Data on the motion of objects has become common in many fields of science, engineering, and general human experience. Since the time of Newton mechanical motion has been described analytically by differential equations. Deterministic differential equations theory and application have developed into corresponding study of stochastic differential equations (SDEs). In a variety of practical situations it has been unclear how to select the drift function of an SDE. This research will investigate the use of a potential function to provide a formula for drift. In particular it will be assumed that the motion of interest is governed by a potential function. The drift is then the potential function's gradient. This structure is referred to as a stochastic gradient system. Being real-valued a potential function is easier to model than a vector-valued drift function. An estimated potential function may be used for simple description, summary, comparison, simulation, prediction, model appraisal, bootstrapping, and employed for estimating quantities of interest. The work will include study of models based on functional stochastic differential equations. This will allow the inclusion of time history in the description. Specific analytic problems to be studied include: unequally spaced times of observation, development of simulation methods to display variability, allowing model appraisal and making predictions.
The theoretical structure investigated in the particle motion research will be applied to a variety of biological, ecological and other motion situations. In particular Hawaiian monk seal GPS data will be modeled and questions asked by the concerned marine biologists addressed. These data are important because the monk seal is America's most endangered marine mammal. Only about 1300 remain. The study of migration routes of northern elephant seal will also be modeled. This animal is a protected species under the Marine Mammal Act of 1972. Continuing, paths of animals in the Starkey Experimental Reserve in Oregon will be included in the work. These data have been collected to study the management of habitats shared by wild animals, cows and people. Lastly, work will continue on risk models concerning wildfires at the urban-wildfire interface will be modeled employing data from the San Diego County fires of 2003.
