{"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}$. We provide a su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01243","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":""},"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"}