{"paper":{"title":"Functional limit theorem for the self-intersection local time of the fractional Brownian motion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Arturo Jaramillo, David Nualart","submitted_at":"2017-01-19T03:56:27Z","abstract_excerpt":"Let $\\{B_{t}\\}_{t\\geq0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $0<H<1$, where $d\\geq2$. Consider the approximation of the self-intersection local time of $B$, defined as \\begin{align*} I_{T}^{\\varepsilon}\n  &=\\int_{0}^{T}\\int_{0}^{t}p_{\\varepsilon}(B_{t}-B_{s})dsdt, \\end{align*} where $p_\\varepsilon(x)$ is the heat kernel. We prove that the process $\\{I_{T}^{\\varepsilon}-\\mathbb{E}\\left[I_{T}^{\\varepsilon}\\right]\\}_{T\\geq0}$, rescaled by a suitable normalization, converges in law to a constant multiple of a standard Brownian motion for $\\frac{3}{2d}<H\\leq\\frac{3}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05289","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"}