In quantum mechanics, for a particle in a potential $V(x)$ we postulate that the propagator - which is the probability of transition between two points- is given by the Feynmann path integral:
$$K(x_b,t_b|x_a,t_a)=\int_{x_{t_a}=x_a}^{x_{t_b}=x_b}Dx(t)\exp\left\{-\frac{i}{h}\int_{t_a}^{t_b}dtV(x(t))\right\}$$
with measure of integration $$\begin{aligned} \int_{x_i}^{x_f} \mathcal{D} x(\tau) \ldots=\lim _{N \rightarrow \infty} \int d x_{1} \ldots d x_{N-1}\left(\frac{2 \pi i \hbar \varepsilon}{m}\right)^{-N / 2} \times \\ \prod_{j=1}^{N} \exp \left\{\frac{i m}{2 \hbar \varepsilon}\left(x_{j}-x_{j-1}\right)^{2}\right\} \ldots \end{aligned} $$
I'm novice in measure theory and my understanding is that the measure in Feynmann integration is the measure induced by Brownian motion as a Wiener process : Wiener measure. Hence we say that it "corresponds to integration over all Brownian paths". Is this correct? Is there another reason-way to see why the above is an integral over brownian paths?
Now I am studying Laskin's Fractional Quantum Mechanics where the above expression is replaced by an "integral over Levy flights" or "the measure generated by the Levy flight-process". However I can't find any proof as to how this expression (found in the original paper by Laskin, as well as in this one available in arxiv and his book "Fractional Quantum Mechanics"): $$\begin{aligned} K\left(x_{b} t_{b} | x_{a} t_{a}\right)=& \lim _{N \rightarrow \infty} \int_{-\infty}^{\infty} d x_{1} \ldots d x_{N-1} \frac{1}{(2 \pi \hbar)^{N}} \int_{-\infty}^{\infty} d p_{1} \ldots d p_{N} \\ & \times \exp \left\{\frac{i}{\hbar} \sum_{j=1}^{N} p_{j}\left(x_{j}-x_{j-1}\right)\right\} \\ & \times \exp \left\{-\frac{i}{\hbar} D_{\alpha} \varepsilon \sum_{j=1}^{N}\left|p_{j}\right|^{\alpha}-\frac{i}{\hbar} \varepsilon \sum_{j=1}^{N} V\left(x_{j}\right)\right\} \end{aligned} $$ corresponds to "an integral over Levy Paths". Additionaly Levy walks are not a Wiener process since the variance is infinite, so we don't have the same way of inducing a measure on continuous functions, so why and how is the above expression (without the term containing $V$) "a measure generated by Levy flights".
P.S. I'm not sure if this should be on physics-stack, but I think I'm more likely to find an answer here.