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Under additional mild assumptions on $J(x,y)$ for $|x-y|<1$, we show that $C\\sqrt{r\\log \\log r}$ with some constant $C>0$ is an upper rate function of the process $X$, which enjoys the same form as that for Brownian motions. The approach is based on heat kernel estimates of large time for the process $X$. As a by-product, we also obtain two-sided he"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.03942","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-13T00:46:38Z","cross_cats_sorted":[],"title_canon_sha256":"655e0d19de1aa174bc086c13dc56668b3a2ecc580cc31e29830ab8ce030c0626","abstract_canon_sha256":"a27dfdf03f430e30ada0352a98b0115724bedb3962abe2fb0987a443620d8b25"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:23.057497Z","signature_b64":"OjsmXi+cAnTNFisvUkfsHFcATbgp32BRf8q3WjhffeurpxCHvlZk8i4NCHNeB+PNYXy1LxTzzrkZLH335uYWBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b5adf7d017c39f473e6e728fa595ef775961636d2f448fa77f306b3f335488b","last_reissued_at":"2026-05-18T00:40:23.056995Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:23.056995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper Rate Functions of Brownian Motion Type for Symmetric Jump Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jian Wang, Yuichi Shiozawa","submitted_at":"2017-07-13T00:46:38Z","abstract_excerpt":"Let $X$ be a symmetric jump process on $\\R^d$ such that the corresponding jumping kernel $J(x,y)$ satisfies $$J(x,y)\\le \\frac{c}{|x-y|^{d+2}\\log^{1+\\varepsilon}(e+|x-y|)}$$ for all $x,y\\in\\R^d$ with $|x-y|\\ge1$ and some constants $c,\\varepsilon>0$. 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