The influence of spatially anisotropic randomness on the solution of one-dimensional stochastic differential and integral equations

We demonstrate that the concept of a one-dimensional stochastic problem, in which only the statistical properties of the medium in one direction are used, is an unphysical situation. Even though the statistically averaged quantities, such as mean value and covariance, may depend only on one space dimension, the statistical properties of the medium in the other two directions must be included. A simple example, based on a second order differential equation, is used to illustrate the point and is supported by numerical calculations. The relevance of this matter to radiation and neutron transport in spatially stochastic media is made clear.