How to compute the levy path integral with zero potential? In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as
$$K(x_b,t_b|x_a,t_a)=\int_{x_{t_a}=x_a,x_{t_b}=x_b}Dx(t)\exp\left\{-\frac{i}{h}\int_{t_a}^{t_b}dtV(x(t))\right\}$$
where h is the Planck constant. It is known that in Feynman functional measure (generated by the process of the Brownian motion) and with zero potential ($V(x)=0$), the amplitude can be computed exactly, but what is the case with the non-Gaussian case? In the paper of Prof.Nikolai Laskin it said it can be computed with the measure generated by the $\alpha$ stable Levy motion ($1<\alpha<2$), but in this case the probability density function is so different from the Brownian case, so how to compute the amplitude?

It turns out that for the Levy path integral, the calculation of the amplitude of a quantum particle uses the Fourier translation of the probability density function, since this representation is an integral of an exponential function which is easy to compute.  
