The derivation of fractional equations

Fractional derivatives are nonlocal, but actions are usually assumed to be local.

When I've seen fractional derivatives I've assumed that one place where they would naturally arise is in physical situations where there's a fractional dependency on time.

For example, random walks typically result in movement proportional to $\sqrt{t}$. Googling for "fractional+derivative+random+walk" on arxiv.org finds some papers that explore this:

http://www.google.com/search?q=fractional+derivative+random+walk+site%3Aarxiv.org

So I'm wondering if there's a way of relating some of the diffusion versions of QM with fractional derivatives.