{"paper":{"title":"A probabilistic solution to the Stroock-Williams equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Goran Peskir","submitted_at":"2013-06-28T22:54:00Z","abstract_excerpt":"We consider the initial boundary value problem \\begin{eqnarray*}u_t=\\mu u_x+\\tfrac{1}{2}u_{xx}\\qquad (t>0,x\\ge0),\\\\u(0,x)=f(x)\\qquad (x\\ge0),\\\\u_t(t,0)=\\nu u_x(t,0)\\qquad (t>0)\\end{eqnarray*} of Stroock and Williams [Comm. Pure Appl. Math. 58 (2005) 1116-1148] where $\\mu,\\nu\\in \\mathbb{R}$ and the boundary condition is not of Feller's type when $\\nu<0$. We show that when $f$ belongs to $C_b^1$ with $f(\\infty)=0$ then the following probabilistic representation of the solution is valid: \\[u(t,x)=\\mathsf{E}_x\\bigl[f(X_t)\\bigr]-\\mathsf{E}_x\\biggl[f'(X_t)\\int_0^{\\ell_t^0(X)}e^{-2(\\nu-\\mu)s}\\,ds\\big"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0046","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"}