The forward evolution equation () and the backward
equation () are connected
through a duality relation. For any time *t*, we compute () as

For now, the main point is that the sum on the bottom line does not depend
on *t*. Given the constancy of this sum and the *u* evolution equation
(4), we can give another derivation of the *f* evolution equation (7).
Start with

Then use (4) on the left side and rearrange the sum:

Now, if this is going to be true for any *u*(*y*,*t*), the coefficients of
*u*(*y*,*t*) on the left and right sides must be equal for each *y*.
This gives (7). Similarly, it is possible to derive (4) from (7) and
the constancy of the expected value.

The evolution equations (4) and (7) have some qualitative properties in
common. The main one being that they preserve positivity. If
for all , then for
all also. Likewise, if for all *x*,
then for all *x*. These properties are simple consequences
of (4) and (7) and the positivity of the *p*(*x*,*y*). Positivity preservation
does not work in reverse. It is possible, for example, that
*f*(*x*,*t*+1) ;*SPMlt*; 0 for some *x* even though for all *x*.

The probability evolution equation (4) has a conservation law not shared by (7). It is

independent of *t*. This is natural if *u* is a probability distribution,
so that the constant is 1. The expected value evolution equation (7)
has a *maximum principle*

This is a natural consequence of the interpretation of *f* as an expectation
value. The probabilities, *u*(*x*,*t*) need not satisfy
a maximum principle either forward of backward in time.

This duality relation has is particularly transparent in matrix terms.
The formula () is expressed explicitly in terms of the
probabilities at time *t* as

which has the matrix form

Written in this order, the matrix multiplication is compatible; the other order, , would represent an matrix instead of a single number. In view of (), we may rewrite this as

Because matrix multiplication is associative, this may be rewritten

for any *t*. This is the same as saying that
is independent of *t*, as we already saw.

In linear algebra and functional analysis, ``adjoint'' or ``dual''
is a fancy generalization of the transpose operation of matrices. People
who don't like to think of putting the vector to the left of the matrix
think of as multiplication of (the transpose of)
, on the right, by the transpose (or adjoint or dual) of
*P*. In other words, we can do enough evolution to compute an expected
value either using *P* its dual (or adjoint or transpose). This is the
origin of the term ``duality'' in this context.

Tue Sep 15 17:12:32 EDT 1998