TITLE: Fractional Calculus: Motivation and Foundations.
QUESTION [37 upvotes]: If this is too broad, I apologise; let's keep it focused on the basics if necessary.

What's the motivation and the rigorous foundations behind fractional calculus?

It seems very weird & beautiful to me. Did it arise from some set of applications? If so (and even if not), here's a suitable question concerning its "physical meaning" and history.
The Wikipedia article makes it look quite clear-cut: stick $\Gamma$ into Cauchy's formula for repeated integration. But why can we do that? Why is it listed under "Heuristics"? I know the Gamma function generalises the factorial, but that's as much as I understand.
"Why ask?"
Because I like to see how different areas of Mathematics fit together. I like the way fractional calculus seems to take integration & differentiation and ask, "well, do we really need to do these things a natural number of times?" - and so on. So I'm just curious :)

REPLY [4 votes]: Quite beautifully, fractional/nonlocal operators $\mathcal G$ are the natural notion of differentiation to describe stochastic processes with jumps.
If you write out the generator $\mathcal G$ of a Markov chain $P=\{p_{i,j}\}_{i,j\in \text{State space}}$ (which is intrinsically jumpy) what you obtain is
$$
\mathcal G f(x)=(P-I)f(x)=\sum_{y\in\text{State space}}(f(y)-f(x))p_{x,y}.
$$
The intuition is clear, the infinitesimal jump (unit time in this case) from $x$ to $y$ is assigned intensity/probability $p_{x,y}$. Similarly the fractional Laplacian $\mathcal G=-(-\Delta)^{\frac{\alpha}{2}}$ is the generator of a symmetric-$\alpha$ stable Lévy process, and it is given by
$$
\mathcal G f(x)=-(-\Delta)^{\frac{\alpha}{2}}f(x)=\int_{\mathbb R^{n}}(f(y)-f(x))C_{n,\alpha}\frac{dy}{|y|^{n+\alpha}}.
$$ 
The Caputo derivative $\mathcal G=\ ^{C}D^{\beta}_a$ of order $\beta\in(0,1) $ has the (generator) representation 
\begin{align}
^{C}D^{\beta}_a f(x)&= \int_0^{a-x}(f(x+y)-f(x))\nu(y)dy +(f(a)-f(x))\int_{a-x}^\infty\nu(y)dy,
\end{align}
where $\nu(y):=C_\beta y^{-1-\beta}$, and $x<a$.
Here the intuition is that $^{C}D^{\beta}_a$   is the generator of the the process a stable process $X^{\beta}(s)$ absorbed once crossing the barrier $\{a\}$. A more detailed intuition is: 

consider the process $X^{\beta}$ starting at $x<a$, then the first term  sums up all the intensities $\nu(y)$ for every jump form $x$ to $y$ falling below $a$ ($0\le y\le a-x$) , indeed
  $$\text{sum}_y\ (f(x+y)-f(x)) \nu(y);$$
  The second term case is (i) a standard killing term $-f(x)$ times a (unbounded) coefficient $b(x):=\int_{a-x}^{\infty}\nu(y)dy,$ which contains all the intensity of the jumps that would have fallen above $a$ (for the process starting at $x$), and (ii) a regenerating term $+f(a)b(x)$ with the same coefficient/sum of the intensities, which puts/absorb the process at $\{a\}$.

Hence the Caputo operator, is not only nonlocal, but it contains boundary information! For a probabilistic proof that follows this intuition see the Appendix here.

Hence, as you like to study second order operators such as the Laplacians, you equivalently like to study many  second order fractional differential operators. Using as a motivation the successes of probability in the local operators case, fractional operators offer very concrete probabilistic intuition (think about the intuition the Brownian motion provides for the heat equation), which makes  models worth being applied. Also  new interesting properties arise from nonlocality and boundary information (like memory and trapping phenomena); interesting properties for both theory and applications, (and often their probabilistic features is what allows you to both uncover these properties and give intuition).

(Sorry for possibly  sounding too probabilistically enthusiastic, but the answers above seem to ignore the probabilistic aspects, hence I thought it would be fair to to push it a bit)