$\alpha$-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}}.

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