Discrete stochastic calculus and commutators
The basic fact needed to address the question is that we have a set 0-dimensional objects and 1-dimensional objects that obey the following geometrically-motivated multiplication rules:
To see the geometrical meaning of these multiplications, it might help to consider discrete 0-forms
Multiplication 1.) above implies
which is completely reminiscent of multiplication of functions where we think of as the value of the function at the “node”
Multiplications 2.) and 3.) introduce new concepts, but whose geometrical interpretation is quite intuitive. To see this consider
Multiplying the function on the left of the “directed edge” projects out the value of the function at the beginning of the edge and multiplying on the right projects out the value of the function at the end. Hence, the product of functions and edges do not commute.
In the paper, it was shown that the exterior derivative of a discrete 0-form is given by
Here, we show that this may be expressed as the commutator with the “graph operator”
The result is quite simple and follows directly from and the multiplication rules, i.e.