Consider the diffusion equation $\partial_t p=D\frac{\partial}{\partial |x|}p$ where for simplicity we write p=p(x,t). By using the Fourier transform on both sides we get $\partial_t \tilde{p}=-D|k|\tilde{p}$ (as suggested here) with $\tilde{p}=\tilde{p}(k,t)$ and thus $\partial_t[e^{D|k|t}\tilde{p}]=0$. This means that we can write $\tilde{p}(k,t)=\tilde{f}(k)e^{-D|k|t}$. If we choose the initial condition $\tilde{p}(k,0)=\tilde{f}(k)=1$ (corresponding to a Dirac delta on the $x$ space) we obtain, by antitransforming:
$p_1(x,t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dk e^{-D|k|t+ikx}=\frac{1}{\pi D t \big{(}1+(\frac{x}{Dt})^2\big{)}}$
i.e. I'm obtaining a Cauchy process. By plugging in the solution in the initial diffusion equation, though, I find out that the solution I've found is not actually a solution!
We can do similar calculations in the $x$ space, obtaining $p_2(x,t)=e^{Dt\cdot sgn(x)\partial_x}f(x)=f(x+sgn(x)Dt)$ (where we recognised the translation operator) which is a solution of the diffusion equation for every function $f$ (regular enough); on the other hand, $p_1(x,t)$ can not be written in that form, which is another tell that something went wrong.
Hence I have two doubts here:
- Why does the first method does not work?
- Is the Cauchy process the solution of a Lèvy flight equation, possibly more general, for instance of the form $\partial_t p(x,t)=D\frac{\partial^n}{\partial |x|^n}p(x,t)-\partial_x(g(x,t)p(x,t))$, for some integer $n$?