Derivation of the Probability Density Function for a Stochastic Nonlinear Advection Equation

摘要

The report considers the solution of a stochastic advection-reaction equation. Interest in theis subject arose out of a study of models of atmospheric pollution, where diffusion also plays an important role. The results derived may be viewed as a model with the diffusion neglected, but it is emphasized that, although molecular diffusion is a small effect in the stmosphere, it cannot be ignored in stochastic problems. The main result consists of a limit theorem for an advection-reaction equation, in the limit of small fluctuations in the advecting vector field. The reactive problem is reduced to a nonreactive one by introducing the probability density for the stochastic solution. The corresponding limit theorem for the homogeneous advection equation is based on the results obtained by Khas'minskii and Papanicolaou et al. on stochastic ordinary differential equations. In the case of a linear reaction, the well known model of reacting with the mean that is used in current pollution models is recovered; however, when the reaction is nonlinear, reacting with the mean is shown to be invalid. No attempt at rigor is made, since main results are purely formal. (ERA citation 04:024464)