{"paper":{"title":"Kolmogorov equations for evaluating the boundary hitting of degenerate diffusion with unsteady drift","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The finite difference method for linear and nonlinear Kolmogorov equations of unsteady Jacobi diffusion yields unique numerical solutions due to discrete ellipticity when the discount is positive.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Hidekazu Yoshioka","submitted_at":"2025-01-06T02:57:44Z","abstract_excerpt":"Jacobi diffusion is a representative diffusion process whose solution is bounded in a domain under certain drift and diffusion coefficient conditions. However, the process without such conditions has not been thoroughly investigated. We explore a Jacobi diffusion whose drift coefficient is affected by another deterministic process, causing the process to hit the boundary of a domain in finite time. The Kolmogorov equation (a degenerate elliptic partial differential equation) for evaluating the boundary hitting of the proposed Jacobi diffusion is then presented and analyzed, with several condit"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The finite difference method for the linear and nonlinear Kolmogorov equations yields a unique numerical solution because of discrete ellipticity if the discount is positive.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The accuracy of the finite difference method critically depends on the regularity of the boundary condition, and the use of high-order discretization is not always effective.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Develops Kolmogorov equations and finite difference methods for boundary hitting in degenerate Jacobi diffusions with unsteady drift, with a mean-field tourism application.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The finite difference method for linear and nonlinear Kolmogorov equations of unsteady Jacobi diffusion yields unique numerical solutions due to discrete ellipticity when the discount is positive.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1316f5ce2812d4a6881b83c92c5244f80b73c2849edbf54db45889ea039dd449"},"source":{"id":"2501.02729","kind":"arxiv","version":5},"verdict":{"id":"1e37a8a1-7e54-4783-9e4a-930b3f3b6333","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T06:33:16.871676Z","strongest_claim":"The finite difference method for the linear and nonlinear Kolmogorov equations yields a unique numerical solution because of discrete ellipticity if the discount is positive.","one_line_summary":"Develops Kolmogorov equations and finite difference methods for boundary hitting in degenerate Jacobi diffusions with unsteady drift, with a mean-field tourism application.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The accuracy of the finite difference method critically depends on the regularity of the boundary condition, and the use of high-order discretization is not always effective.","pith_extraction_headline":"The finite difference method for linear and nonlinear Kolmogorov equations of unsteady Jacobi diffusion yields unique numerical solutions due to discrete ellipticity when the discount is positive."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.02729/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"}