STOCHASTIC GREEN'S FUNCTIONS

BY

G. ADOMIAN

1. Introduction. The purpose of this paper is a brief introduction to an approach

[1] to the theory of random equations which appears fruitful. Such equations

arise in many connections. Physical constants for equations representing

dynamical systems are usually the mean value of a set of experimental observations

so in reality are not constants at all but random variables whose values are subject

to a probability distribution. Often it may be that the random equation actually

is more likely to bring theory and experiment into agreement but the problem has

been simplified to avoid the complication of randomness. In many-body problems

it may be desirable to consider a stochastic Hamiltonian operator to represent

the effects of many other bodies on the body of interest trading the complexity of

3n coordinates for 3 coordinates but with randomness in the operator. In wave

propagation problems with random media and in optimization problems with

complex systems involving random parameters, and in many other connections,

such random equations arise. In the work to follow, we intend to be more con-

cerned with vigor than rigor although a rigorous reformulation more in line with

the work of the Prague probabilists [2; 3] and methods of probabilistic functional

analysis will hopefully follow.

The remainder of this paper is divided into three sections. In §2 we present

a basic summary of necessary ideas and definitions sufficient for the level of

presentation in the remainder of this paper. The paper of Bharucha-Reid in this

volume will present on a rigorous level the definitions and theorems. §§3 and 4

respectively are devoted to a treatment of two types of physical problems. In §3,

a linear transformation or operator theory is developed which allows the operators

as well as the operand to be stochastic. Statistical measures and distributions are

found for the transformed function under general conditions. The effects of a

propagation medium, or of a physical observation or measuring process, or of

some control system with statistically varying parameters can be represented in

terms of such stochastic operators where instead of a Green's function for the

kernel of the operator inverting the differential operator, one is interested in a

stochastic Green's function which maps a desired statistical measure from the space

of the original random function to the space of the transformed function. §4

is devoted to a consideration of the far more difficult problem of the stochastic

differential operator. We are interested in relating this problem to the operator

formulation to give statistical measures for the dependent variable.

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