{"paper":{"title":"Inversion, duality and Doob $h$-transforms for self-similar Markov processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Larbi Alili, Lo\\\"ic Chaumont, Piotr Graczyk, Tomasz \\.Zak","submitted_at":"2016-01-29T11:12:15Z","abstract_excerpt":"We show that any $\\mathbb{R}^d\\setminus\\{0\\}$-valued self-similar Markov process $X$, with index $\\alpha>0$ can be represented as a path transformation of some Markov additive process (MAP) $(\\theta,\\xi)$ in $S_{d-1}\\times\\mathbb{R}$. This result extends the well known Lamperti transformation. Let us denote by $\\widehat{X}$ the self-similar Markov process which is obtained from the MAP $(\\theta,-\\xi)$ through this extended Lamperti transformation. Then we prove that $\\widehat{X}$ is in weak duality with $X$, with respect to the measure $\\pi(x/\\|x\\|)\\|x\\|^{\\alpha-d}dx$, if and only if $(\\theta,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.08056","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}