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Figalli that the linear first-order PDE $$ \\dot{\\rho} + \\mbox{div}_{\\gamma} (\\rho \\cdot {b}) =0, \\ \\ \\rho|_{t=0} = \\rho_0, $$ where $\\rho_0 \\cdot \\gamma $ is a probability measure, admits a weak solution, in particular, under the following assumptions: $$ \\|b\\|_{H} \\in L^p(\\gamma), \\ p>1, \\ \\ \\ \\exp\\bigl(\\varepsilon(\\mbox{\\rm div}_{\\gamma} b)_{-} \\bigr) \\in L^1(\\gamma). $$ Applying transportation of measures via triangular maps we prov"},"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":"1303.7184","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-03-28T17:06:09Z","cross_cats_sorted":[],"title_canon_sha256":"d53a74197f8cd82deb31e6efc8506141eaf18ade6e373e0b12b3ef5fcbd575a3","abstract_canon_sha256":"cb47e587356e4b75daf485cf6564cf422c96c0ed39dfa9444d7f4df6f9ee9f66"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:06.561971Z","signature_b64":"uOYUv1cUctiTcqCksKBq0lblCNKbwcaRLLPRtI4wJDU6o8I+DLHwNK/a1dlOzQikzH1EJ1BFjQbZUTfPUlWzAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90d62b6b4d10153e20c24a51b9fc36c90a92f7c7f7c473534332428866140ad5","last_reissued_at":"2026-05-18T03:04:06.561491Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:06.561491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On continuity equations in infinite dimensions with non-Gaussian reference measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alexander V. 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