Sorry @Swamidass, I was at the beach for a few days and only had sporadic internet access.

Can you show the math or link to it?

Going from a continuous equation (like the Schrodinger Equation of the diffusion equation) to a discrete equation is extremely well-studied, because you can’t solve continuous differential equations on a digital computer! So every continuous equation that is ever solved computationally is first transformed into a discrete equation (e.g. by putting it on a lattice). This subject will be treated in virtually any textbook on differential equations.

Going from discrete to continuous is more complicated, but you can still do it. Here’s a derivation for turning a random walk into the diffusion question:

http://nebula.physics.uakron.edu/dept/faculty/jutta/modeling/diff_eqn.pdf

I don’t think it’s always possible, but diffusion seems very likely to yield similar results if it’s cast into a continuous form.

-Neil