From Osher Doctorow

I pointed out in the last few posts that we are now making contact
with a substantial research literature on adding stochastic term(s) to
the Schrodinger or similar equations, but there are actually two
choices that need to be made.

1) Add a random variable called "error", often written e or e_i or ei,
with or without modifications (such as additional factors) to various
other terms of Schrodinger.

2) Add a probability, for example Probable Influence/Causation (PI)
P(A--B), instead of a random variable in (1).

Readers may be surprised to learn that there is NO literature on (2).

The reason for the absence of research literature on (2) is arguably an
incredible history of Ingenious Imitation in the relationships between
Mathematical Probability-Statistics and Physics. For a previous
example of this as "serious", readers would have to look at Euclidean
Geometry, whose practitioners imitated each other for thousands of
years.

While originally in history mathematicians arguably came first with
Euclidean geometry and then physicists imitated them, in recent
centuries physicists tended to come first with at least
experimental/observational discoveries, after which mathematicians
imitated them and then concentrated on the purely mathematical aspects
of equations mostly but to a small but varying extent on feedback to
physics. Thermodynamics and mechanics came from the physicists, and
then mathematical statisticians imitated them, and mathematical
statisticians in turn were imitated by 3 parties: other mathematical
statisticians, mathematical probability theorists interested in the
pure probability aspects of the equations of the former, and
physicists/engineers trying to incorporate everybody else's results
into somewhat large-scale theories.

One of the little known consequences of all this was a parallel
development in mathematical statistics called Regression Analysis,
especially Linear or Multilinear Regression, in which an "error" term e
as a random variable was added to one side of a deterministic equation
and by a rather interesting "trick" another term was transformed to
something in conditional probability called a conditional expectation
or conditional mean. Statisticians then went back to this and mostly
dealt with the sample form of the equation(s) and how it fit into their
usual obsession with the difference between population and sample
quantities. Probability theorists then took the statisticians'
viewpoint as a "given" and spent their time examining the probabilistic
aspects of the "given" equations without changing their basic forms
except in a rather curious way of relating them to differential
equations in what became the subfield of Stochastic Differential
Equations in Probability and lately in Statistics.

Why nobody thought of simply adding a probability or probability
density function (pdf) or cumulative distribution function (cdf) or
Probable Influence/Causation instead of a random variable called
"error" (e), was simply due to the fact that the imitation was not
explicitly discussed early on, and later people just imitated the early
people in not doing it.

But why should probability rather than a random variable be added to
the Schrodinger or similar scenarios? Because probabilities are
SIMPLE, while random variables are extremely complicated. For
example, a random variable describing a person's height as randomly
sampled is not an ordinary function or measure, unlike a probability
which is a measure (except for conditional probability which is a
relative measure) and which can be expressed as function. A random
variable has a value, say 60" (sixty inches) for height, but it doesn't
asssign this value x in the range of the random variable to an element
w in the domain of the random variable. It is in a sense "multivalued"
in that it doesn't take on the value 60" or x deterministically but
with a certain probability, or more precisely for continuous random
variables the probability is called a cumulative distribution function
F(x):

3) F(x) = P(random variable X = x) = P{w: X(w) = x}

where w are elements of the probability space or domain.

The result of all this is that it is super-ponderous to attempt to
progress in stochastic differential equations, for example, whereas if
you add a probability P to a deterministic equation (say, to the left
hand side), you don't introduce any complications since you don't keep
track separately of both values of P and probabilities of being =
those values, unlike for random variables. Your new equation is
treated as just having a new function P, which may be a function of
time as in P(t) or P_t or Pt, or even additional arguments. The other
"deterministic" variables when normalized are now regarded as either
probabilities or functions of probabilities which become deterministic
only when the probabilities are fixed or constant for some reason.

Stochastic differential equations are far more difficult to solve in
general than "deterministic" differential equations. You have to
literally learn machinery called Ito and/or Stratonovich integrals and
their differential or derivative aspects if any, and a typical paper in
stochastic differential equations satisfies itself with just solving a
particular stochastic differential equation or a rather small class of
stochastic differential equations, somewhat analogous to the way that
the complications of algebraic geometry and algebraic topology have
removed theoretical physics largely from the realm of easy and elegant
and quick solutions and theories into the realm of ponderous, slow
progress requiring almost endless "machinery" or sub-theorems and
sub-subtheorems and definitions and so on.

Osher Doctorow