On diffusion hemigroups of probability measures on an Abelian locally compact group
Let \(G\) be a locally compact Abelian group satisfying 2nd countability axiom, and let \(\widehat{G}\) be its dual. For a \(G\)-valued independent increments stochastic process \(X(t)\), \(t \in [0,T]\), a two-parameter convolution semigroup (``hemigroup'') \((\mu_{s,t} := P_{X(t) - X(s)}\), \(0 \leq s <t)\) corresponds. A canonical representation formula of \((\widehat{\mu}_{s,t}(\chi), \chi \in \widehat{G})\) is given. The correspondences between the following properties are the main result of the paper: (A) the convolution hemigroup \(\mu_{s,t}\) is a ``diffusion hemigroup''; (B) the Lévy measure in the canonical representation vanishes; (C) \(X(t)\), \(t \geq 0\), is Gaussian; (D) \(X(t)\), \(t \geq 0\), has a.s. continuous path. The properties (B), (C) and (D) are equivalent, and (A) implies (D). It is shown that when \(\mu_{s,t}\) is Lipschitz continuous in the sense of ``bounded variation relative to the local inner product'' in a sense, (D) implies (A). Thus, under this condition, properties (A), (B), (C) and (D) are equivalent. This condition is indispensable to the inclusion (D) to (A).
scientific article; zbMATH DE number 1484876