# $\alpha$-stable process and its generator

We hereby recall only the concept of special Levy process and its generator taken from [Applebaum09].

Let ${X}$ be a isotropic ${\alpha}$-stable process, which has its Levy symbol, for ${\alpha\in (0,2)}$

$\displaystyle \eta(u) = - |u|^{\alpha}$

or equivalently

$\displaystyle \eta(u_{j}: j= 1, \ldots d) = - (\sum_{j=1}^{d} u_{j}^{2})^{\alpha/2}.$

See Ex 3.3.8 of [App09]. By convention, the generator ${A}$ associated to ${X}$ is denoted by, replacing ${u_{j} \rightarrow - i \partial_{j}}$ in ${\eta}$

$\displaystyle A = \eta(- i \partial_{j}: j = 1, \ldots d) = - (\sum_{j = 1}^{d} - \partial_{j}^{2})^{\alpha/2} = - (- \Delta)^{\alpha/2}.$

Its Levy-Khintchine representation (see Ex3.5.9 of [App09]) is

$\displaystyle \eta(u) = - |u|^{\alpha} = K(\alpha) \int_{\mathbb R^{d} \setminus \{0\}} (e^{i(u,y)} - 1 - i u \cdot y I_{B_{1}}) \frac{dy}{|y|^{d + \alpha}}$

for some constant ${K(\alpha)>0}$. The precise meaning of the generator ${A}$ is then

$\displaystyle A f(x) = \int_{\mathbb R^{d} \setminus\{0\}} [f(y+x) - f(x) - y^{j} \partial_{j} f(x) I_{B_{1}} ] \frac{dy}{|y|^{d + \alpha}}.$