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In this paper, we study the following Cauchy problem related to the stochastic process $X$:\n  $\\label{main eqn} \\frac{\\partial u}{\\partial t}(t,x) = \\cA(t)u(t,x) +f(t,x), \\quad u(0,\\cdot)=0, \\quad (t,x) \\in (0,T) \\times \\mathbf{R}^d, \\end{align} where $f \\in L_p( (0,T) ; L_p(\\mathbf{R}^d))=L_p( (0,T) ; L_p)$ and \\begin{align*} \\cA(t)u(t,x) = \\lim_{h \\downarrow 0}\\frac{\\mathbb{E}\\left[u(t,x+X_{t+h}-X_t)-u(t,x)\\right]}{h}$. We provide a su"},"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":"1705.01243","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-03T03:13:17Z","cross_cats_sorted":[],"title_canon_sha256":"37f17dff04c491d4c75fd35ba280cd6c1a47655c7ec7a5ec52eb68485ed7c2c3","abstract_canon_sha256":"cdc6af7df1ded28f62e9c645f34311da65577abf2f16ce3182126bc5b4c260b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:55.891854Z","signature_b64":"uxO/eOPuVJBQyx+hJafihphln+rl+nqNKGWIwpnD/tJWnfD27W9U+4SZQiNJNREwuebtO5KldK+m6I7NwvfOBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"114a0f5f3b37a45d29e948706a64a3d91c9daf261e17021b6c944ab534450a0f","last_reissued_at":"2026-05-18T00:31:55.891356Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:55.891356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An $L_p$-theory for diffusion equations related to stochastic processes with non-stationary independent increment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ildoo Kim, Kyeong-Hun Kim, Panki Kim","submitted_at":"2017-05-03T03:13:17Z","abstract_excerpt":"Let $X=(X_t)_{t \\ge 0}$ be a stochastic process which has an (not necessarily stationary) independent increment on a probability space $(\\Omega, \\mathbb{P})$. In this paper, we study the following Cauchy problem related to the stochastic process $X$:\n  $\\label{main eqn} \\frac{\\partial u}{\\partial t}(t,x) = \\cA(t)u(t,x) +f(t,x), \\quad u(0,\\cdot)=0, \\quad (t,x) \\in (0,T) \\times \\mathbf{R}^d, \\end{align} where $f \\in L_p( (0,T) ; L_p(\\mathbf{R}^d))=L_p( (0,T) ; L_p)$ and \\begin{align*} \\cA(t)u(t,x) = \\lim_{h \\downarrow 0}\\frac{\\mathbb{E}\\left[u(t,x+X_{t+h}-X_t)-u(t,x)\\right]}{h}$. 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