochastic fluid equations or stochastically forced reaction diffusion equations will be a focus of the proposed work. These systems are often agitated at one scale and it is of both mathematical and modeling interest to understand the effect of this agitation at other scales. In particular, this proposal hopes to shed light on the transfer of randomness between different scales in turbulent fluid flow. In addition to SPDEs, the proposal will study related questions in large chemical networks arising in cell regulation and nutrient flow in forest environments. The emphasis will be on the pathwise dynamics, novel stochastic numerical methods, and estimation problems.
Quantifying and modeling uncertainty is increasingly important in our attempts to understand and predict our complex and changing world. From the financial markets, to weather prediction, to environmental engendering, to neuroscience, to aeronautics, models including random influences have become central to the physical sciences, to the social sciences and to engineering. Of particular significance is understanding how pure randomness is shaped into the structures we observe, especially when the randomness is transferred across disparate spatial scales. This understanding is the key to producing effective models with which to predict complex real systems. This proposal specifically addresses questions central to turbulent water flow and complicated biochemical pathways in cells. More broadly, applied mathematicians are on the leading edge of our scientific and technological future. Because of their broad training, they are the best equipped to move into new, non-traditional fields as they appear, build the bridges between classical disciplines, and disseminate the understanding gained in purer mathematical studies to larger scientific and social endeavors. America's leadership in science and technology requires us to continue producing world-class researchers and students in applied mathematics. This proposal includes an undergraduate research component to encourage interested students to choose a science-related career. It includes outreach to local high schools to expose students at an earlier age to modern mathematics and the myriad of career possibilities it presents.
