{"paper":{"title":"A dyadic construction of a three-dimensional attractive point interaction Markov family","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Iterated Doob transforms along dyadic partitions construct a Markov family for three-dimensional attractive point interactions.","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Barkat Mian","submitted_at":"2026-05-10T19:07:12Z","abstract_excerpt":"We discuss a probabilistic approximation framework for the three-dimensional attractive point interaction on a finite time horizon. By iterating the Doob transforms of the explicit heat kernel associated with the singular Schr\\\"odinger operator formally given by \\[ \\frac12\\Delta \\,+\\, \\frac{\\beta}{2}\\, \\delta_0(\\cdot), \\qquad \\beta>0, \\] we obtain sub-probability kernels along finite partitions on the punctured domain \\[ E_\\varepsilon=\\{x\\in\\mathbb R^3:\\ |x|>\\varepsilon\\}, \\] which yield a limiting sub-probability kernel via refinement along global dyadic partitions, and we extend this limit t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By iterating the Doob-transforms of the fundamental solution of the corresponding singular heat equation, we obtain sub-probability kernels along finite partitions which yield a limiting sub-probability kernel via refinement along global dyadic partitions, and we extend this limit to a transition probability kernel on an enlarged space obtained by adjoining a cemetery state. These kernels determine a time-inhomogeneous Markov process on the set of dyadic times, and its step-function interpolations yield càdlàg processes with consistent finite-dimensional distributions and partial tightness properties.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence and regularity of the fundamental solution to the singular heat equation on the punctured domain E_ε together with the convergence of the iterated Doob-transformed kernels under dyadic refinement as the partition mesh tends to zero.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Iterated Doob transforms along dyadic refinements produce a time-inhomogeneous Markov process on dyadic times that approximates the 3D attractive point interaction via càdlàg paths with consistent finite-dimensional distributions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Iterated Doob transforms along dyadic partitions construct a Markov family for three-dimensional attractive point interactions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a96ed308fa6c17f2e378d1edefdc99933ba0c017d3fb6ac9823120ed9db8618d"},"source":{"id":"2605.09706","kind":"arxiv","version":2},"verdict":{"id":"044633c8-15a4-4990-bb5c-b937514b7924","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T03:36:09.388796Z","strongest_claim":"By iterating the Doob-transforms of the fundamental solution of the corresponding singular heat equation, we obtain sub-probability kernels along finite partitions which yield a limiting sub-probability kernel via refinement along global dyadic partitions, and we extend this limit to a transition probability kernel on an enlarged space obtained by adjoining a cemetery state. These kernels determine a time-inhomogeneous Markov process on the set of dyadic times, and its step-function interpolations yield càdlàg processes with consistent finite-dimensional distributions and partial tightness properties.","one_line_summary":"Iterated Doob transforms along dyadic refinements produce a time-inhomogeneous Markov process on dyadic times that approximates the 3D attractive point interaction via càdlàg paths with consistent finite-dimensional distributions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence and regularity of the fundamental solution to the singular heat equation on the punctured domain E_ε together with the convergence of the iterated Doob-transformed kernels under dyadic refinement as the partition mesh tends to zero.","pith_extraction_headline":"Iterated Doob transforms along dyadic partitions construct a Markov family for three-dimensional attractive point interactions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.09706/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T07:22:01.339250Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T16:37:31.154118Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T12:31:18.230698Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:00:11.761163Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"126e8107c7ef82104394edd3013c50a383eddc8fb2a92d7587f31e5a3ee321bd"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"c2c75ab65df4c6adb46c61c07de50201bd32afcc5d0584a66a70ec8a26ef31d4"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}