Stochastic Green’s functions

A new approach is presented to the theory of random equations for use in physical problems. After a basic summary of necessary ideas and definitions relevant to stochastic processes, the paper develops a linear transformation theory which allows the operators as well as the operand to be stochastic. The statistics of a transformed process are found in terms of the known statistics of the original process and a 'stochastic Green's function'. The problem of determining a stochastic Green's function for a linear stochastic differential operator is studied. (Detailed results for the random sampling of a random process have been obtained in the author's dissertation but only an initial form is shown in the paper to show the nature of the integral kernel or stochastic Green's function.) Interesting applications to 'systems' problems, quantum theory, and propagation in random media are briefly suggested.