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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