{"paper":{"title":"First exit and Dirichlet problem for the nonisotropic tempered $\\alpha$-stable processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Weihua Deng, Xing Liu","submitted_at":"2019-01-08T11:13:54Z","abstract_excerpt":"This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered $\\alpha$-stable process $X_t$. The upper bounds of all moments of the first exit position $\\left|X_{\\tau_D}\\right|$ and the first exit time $\\tau_D$ are firstly obtained. It is found that the probability density function of $\\left|X_{\\tau_D}\\right|$ or $\\tau_D$ exponentially decays with the increase of $\\left|X_{\\tau_D}\\right|$ or $\\tau_D$, and $\\mathrm{E}\\left[\\tau_D\\right]\\sim \\left|\\mathrm{E}\\left[X_{\\tau_D}\\right]\\right|$,\\ $\\mathrm{E}\\left[\\tau_D\\right]\\sim\\mathrm{E}\\left[\\left|X_{\\tau_D}-\\mathrm{E}\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.03204","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"}