{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:UT7BLUQRJAATER25E6IGOPIBU2","short_pith_number":"pith:UT7BLUQR","canonical_record":{"source":{"id":"2007.01987","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-07-04T02:25:33Z","cross_cats_sorted":[],"title_canon_sha256":"4718f5074e4c05736a23d3f4a15f7f4f610a1e26e8cb2812e7706ff5e1d1c6b7","abstract_canon_sha256":"4d22c29410475d168238bcca871c08cf629170af9d28b99d80086bfa56b72090"},"schema_version":"1.0"},"canonical_sha256":"a4fe15d211480132475d2790673d01a6b69e7e8c49fb87e1c874298910ca5a50","source":{"kind":"arxiv","id":"2007.01987","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.01987","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"arxiv_version","alias_value":"2007.01987v2","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.01987","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"pith_short_12","alias_value":"UT7BLUQRJAAT","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"pith_short_16","alias_value":"UT7BLUQRJAATER25","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"pith_short_8","alias_value":"UT7BLUQR","created_at":"2026-07-05T02:34:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:UT7BLUQRJAATER25E6IGOPIBU2","target":"record","payload":{"canonical_record":{"source":{"id":"2007.01987","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-07-04T02:25:33Z","cross_cats_sorted":[],"title_canon_sha256":"4718f5074e4c05736a23d3f4a15f7f4f610a1e26e8cb2812e7706ff5e1d1c6b7","abstract_canon_sha256":"4d22c29410475d168238bcca871c08cf629170af9d28b99d80086bfa56b72090"},"schema_version":"1.0"},"canonical_sha256":"a4fe15d211480132475d2790673d01a6b69e7e8c49fb87e1c874298910ca5a50","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:34:40.487294Z","signature_b64":"+DmF1MBn9aAyzE0VthLSU4QTTTjjjvhmRmxOczT9n+MoeoL8LHj221iEWc+GSxIVwS0Libx/mFNchdELfywWBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4fe15d211480132475d2790673d01a6b69e7e8c49fb87e1c874298910ca5a50","last_reissued_at":"2026-07-05T02:34:40.486815Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:34:40.486815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2007.01987","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:34:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M5iYoCNOsZVJqQjVgQxPlTDikYer6fedOQZLi3c8327nIoRZ7Tv2uyJM4ifHMn69B+khAVhz2awKEafLwPhMBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T15:56:00.447879Z"},"content_sha256":"411de466350c10ba0592d5803acd1d8c22b37916ddb97614c425dbf9e3490457","schema_version":"1.0","event_id":"sha256:411de466350c10ba0592d5803acd1d8c22b37916ddb97614c425dbf9e3490457"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:UT7BLUQRJAATER25E6IGOPIBU2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $d\\geq 1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Davar Khoshnevisan, David Nualart, Fei Pu","submitted_at":"2020-07-04T02:25:33Z","abstract_excerpt":"Suppose that $\\{u(t\\,, x)\\}_{t >0, x \\in\\mathbb{R}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies Dalang's condition. Let $\\boldsymbol{p}_t(x):=(2\\pi t)^{-d/2}\\exp\\{-\\|x\\|^2/(2t)\\}$ denote the standard Gaussian heat kernel on $\\mathbb{R}^d$. We prove that for all $t>0$, the process $U(t):=\\{u(t\\,, x)/\\boldsymbol{p}_t(x): x\\in \\mathbb{R}^d\\}$ is stationary using Feynman-Kac's formula, and is er"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.01987","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2007.01987/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:34:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U0DIurWN4qzwbHIpmlMKSQTQZCSwHUM+dEq7D5262ipmXIyjT/R8AHU9fr5NyYt+Iddlt43cVSpYmqHr68ixCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T15:56:00.448261Z"},"content_sha256":"40d3547f245821d13741ae09d053a0e3b1da8ad08d6f3227cc66115ae9e8c230","schema_version":"1.0","event_id":"sha256:40d3547f245821d13741ae09d053a0e3b1da8ad08d6f3227cc66115ae9e8c230"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UT7BLUQRJAATER25E6IGOPIBU2/bundle.json","state_url":"https://pith.science/pith/UT7BLUQRJAATER