Decomposition of Convolution Semigroups on Groups and the 0-1 Law

Let (X(t))t>0, be a stochastically continuous symmetric Lévy process with values in a complete separable group G. We denote by (μt)t>0 the semigroup of one-dimensional distributions of X(t). Suppose that H is a Borel subgroup of G such that μt (H) > 0 for all t > 0. We obtain a decomposition of the generator of the process X ( t ) into a bounded part concentrated on Hc and the generator of a semigroup concentrated on H. This yields the 0-1 law for such processes. We also examine the differentiation of transition probability of the induced Markov process π (X (t)) on the homogeneous space G/H.