{"paper":{"title":"A geometric interpretation of the transition density of a symmetric L\\'evy Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"N. Jacob, R.L. Schilling, S. Landwehr, V. Knopova","submitted_at":"2011-05-25T13:18:06Z","abstract_excerpt":"We study for a class of symmetric L\\'evy processes with state space $\\rn$ the transition density $p_t(x)$ in terms of two one-parameter families of metrics, $(d_t)_{t>0}$ and $(\\delta_t)_{t>0}$. The first family of metrics describes the diagonal term $p_t(0)$; it is induced by the characteristic exponent $\\psi$ of the L\\'evy process by $d_t(x,y)=\\sqrt{t\\psi(x-y)}$. The second and new family of metrics $\\delta_t$ relates to $\\sqrt{t\\psi}$ through the formula\n  $$\n  \\exp(-\\delta_t^2(x,y))\n  = \\Ff[\\frac{e^{-t\\psi}}{p_t(0)}](x-y)\n  $$\n  where $\\Ff$ denotes the Fourier transform. Thus we obtain the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.5016","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"}