An explicit Lévy-Hinčin formula for convolution semigroups on locally compact groups

Abstract

LetG be a locally compact group and (μ_t)_{t ⩾ 0} a continuous convolution semigroup of probability measures onG. We show that an operatorN is the infinitesimal generator of (μ_t)_{t ⩾ 0} iffN is defined at least on the spaceC ₂(G) of twice right differentiable functions and if

$$\begin{gathered} Nf(x) = \sum\limits_{i \in I} {r_i X_i f(x) + } \sum\limits_{i, j \in I} {a_{\ddot y} X_i X_j f(x)} \hfill \\ + \int_{G\backslash \{ e\} } {\left[ {f(xy) - f()y - \sum\limits_{i \in I} {k_i (y)X_i f(x)} } \right]} \eta (dy) \hfill \\ \end{gathered} $$

holds for allf∈C ₂(G) andx∈G. Here (X _i) _iεI is a projective basis of the Lie algebra ofG and (k _i) _i∈I is a weak coordinate system ofG with respect to (X _i) _i∈I . In analogy to the case of Lie groups,N is determined by a vector r ∈ℝ¹, a positive semidefinite symmetric real-valuedI×I-matrix α and a Lévy measure η.