Excursion theory for rotation invariant Markov processes

Abstract

Let (Xt,Px) be a rotation invariant (RI) strong Markov process onRd{0} having a skew product representation [|Xt|,\(\theta _{A_t }\)], where (θ t ) is a time homogeneous, RI strong Markov process onSd−1, |Xt|, andθ t are independent underPx andAt is a continuous additive functional of |Xt|. We characterize the rotation invariant extensions of (Xt,Px) toRd. Two examples are given: the diffusion case, where especially the Walsh's Brownian motion (Brownian hedgehog) is considered, and the case where (Xt,Px) is self-similar.