25E6IGOPIBU2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UT7BLUQRJAATER25E6IGOPIBU2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-05T15:56:00Z","links":{"resolver":"https://pith.science/pith/UT7BLUQRJAATER25E6IGOPIBU2","bundle":"https://pith.science/pith/UT7BLUQRJAATER25E6IGOPIBU2/bundle.json","state":"https://pith.science/pith/UT7BLUQRJAATER25E6IGOPIBU2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UT7BLUQRJAATER25E6IGOPIBU2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:UT7BLUQRJAATER25E6IGOPIBU2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4d22c29410475d168238bcca871c08cf629170af9d28b99d80086bfa56b72090","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-07-04T02:25:33Z","title_canon_sha256":"4718f5074e4c05736a23d3f4a15f7f4f610a1e26e8cb2812e7706ff5e1d1c6b7"},"schema_version":"1.0","source":{"id":"2007.01987","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.01987","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"arxiv_version","alias_value":"2007.01987v2","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.01987","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"pith_short_12","alias_value":"UT7BLUQRJAAT","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"pith_short_16","alias_value":"UT7BLUQRJAATER25","created_at":"2026-07-05T02:34:40Z"},{"alias_kind":"pith_short_8","alias_value":"UT7BLUQR","created_at":"2026-07-05T02:34:40Z"}],"graph_snapshots":[{"event_id":"sha256:40d3547f245821d13741ae09d053a0e3b1da8ad08d6f3227cc66115ae9e8c230","target":"graph","created_at":"2026-07-05T02:34:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2007.01987/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Suppose that $\\{u(t\\,, x)\\}_{t >0, x \\in\\mathbb{R}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies Dalang's condition. Let $\\boldsymbol{p}_t(x):=(2\\pi t)^{-d/2}\\exp\\{-\\|x\\|^2/(2t)\\}$ denote the standard Gaussian heat kernel on $\\mathbb{R}^d$. We prove that for all $t>0$, the process $U(t):=\\{u(t\\,, x)/\\boldsymbol{p}_t(x): x\\in \\mathbb{R}^d\\}$ is stationary using Feynman-Kac's formula, and is er","authors_text":"Davar Khoshnevisan, David Nualart, Fei Pu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-07-04T02:25:33Z","title":"Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $d\\geq 1$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.01987","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:411de466350c10ba0592d5803acd1d8c22b37916ddb97614c425dbf9e3490457","target":"record","created_at":"2026-07-05T02:34:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4d22c29410475d168238bcca871c08cf629170af9d28b99d80086bfa56b72090","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-07-04T02:25:33Z","title_canon_sha256":"4718f5074e4c05736a23d3f4a15f7f4f610a1e26e8cb2812e7706ff5e1d1c6b7"},"schema_version":"1.0","source":{"id":"2007.01987","kind":"arxiv","version":2}},"canonical_sha256":"a4fe15d211480132475d2790673d01a6b69e7e8c49fb87e1c874298910ca5a50","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a4fe15d211480132475d2790673d01a6b69e7e8c49fb87e1c874298910ca5a50","first_computed_at":"2026-07-05T02:34:40.486815Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:34:40.486815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+DmF1MBn9aAyzE0VthLSU4QTTTjjjvhmRmxOczT9n+MoeoL8LHj221iEWc+GSxIVwS0Libx/mFNchdELfywWBw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:34:40.487294Z","signed_message":"canonical_sha256_bytes"},"source_id":"2007.01987","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:411de466350c10ba0592d5803acd1d8c22b37916ddb97614c425dbf9e3490457","sha256:40d3547f245821d13741ae09d053a0e3b1da8ad08d6f3227cc66115ae9e8c230"],"state_sha256":"f93281ac02b1e0d7bec0b96eb2d186d2235a608d9698b8d5be2fac0b1c723bfc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3zVHglJOTO22q/fbNK5feg+kXkbjbCG7E2WCmF2jkCYP0iXng9m4Y7yagZA+HyYruPb3IZiEqjiJhO88WZIAAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T15:56:00.450521Z","bundle_sha256":"f589baca652660258c1a919f2272fda584e61d5701bd9ef7088a306af143793e"}}