The Lévy-Khintchine-formula for dissipative distributions
Let \((\mu_ t:\) \(t\geq 0)\) be a convolution semigroup of complex measures on a locally compact group G with \(\| \mu_ t\| \leq 1\) for \(t\geq 0\). Then it is well known that \(\mu_ 0=\chi \omega_ K\) (where K is a compact subgroup of G and \(\chi\) is a character of K) and that the semigroup is uniquely characterized by its infinitesimal generator \(T\in {\mathcal D}(G)\) defined by \(T(f)=\lim_{t\to 0}t^{-1}(\mu_ t(f)-\mu_ 0(f))\) for every test function f on G. Then T is a \(\chi\)-dissipative distribution on G in the sense that \(T=\chi \omega_ K*T*\chi \omega_ K\) and for every \(\chi\)-invariant test function f on G with \(\| f\|_{\infty}=f(1)\) we have Re(T(f))\(\leq 0.\) Using results of the author's article in Math. Z. 189, 529-554 (1985; Zbl 0547.60012), it is proved that T decomposes into the following four summands: a nonpositive multiple of \(\chi \omega_ K\) (corresponding to the decay rate of the norm of \(\mu_ t)\), a primitive distribution (corresponding to the drift part of \((\mu_ t:\) \(t\geq 0))\), a quadratic distribution (corresponding to the diffusion part of the semigroup) and the Lévy term. The last of these terms is of the form \[ f\mapsto \int_{G\setminus K}f_ 0 d\eta +iP(f),\quad f_ 0={\tilde \chi}\omega_ K*f*{\tilde \chi}\omega_ K-\Gamma ({\tilde \chi}\omega_ K*f*{\tilde \chi}\omega_ K) \] where \(\Gamma\) is a Lévy mapping (fixed and independent of T), \(\eta\) is the Lévy measure of T, a complex and in general unbounded measure on \(G\setminus K\) satisfying certain conditions, and P, a primitive distribution depending on \(\Gamma\) and \(\eta\), is a correction term. For a given Lévy mapping \(\Gamma\) this decomposition is unique, and every of these four summands generates a convolution semigroup.
scientific article; zbMATH DE number 3969755